KP solitons in shallow water

KP SOLITONS IN SHALLO W W A TER YUJI KOD AMA Abstract. The main purp ose of the pap er is to pro vide a surv ey of our recent studies on soliton solutions of the Kadomtsev-P etviash vili (KP) equation. The KP equation describ es weakly dis- persive and small amplitude w av e propagation in a quasi-t w o dimensional framew ork. Recently a large v ariet y of exact soliton solutions of the KP equation has b een found and classiﬁed. These solutions are lo calized along certain lines in a two-dimensional plane and deca y exp onen tially ev- erywhere else, and are called line-solitons. The classiﬁcation is based on the far-ﬁeld patterns of the solutions which consist of a ﬁnite num ber of line-solitons. Each soliton solution is then deﬁned by a p oint of the totally non-negative Grassmann v ariet y whic h can be parametrized by a unique derangement of the symmetric group of permutations. Our study also includes certain numerical stability problems of those soliton solutions. Numerical simulations of the initial v alue problems indicate that certain class of initial wa v es asymptotically approac h to these exact solutions of the KP equation. W e then discuss an application of our theory to the Mach reﬂection problem in shallow water. This problem describ es the resonant in teraction of solitary wa v es appearing in the reﬂection of an obliquely incident w a ve on to a v ertical wall, and it predicts an extra-ordinary four-fold ampliﬁcation of the wa v e at the w all. There are sev eral numerical studies conﬁrming the prediction, but all indicate disagreements with the KP theory . Contrary to those previous numer- ical studies, w e ﬁnd that the KP theory actually pro vides an excellen t model to describ e the Mach reﬂection phenomena when the higher order corrections are included to the quasi-two dimensional approximation. W e also presen t lab oratory exp eriments of the Mach reﬂection recen tly carried out by Y eh and his colleagues, and show how precisely the KP theory predicts this wa ve b ehavior. Contents 1. In tro duction 1 2. Shallo w water w av es: Basic equations 3 3. The KP equation 5 3.1. Quasi-t wo dimensional appro ximation and one soliton solution 6 3.2. Soliton solutions in the W ronskian determinant 7 4. T otally nonnegative Grassmannian Gr + ( N , M ) 10 4.1. The Grassmannian Gr( N , M ) 10 4.2. The Pl ¨ uc ker co ordinates and total non-negativity 13 4.3. The τ -function as a p oint on Gr + ( N , M ) 14 5. Classiﬁcation of soliton solutions 15 5.1. Asymptotic line-solitons 16 5.2. Characterization of the line-solitons 22 6. (2 , 2)-soliton solutions 25 6.1. O-t yp e soliton solutions 28 6.2. (3142)-t yp e soliton solutions 30 6.3. T-t yp e soliton solutions 33 7. Numerical simulation and the stability of the soliton solutions 35 7.1. Regular reﬂection: κ > 1 37 7.2. The Mach reﬂection: κ < 1 39 7.3. T-t yp e interaction with X-shap e initial wa v e 40 8. Shallo w water w av es: The Mach reﬂection 41 8.1. Previous numerical results of the Boussinesq-type equations 42 KP SOLITONS IN SHALLOW W A TER 1 8.2. Exp erimen ts 44 References 47 1. Introduction It is a quite well-kno wn story that in August 1834 Sir John Scott Russel observed a large solitary w av e in a shallow w ater channel in Scotland. He noted in his ﬁrst pap er (1838) on the sub ject that I was observing the motion of a b o at which was r apid ly dr awn along a narr ow channel by a p air of horses, when the b o at suddenly stopp e d - not so the mass of water in the channel which it had put in motion; it ac cumulate d r ound the pr ow of the vessel in a state of violent agitation, then suddenly le aving it b ehind, r ol le d forwar d with gr e at velo city, assuming the form of a lar ge solitary elevation, a r ounde d, smo oth and wel l deﬁne d he ap of water, which c ontinue d its c ourse along the channel app ar ently without change of form or diminution o f sp e e d .... . This solitary wa ve is no w kno wn as an example of a soliton , and is describ ed b y a solution of the Kortew eg-de V ries (KdV) equation. The KdV equation describ es one-dimensional wa ve propagation suc h as beach w av es parallel to the coast line or w av es in narro w canal, and is obtained in the leading order appro ximation of an asymptotic perturbation theory under the assumptions of weak nonlinear- it y (small amplitude) and weak disp ersion (long wa ves). The KdV equation has rich mathematical structure including the existence of N -soliton solutions and the Lax pair for the inv erse scattering metho d, and it is a prototype equation of the 1 + 1 dimensional in tegrable systems. In particular, the initial v alue problem of the KdV equation has b een extensively studied by means of the metho d of in v erse scattering transform (IST). It is w ell known that a general initial data decaying rapidly in the spatial v ariable evolv es to a n umber of individual solitons and weakly disp ersive wa ve trains separate from the solitons (see for examples, [1, 31, 34, 44]). In 1970, Kadomtsev and Petviash villi [17] prop osed a 2 + 1 dimensional disp ersiv e w av e equation to study the stability of the one-soliton solution of the KdV equation under the inﬂuence of weak transv erse p erturbations. This equation is no w referred to as the KP equation. It turns out that the KP equation has muc h ric her structure than the KdV equation, and migh t b e considered as the most fundamen tal integrable system in the sense that many known integrable systems can be deriv ed as sp ecial reductions of the so-called KP hierarch y which consists of the KP equation together with its inﬁnitely man y symmetries. The KP equation can be also represented in the Lax form, that is, there exists a pair of linear equations asso ciated with an eigenv alue problem and an evolution of the eigenfunction, which enables the metho d of IST. How ever, unlike the case of the KdV equation, the IST for the KP equation do es not seem to provide a practical method of solving the initial v alue problem for initial wa v es consisting of line-solitons in the far ﬁeld. It is quite imp ortan t to recognize that the resonant in teraction pla ys a fundamental role in multi- dimensional wa ve phenomenon. The original description of the soliton interaction for the KP equa- tion was based on a tw o-soliton solution found in Hirota bilinear form, whic h has the shape of “X”, describing the intersection of tw o lines with oblique angle and a phase shift at the intersection p oin t. This X-shap e solution is referred to as the “O”-type soliton, where “O” stands for original . In his study of 1977 on an oblique in teraction of tw o line-solitons, Miles [28] p oin ted out that the O-t yp e solution b ecomes singular if the angle of the intersection is smaller than certain critical v alue dep ending on the amplitudes of the solitons. Miles then found that at the critical angle, the tw o line-solitons of the O-t yp e solution interact resonantly , and a third wa ve is created to make a “Y- shap ed” w av e form. Indeed, it turns out that such Y-shaped resonan t w av e forms are exact solutions of the KP equation (see also [32]). Miles applied his theory to study the Mach reﬂection of an inci- den t wa ve onto a vertical wall, and predicted that the third wa ve, called the Mach stem , created by the resonan t interaction can reac h four-fold ampliﬁcation of the incidence wa ve. Several lab oratory 2 YUJI K ODAMA and numerical exp erimen ts attempted to v alidate his prediction of four-fold ampliﬁcation, but with no deﬁnitive success (see for examples [15, 21, 41] for numerical exp erimen ts, and [36, 27, 46] for lab oratory exp erimen ts). After the disco very of the resonant phenomena in the KP equation, several numerical and exp eri- men tal studies w ere p erformed to inv estigate resonan t in teractions in other ph ysical tw o-dimensional equations such as the ion-acoustic and shallow water w av e equations under the Boussinesq approxi- mation (see for examples [18, 19, 13, 33, 15, 41, 29, 35, 43]). Ho wev er, apart from these activities, no signiﬁcant progress has been made in the study of the solution space or real applications of the KP equation. It w ould app ear that the general p erception was that there were not man y new and signiﬁcan t results left to b e uncov ered in the soliton solutions of the KP theory . Ov er the past several years, we hav e b een working on the classiﬁcation problem of the soliton solutions of the KP equation and their applications to shallo w w ater w av es. Our studies hav e rev ealed a large v ariet y of solutions that were totally ov erlo oked in the past [4, 22, 6, 7, 8, 9] 1 , and w e found that some of those exact solutions can b e applied to study the Mach reﬂection problem [8, 23, 47, 46]. Our numerical study [20] indicates that the solution to the initial v alue problem of the KP equation with certain class of initial wa ves asso ciated with the Mach reﬂection problem con verges asymptotically to some of these new exact solutions, that is, a separation of disp ersiv e radiations from the soliton solution similar to the case of the KdV soliton. The main purpose of this pap er is to presen t a survey of our studies on the soliton solutions of the KP equation. The pap er also presents several results for recent lab oratory exp erimen ts done by Harry Y eh and his colleagues at Oregon State Universit y . The pap er is organized as follows: In Section 2, w e presen t the deriv ation of the Boussinesq-type equation from the three-dimensional Euler equation for the irrotational and incompressible ﬂuid under the assumptions of w eak nonlin- earit y and weak disp ersion. The purp ose of this section is to give a precise physical meaning to those assumptions and to explain the existence of a solitary w av e solution in the form of the KdV soliton. In Section 3, we explain the quasi-tw o dimensional approximation to derive the KP equation, and discuss ph ysical interpretation of the KP soliton in terms of the KdV soliton. In order to describ e the general soliton solutions, we here introduce the τ -function whic h is expressed b y a W ronskian determinant for a set of N linearly indep enden t functions { f i : i = 1 , . . . , N } . Eac h function f i is a linear combination of the exp onential functions { E j : j = 1 , . . . , M } , where E j = e θ j with θ j = k j x + k 2 j y − k 3 j t for some k j ∈ R . The τ -function in the W ronskian form w as found in [25, 40, 14] (see also [16]). Setting f i = M P j =1 a ij E j , eac h solution is parametrized by the N × M co eﬃcien t matrix A = ( a ij ) of rank N . This representation naturally leads to the notion of the Grassmann v ariet y Gr( N , M ), the set of N -dimensional subspaces giv en by Span R { f i : i = 1 , . . . , N } of R M =Span R { E j : j = 1 , . . . , M } , and eac h point of Gr( N , M ) is marked by this A -matrix [39, 22, 8]. In Section 4, we provide a brief summary of the totally nonnegative (TNN) Grassmann v ariet y , denoted by Gr + ( N , M ), which provides the foundation of the classiﬁcation theorem for the r e gular soliton solutions of the KP equation discussed in the next Section. The main purp ose of this section is to show that the τ -function in the W ronskian determinan t form describ ed in Section 3 can b e iden tiﬁed as a p oin t of the TNN Grassmannian cell. That is, a classiﬁcation of the regular soliton solutions of the KP equation is equiv alen t to a parametrization of TNN Grassmannian cells [6, 8, 9]. In Section 5, we present a classiﬁcation theorem which states that the τ -function identiﬁed as a p oin t of Gr + ( N , M ) generates a soliton s olution of the KP equation that has asymptotically M − N line-solitons for y  0 and N line-solitons for y  0. Moreov er, these solutions can b e parametrized 1 The lo wer dimensional solutions, called (2 , 2)-soliton solutions, hav e been found b y the binary Darboux transfor- mation in [5]. KP SOLITONS IN SHALLOW W A TER 3 b y the derangemen ts (the p erm utations without ﬁxed points) of the symmetric group S M . This type of solutions is called ( M − N , N )-soliton solution. The derangements then give a parametrization of the TNN Grassmannian cells, and each derangement is expressed by a unique chor d diagr am . The c hord diagram is particularly useful to describ e the far-ﬁeld structure of the corresponding soliton solution. (See [6, 8, 9].) In Section 6, w e explain all the soliton solutions generated by the τ -functions on Gr + (2 , 4) (see also [5]). Some of these solutions are useful to describ e the Mach reﬂection problem in shallow w ater. W e show that the A -matrix determine the detailed structure of those solutions, suc h as the asymptotic lo cations of solitons and lo cal interaction patterns. (See [8, 20, 9].) In Sec tion 7, we present the n umerical study of the KP equation for certain types of initial w av es. In particular, we consider an initial v alue problem where the initial wa ve consists of t wo semi-inﬁnite line-solitons forming a V-shap e pattern. Those initial wa ves were considered in the study of the generation of large amplitude wa ves in shallow water [37, 43]. The main result of this section is to show that the solutions of this particular initial v alue problem c onver ge asymptotically to some of the exact (2,2)-soliton solutions. These results demonstarte a separation of the (exact) soliton solution from disp ersive radiation in the manner similar to the KdV case. (See [8, 23, 20].) In Section 8, w e discuss the Mach reﬂection problem in terms of the KP solitons whic h is equiv alent to Miles’ theory (assuming quasi-tw o dimensionality). W e ﬁrst show that the previous numerical results (see for examples [15, 41]), which rep orted a large discrepancy with the theory , are actually in a go od agreement with the predictions given by the KP theory . How ever, here one needs to give a prop er ph ysical interpretation of the theory when one compares it with the numerical results. W e also present some lab oratory exp erimen ts of shallow water wa ves [46, 47]. W e show that the exp erimen tal results are all in go o d agreement with the predictions of the KP theory which can describ e the evolution of the wa v e-proﬁle. Finally we demonstrate that the most complex (2,2)- soliton solution asso ciated with the τ -function on Gr + (2 , 4), referred to as T-type solution, can b e realized in an exp eriment. Sir John Scott Russel contin ued on to say in his b o ok (1865) that This is a most b e autiful and extr aor dinary phenomenon: the ﬁrst day I saw it was the happiest day of my life. Nob o dy has ever had the go o d fortune to se e it b efor e or, at al l events, to know what it me ant. It is now known as the solitary wave of tr anslation. No one b efor e had fancie d a solitary wave as a p ossible thing. I hop e this surv ey is succ essful to con vince the readers that the tw o-dimensional wa v e pattern gener- ated by the soliton solutions of the KP equation is a most b e autiful and extr aor dinary phenomena of t wo-dimensional shallow water wa ves, and one should hav e no doubt that observing these patterns at a b each will bring the happiest moment of one’s life. 2. Shallow w a ter w a ves: Basic equa tions Let us start with the physical background of the KP equation: W e consider a surface w av e on w ater whic h is assumed to be irrotational and incompressible (see for examples [44, 1]). Then the surface wa v e may b e describ ed b y the three-dimensional Euler equation, (2.1) ˜ ∆ ˜ φ = 0 , for 0 < ˜ z < h 0 + ˜ η , ˜ φ ˜ z = 0 , at ˜ z = 0 , ˜ φ ˜ t + 1 2    ˜ ∇ ˜ φ    2 + g ˜ η = 0 , ˜ η ˜ t + ˜ ∇ ⊥ ˜ φ · ˜ ∇ ⊥ ˜ η = ˜ φ ˜ z ,    at ˜ z = ˜ η + h 0 , 4 YUJI K ODAMA where ˜ φ is the v elo cit y potential with the Laplacian ˜ ∆ = ∂ 2 ˜ x + ∂ 2 ˜ y + ∂ 2 ˜ z , g = 980 cm / sec 2 the gra vitational constant, and h 0 the av erage depth. The ﬁrst tw o equations of (2.1) implies, ˜ φ ( ˜ x, ˜ y , ˜ z , ˜ t ) = cos( ˜ z q ˜ ∆ ⊥ ) ˜ ψ ( ˜ x, ˜ y , ˜ t ) with ˜ ∆ ⊥ = ∂ 2 ˜ x + ∂ 2 ˜ y , where ψ ( ˜ x, ˜ y , ˜ t ) = φ ( ˜ x, ˜ y , 0 , ˜ t ). In the linear limit, the system can b e written in the form,    cos( h 0 p ˜ ∆ ⊥ ) ˜ ψ ˜ t + g ˜ η = 0 ˜ η ˜ t = p ˜ ∆ ⊥ sin( h 0 p ˜ ∆ ⊥ ) ˜ ψ . This then gives the disp ersion relation, ω 2 = g k tanh h 0 k = c 2 0 k 2  1 − 1 3 h 2 0 k 2 + · · ·  , (2.2) where k := q k 2 x + k 2 y and the sp eed of the surface wa v e c 0 = √ g h 0 (e.g. c 0 = 70 cm/sec when h 0 = 5 cm). Let us denote the following scales: λ 0 ∼ horizontal length scale h 0 ∼ vertical length scale = w ater depth a 0 ∼ amplitude scale The non-dimensional v ariables { x, y , z , t, η , φ } are deﬁned as (2.3)      ˜ x = λ 0 x, ˜ y = λ 0 y , ˜ z = h 0 z , ˜ t = λ 0 c 0 t, ˜ η = a 0 η , ˜ φ = a 0 h 0 λ 0 c 0 φ. Then the shallow water equation in the non-dimesional form is given by φ z z + β ∆ ⊥ φ = 0 , for 0 < z < 1 + αη , φ z = 0 , at z = 0 , φ t + 1 2 α |∇ ⊥ φ | 2 + 1 2 α β φ 2 z + η = 0 , η t + α ∇ ⊥ φ · ∇ ⊥ η = 1 β φ z ,        at z = 1 + αη , where the parameters α and β are given by α = a 0 h 0 and β =  h 0 λ 0  2 . The w eak nonlinearity implies α  1, and the weak disp ersion (or long wa ve assumption) implies β  1. With a small parameter   1, we assume α ∼ β = O (  ) . As in the previous manner, φ can b e written formally in the form, φ ( x, y, z , t ) = cos  z p β ∆ ⊥  ψ ( x, y , t ) , whic h leads to the expansion, φ = ψ − β z 2 2 ∆ ⊥ ψ + O (  2 ) . KP SOLITONS IN SHALLOW W A TER 5 Then the equations at the surface gives the follo wing system of equations, sometimes called the Boussinesq-t yp e equation, (2.4)      η + ψ t + α 2 |∇ ψ | 2 − β 2 ∆ ψ t = O (  2 ) η t + ∆ ψ − α ∇ · ( ψ t ∇ ψ ) − β 6 ∆ 2 ψ = O (  2 ) . Here we hav e omitted ⊥ sign. Eliminating η in (2.4), we obtain the so-called isotropic Benney-Luke equation [2], ψ tt − ∆ ψ + α ( ∇ ψ · ∇ ψ t + ∇ · ( ψ t ∇ ψ )) − β 2  ∆ ψ tt − 1 3 ∆ 2 ψ  = O (  2 ) . One can then write this equation in the following form up to the same order, (2.5)  1 − β 3 ∆  ψ tt − ∆ ψ + α ( ∇ ψ · ∇ ψ t + ∇ · ( ψ t ∇ ψ )) = O (  2 ) . This is a regularized form of the t wo-dimensional Boussinesq-type equation for the shallow water w av es. Note here that the disp ersion relation of this equation is given by ω 2 = k 2 1 + β 3 k 2 = k 2  1 − β 3 k 2 + . . .  . whic h agrees with (2.2) up to O (  2 ). It is also w ell-known that the Boussinesq-type equation can b e reduced to the KdV equation for a far ﬁeld with a unidirectional approximation: Let χ b e the co ordinate whic h is p erp endicular to the wa v e crest of a linear shap e solitary wa ve, i.e. χ = ˜ x cos Ψ 0 + ˜ y sin Ψ 0 , where (cos Ψ 0 , sin Ψ 0 ) is the unit v ector in the propagation direction. Then using the far-ﬁeld co ordinates in the propgation direction, i.e. ξ := χ − t, τ = t, the Boussinesq-type equation (2.5) b ecomes the KdV equation, ψ τ ξ + 3 α 2 ψ ξ ψ ξξ + β 6 ψ ξξ ξ = 0 . F rom the ﬁrst equation of (2.4), we hav e η = ψ χ + O (  ). Then the KdV equation can b e expressed in the form with physical co ordinates, (2.6) ˜ η ˜ t + c 0 ˜ η χ + 3 c 0 2 h 0 ˜ η ˜ η χ + c 0 h 2 0 6 ˜ η χχχ = 0 . The one-soliton solution of the KdV equation is then given by (2.7) ˜ η = ˆ a 0 sec h 2 s 3ˆ a 0 4 h 3 0  χ − c 0  1 + ˆ a 0 2 h 0  ˜ t − χ 0  , where ˆ a 0 > 0 and χ 0 are arbitrary constants. One should note that an y line-solitary w av e in the Euler equation (2.1) can be (at least lo cally) approximated b y this soliton under the assumption of w eak disp ersion and weak nonlinearity . This remark will be imp ortan t when w e compare any n umerical results of the Euler equation or the Boussinesq-type equation with those of the KP equation. 3. The KP equa tion In this section, we give some basic prop ert y of the KP equation, and in tro duce the soliton solutions relev an t to shallow water wa ve problem. In particular, we discuss the ph ysical asp ect of the KP equation which is derived by a further assumption called quasi -tw o dimensional approximation (see for examples [17, 1]). 6 YUJI K ODAMA 3.1. Quasi-t w o dimensional appro ximation and one soliton solution. Let us now assume a quasi-tw o dimensionality with a weak dep endence in the y -direction, and we introduce a small parameter γ so that the y -co ordinate is scaled as (3.1) ζ := √ γ y , with γ = O (  ) . Then the system of equations (2.4) b ecomes      η + ψ t + α 2 ψ 2 x − β 2 ψ xxt = O (  2 ) η t + ψ xx − α ( ψ t ψ x ) x − β 6 ψ xxxx + γ ψ ζ ζ = O (  3 ) . No w we consider a far ﬁeld expressed with the scaling, (3.2) ξ = x − t and τ = t. Then the ab ov e equations hav e the expansions,      η − ψ ξ + ψ τ + α 2 ψ 2 ξ + β 2 ψ ξξ ξ = O (  2 ) − η ξ + ψ ξξ + η τ + α  ψ 2 ξ  ξ − β 6 ψ ξξ ξ ξ + γ ψ ζ ζ = O (  2 ) . Eliminating η , we obtain 2 ψ τ ξ + 3 αψ ξ ψ ξξ + β 3 ψ ξξ ξ ξ + γ ψ ζ ζ = O (  2 ) Noting η = ψ ξ + O (  ), we hav e the KP equation for η at the leading order,  2 η τ + 3 αη η ξ + β 3 η ξξ ξ  ξ + γ η ζ ζ = O (  2 ) (3.3) In terms of physical co ordinates, the KP equation is given by  ˜ η ˜ t + c 0 ˜ η ˜ x + 3 c 0 2 h 0 ˜ η ˜ η ˜ x + c 0 h 2 0 6 ˜ η ˜ x ˜ x ˜ x  ˜ x + c 0 2 ˜ η ˜ y ˜ y = 0 . As a particular solution, we hav e one line-soliton solution in the form, (3.4) ˜ η = a 0 sec h 2 s 3 a 0 4 h 3 0  ˜ x + ˜ y tan Ψ 0 − c 0  1 + a 0 2 h 0 + 1 2 tan 2 Ψ 0  ˜ t − ˜ x 0  , where a 0 > 0 , Ψ 0 and ˜ x 0 are arbitrary constants. One should note that the KP equation is deriv ed under the assumption of quasi-tw o dimensionality , that is, the angle Ψ 0 should b e small of order O (  ), and the solution (3.4) becomes unphysical for the case with a large angle. This can be seen explicitly by writing it in the follo wing form in the co ordinate p erp endicular to the w a v e crest, i.e. χ = ˜ x cos Ψ 0 + ˜ y sin Ψ 0 , (3.5) ˜ η = a 0 sec h 2 s 3 a 0 4 h 3 0 cos 2 Ψ 0  χ − c 0 cos Ψ 0  1 + a 0 2 h 0 + 1 2 tan 2 Ψ 0  ˜ t − χ 0  . Noting that cos Ψ 0 = 1 − 1 2 tan 2 Ψ 0 + O (  4 ) with Ψ 0 = O (  ), the velocity of the soliton has the corrected form up to O (  2 ), i.e. cos Ψ 0  1 + a 0 2 h 0 + 1 2 tan 2 Ψ 0  = 1 + a 0 2 h 0 + O (  3 ) , whic h do es not dep end on the angle up to O (  2 ). This is consistent with the assumption of the quasi-t wo dimensionalit y . W e also note that the line-soliton of (3.4) do es not satisfy the KdV equation (2.6) except the case with Ψ 0 = 0. Now comparing the KP soliton (3.4) with the KdV soliton (2.7), one can ﬁnd KP SOLITONS IN SHALLOW W A TER 7 the correction to the quasi-tw o dimensional appro ximation, that is, the amplitude a 0 in (3.4) is now corrected to (3.6) ˆ a 0 = a 0 cos 2 Ψ 0 . With this correction, the KP soliton (3.4) no w satisﬁes the KdV equation (2.6) up to O (  2 ). This correction b ecomes quite imp ortant when we compare our KP results with numerical results of the Euler or Boussinesq-type equations, that is, (3.6) giv es the relation betw een the amplitudes of the KP soliton and the KdV soliton (see Section 8 for the details). 3.2. Soliton solutions in the W ronskian determinan t. In order to giv e a general scheme to discuss the soliton solutions, we ﬁrst put the KP equation (3.3) in the standard form, (3.7) (4 u T + 6 uu X + u X X X ) X + 3 u Y Y = 0 , where the new v ariables ( X , Y , T ) and u are related to the physical ones with (3.8) ˜ η = 2 h 0 3 u, ˜ x − c 0 ˜ t = h 0 X, ˜ y = h 0 Y , ˜ t = 3 h 0 2 c 0 T . Hereafter we use the lo wer case letters ( x, y , t ) for ( X , Y , T ) (w e do not use the non-dimensional v ariables in (2.3), and the KP v ariables can b e conv erted to the physical v ariables directly through the relations (3.8)). W e write the solution of the KP equation (3.7) in the τ -function form, (3.9) u ( x, y, t ) = 2 ∂ 2 x ln τ ( x, y , t ) . where the τ -function is assumed to b e the W ronskian determinan t with N functions f i ’s (see for examples [40, 25, 14, 16]), (3.10) τ = W r( f 1 , f 2 , . . . , f N ) :=           f 1 f (1) 1 · · · f ( N − 1) 1 f 2 f (1) 2 · · · f ( N − 1) 2 . . . . . . . . . . . . f N f (1) N · · · f ( N − 1) N           , with f ( n ) i = ∂ n x f i . The functions { f i : i = 1 , . . . , N } satisfy the linear equations ∂ y f i = ∂ 2 x f i and ∂ t f i = − ∂ 3 x f i , and for the soliton solution, we take (3.11) f i = M P j =1 a ij E j , with E j = e θ j := exp  k j x + k 2 j y − k 3 j t  . Here A := ( a ij ) deﬁnes an N × M matrix A = ( a ij ) of rank N , and we assume the real parameters { k j : j = 1 , . . . , M } to b e ordered, k 1 < k 2 < · · · < k M . One should emphasize here that we hav e a parametrization of each soliton solution of the KP equation in terms of the k -parameters and the A -matrix. Then the classiﬁcation of the soliton solutions is to giv e a complete characterization of the τ -function (3.10) with the exp onential functions in (3.11). This is the main theme in [22, 3, 8, 9] and will b e discussed in the following sections. In terms of the τ -function, the KP equation (3.7) is written in the bilinear form, (3.12) 4( τ τ xt − τ x τ t ) + τ τ xxxx − 4 τ x τ xxx + 3 τ 2 xx + 3( τ τ y y − τ 2 xx ) = 0 . T o sho w that the τ -function (3.10) satisﬁes this equation, w e express the deriv atives of the τ -function using the Y oung diagram. Let Y b e giv en by the partition Y = ( λ 1 ≥ . . . ≥ λ n ) where λ j ’s represent the num b ers of b o xes in Y , and | Y | denote the total num b er of b o xes, i.e. | Y | = n P i =1 λ i . Denote τ in the N -tablet, τ = τ ∅ = (0 , 1 , 2 , . . . , N − 1) , 8 YUJI K ODAMA where the n um b ers describe the orders of the deriv ative of the column v ector ( f 1 , . . . , f N ) T in the τ -function (3.10). Then the num b er of b o xes in each ro w of Y can b e found by counting the missing n umbers which are less than the corresp onding num b er in τ Y . F or example, τ represen ts τ = (0 , 1 , 2 , . . . , N − 3 , N − 1 , N + 1) . F or the num b er N + 1, t wo num b ers N − 2 and N are missing, and this giv es . F or the num b er N − 1, one num b er N − 2 is missing and this gives . In terms of the determinant, τ represents, τ =           f 1 · · · f ( N − 3) 1 f ( N − 1) 1 f ( N +1) 1 f 2 · · · f ( N − 3) 2 f ( N − 1) 2 f ( N +1) 2 . . . . . . . . . . . . f N · · · f ( N − 3) N f ( N − 1) N f ( N +1) N           With this notation and using the equations for f i , i.e. ∂ y f i = f (2) i , ∂ t f i = − f (3) i , the deriv atives of the τ -function (3.10) are given by τ = τ x , τ = 1 2 ( τ xx + τ y ) , τ = 1 2 ( τ xx − τ y ) τ = 1 12 ( τ xxxx + 3 τ y y − 4 τ xt ) , τ = 1 3 ( τ xxx − τ t ) Then the bilinear equation (3.12) can b e written in the form, (3.13) τ φ τ − τ τ + τ τ = 0 . This equation is nothing but the Laplace expansion of the following 2 N × 2 N determinant identit y ,                f 1 · · · f ( N − 2) 1 f 1 · · · f ( N − 3) 1 f ( N − 1) 1 f ( N ) 1 f ( N +1) 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . f N · · · f ( N − 2) N f N · · · f ( N − 3) N f ( N − 1) N f ( N ) N f ( N +1) N 0 · · · 0 f 1 · · · f ( N − 3) 1 f ( N − 1) 1 f ( N ) 1 f ( N +1) 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 · · · 0 f N · · · f ( N − 3) N f ( N − 1) N f ( N ) N f ( N +1) N                ≡ 0 , so that (3.13) is identically satisﬁed. The relation (3.13) is the simplest example of the Pl ¨ uc ker relations (see b elow), and it can b e symbolically written by (3.14) ξ (1 , 2) ξ (3 , 4) − ξ (1 , 3) ξ (2 , 4) + ξ (1 , 4) ξ (2 , 3) = 0 , where the Y oung diagrams are expressed by Y = ( j − 2 , i − 1) for the symbol ξ ( i, j ). These symbols ξ ( j 1 , . . . , j N ) are the so-called Pl ¨ uck er co ordinates of the Granssmann manifold Gr( N , M ). In the next section, we outline the basic information for the Grassmannian Gr( N , M ) which will pro vide the foundation of the classiﬁcation theory for the soliton solutions of the KP equation [22, 8]. Example 3.1. Let us express one line-soliton solution (3.4) in our setting. Here we also introduce some notations to describ e the soliton solutions. The soliton solution (3.4) is obtained by the τ - function with M = 2 and N = 1, i.e. τ = f 1 = a 11 E 1 + a 12 E 2 . Since the solution u is given by (3.9), one can assume a 11 = 1 and denote a 12 = a > 0. Then τ = E 1 + aE 2 = 2 √ ae 1 2 ( θ 1 + θ 2 ) cosh 1 2 ( θ 1 − θ 2 − ln a ) , KP SOLITONS IN SHALLOW W A TER 9 with the 1 × 2 A -matrix of the form A = (1 a ). The parameter a in the A -matrix must b e a ≥ 0 for a non-singular solution and it determines the location of the soliton solution. Since a = 0 leads to a trivial solution, we consider only a > 0. Then the solution u = 2 ∂ 2 x (ln τ ) gives u = 1 2 ( k 1 − k 2 ) 2 sec h 2 1 2 ( θ 1 − θ 2 − ln a ) . Th us the solution is lo calized along the line θ 1 − θ 2 = ln a , hence we call it line-soliton solution. W e emphasize here that the line-soliton app ears at the b oundary of tw o regions where either E 1 or E 2 is the dominant exp onen tial term, and b ecause of this w e also call this soliton a [1 , 2]-soliton solution. In Section 5, we will construct more general line-soliton solutions which separates into a n umber of one-soliton solutions asymptotically as | y | → ∞ . W e refer to each of these asymptotic line-solitons as the [ i, j ]-soliton. The [ i, j ]-soliton solution with i < j has the same (lo cal) structure as the one-soliton solution, and can b e describ ed as follows u = A [ i,j ] sec h 2 1 2  K [ i,j ] · x − Ω [ i,j ] t + Θ 0 [ i,j ]  with some constan t Θ 0 [ i,j ] . The amplitude A [ i,j ] , the wa ve-v ector K [ i,j ] and the frequency Ω [ i,j ] are deﬁned by          A [ i,j ] = 1 2 ( k j − k i ) 2 K [ i,j ] =  k j − k i , k 2 j − k 2 i  = ( k j − k i ) (1 , k i + k j ) , Ω [ i,j ] = k 3 j − k 3 i = ( k j − k i )( k 2 i + k i k j + k 2 j ) . The direction of the wa v e-vector K [ i,j ] = ( K x [ i,j ] , K y [ i,j ] ) is measured in the counterclockwise sense from the y -axis, and it is given by K y [ i,j ] K x [ i,j ] = tan Ψ [ i,j ] = k i + k j , that is, Ψ [ i,j ] giv es the angle b etw een the line K [ i,j ] · x = const and the y -axis. Then one line-soliton can b e written in the form with three parameters A [ i,j ] , Ψ [ i,j ] and x 0 [ i,j ] , (3.15) u = A [ i,j ] sec h 2 r A [ i,j ] 2  x + y tan Ψ [ i,j ] − C [ i,j ] t − x 0 [ i,j ]  , with C [ i,j ] = k 2 i + k i k j + k 2 j = 1 2 A [ i,j ] + 3 4 tan 2 Ψ [ i,j ] . In Figure 3.1, we illustrate one line-soliton solution of [ i, j ]-type. In the right panel of this ﬁgure, we sho w a chor d diagr am which represen ts this soliton solution. Here the chord diagram indicates the permutation of the dominan t exp onen tial terms E i and E j in the τ -function, that is, with the ordering k i < k j , E i dominates in x  0, while E j dominates in x  0 (see Section 4 for the precise deﬁnition of the chord diagram). F or each soliton solution of (3.15), the wa ve vector K [ i,j ] and the frequency Ω [ i,j ] satisfy the soliton-disp ersion relation (see (2.2)), (3.16) 4Ω [ i,j ] K x [ i,j ] = ( K x [ i,j ] ) 4 + 3( K y [ i,j ] ) 2 . The soliton velocity V [ i,j ] is along the direction of the w av e-vector K [ i,j ] , and is deﬁned b y K [ i,j ] · V [ i,j ] = Ω [ i,j ] , which yields V [ i,j ] = Ω [ i,j ] | K [ i,j ] | 2 K [ i,j ] = k 2 i + k i k j + k 2 j 1 + ( k i + k j ) 2 (1 , k i + k j ) . Note in particular that s ince C [ i,j ] = k 2 i + k i k j + k 2 j > 0, the x -component of the soliton v elo city is always p ositive, i.e., an y soliton propagates in the positive x -direction. In the physical co ordinates (see (3.8)), this implies that soliton propagates in sup er-sonic (i.e. the sp eed of soliton is faster than c 0 = √ g h 0 , b ecause of its nonlinear eﬀect with ˜ η > 0, see Section 2). On the other hand, one should note that an y small p erturbation propagates in the negativ e x -direction, i.e., the x -comp onent of the 10 YUJI K ODAMA i j π = i j j i Figure 3.1. One line-soliton solution of [ i, j ]-type and the corresp onding chord diagram. The amplitude A [ i,j ] and the angle Ψ [ i,j ] are given by A [ i,j ] = 1 2 ( k i − k j ) 2 and tan Ψ [ i,j ] = k i + k j . The upp er oriented chord represen ts the part of [ i, j ]-soliton for y  0 and the low er one for y  0. group velocity is alwa ys negative. This can b e seen from the disp ersion relation of the line arize d KP equation for a plane wa ve φ = exp( i k · x − iω t ) with the wa ve-v ector k = ( k x , k y ) and the frequency ω , ω = − 1 4 k 3 x + 3 4 k 2 y k x , from which the group velocity of the wa ve is given by v = ∇ ω =  ∂ ω ∂ k x , ∂ ω ∂ k y  = − 3 4 k 2 x + k 2 y k 2 x ! , 3 2 k y k x ! . Ph ysically , this means that the radiations disp erse with sub-sonic sp eeds. This is similar to the case of the KdV equation, and w e exp ect that asymptotically , the soliton separates from small radiations. W e further discuss this issue in Section 7 where we numerically observe the separation. R emark 3.2 . In the form ulas (3.11), if we include the higher times t n in the exponential functions, i.e. E j = exp  ∞ P n =1 k n j t n  , then the τ -function (3.10) gives a solution of the KP hierarch y . The equation for the t n -ﬂo w is a symmetry of the KP equation, and the τ -function with those higher times also satisﬁes the other Pl ¨ uck er relations which are expressed with the Y oung diagrams having larger num b ers of b o xes [30]. 4. Tot all y nonnega tive Grassmannian Gr + ( N , M ) In the previous section, we considered a class of solutions which are expressed by the τ -functions (3.10) with the exp onen tial functions (3.11). Those solutions are determined b y the k -parameters and the A -matrix. Fixing the k -parameters, we hav e a set of M exp onen tials { E j = e θ j : j = 1 , . . . , M } whic h spans R M . Then the set of functions { f i : i = 1 , . . . , N } of (3.11) deﬁnes an N -dimensional subspace of R M . This leads naturally to the notion of Grassmannian Gr( N , M ), the set of all N - dimensional subspaces in R M , and each p oin t of Gr( N , M ) can b e parametrized by the A -matrix in (3.11). Here we giv e a brief review of the Grassmann manifold Gr( N , M ), in particular, w e describ e the totally non-negative part of Gr( N , M ). The main purpose of this section is to explain a mathematical background of r e gular soliton solutions of the KP equation. 4.1. The Gras smannian Gr ( N , M ) . Recall that the set of the functions f i spans an N -dimensional subspace which is parametrized by an N × M matrix A of rank N , i.e. ( f 1 , f 2 , . . . , f N ) = ( E 1 , E 2 , . . . , E M ) A T . KP SOLITONS IN SHALLOW W A TER 11 Since other set of functions ( g 1 , . . . , g N ) = ( f 1 , . . . , f N ) H for some H ∈ GL N ( R ) gives the same subspace, the A -matrix can b e canonically chosen in the r e duc e d r ow e chelon form (RREF). This then gives an explicit deﬁnition of the Grassmannian, Gr( N , M ) = GL N ( R )  M N × M ( R ) , where M N × M ( R ) denotes the set of N × M matrices of rank N . The canonical form of A is distinguished by a set of pivot columns labeled by I = { i 1 , i 2 , . . . , i N } , 1 ≤ i 1 < i 2 < . . . < i N ≤ M suc h that the N × N sub-matrix A I formed by the column set I is the identit y matrix. Each N × M matrix A in RREF uniquely determines an N -dimensional subspace, thus providing a co ordinate for a p oin t of Gr( N , M ). The set W I of all p oin ts in Gr( N , M ) represented b y RREF matrices A which hav e the same piv ot set I is called a Sch ub ert cell which gives the decomp osition of the Grassmannian, the Sch ub ert decomp osition, (4.1) Gr( N , M ) = G 1 ≤ i 1 1 and ( s i s i +1 ) 3 = e . Let P N b e a maximal parab olic subgroup of S M generated by s j ’s without the element s M − N , i.e. P N := h s 1 , . . . , s M − N − 1 , s M − N +1 , . . . , s M − 1 i ∼ = S M − N × S N . Then the pivot set I = { i 1 , i 2 , . . . , i N } parametrizing the Sc h ub ert cell W I can b e uniquely lab eled b y a minimal length represen tative of the coset , S ( N ) M := S M P N = { the reduced words ending with s M − N } . Namely , we hav e the Sch ub ert decomp osition of Gr( N , M ) in terms of the coset S ( N ) M , Gr( N , M ) = G π ∈ S ( N ) M W π . where the dimension of the cell W π is given by the length of the p erm utation, i.e. dim W π =  ( π ). F or example, in the case of Gr(1 , 3), we hav e S (1) 3 = h s 1 , s 2 i / h s 1 i = { e, s 2 , s 1 s 2 } , e =  1 2 3 1 2 3  s 2 − →  1 2 3 1 3 2  s 1 − →  1 2 3 3 1 2  Here i represents a piv ot, so that we ha ve W e = { (0 , 0 , 1) } , W s 2 = { (0 , 1 , ∗ ) } , W s 1 s 2 = { (1 , ∗ , ∗ ) } . Also in the case of Gr(2 , 3), we hav e S (2) 3 = h s 1 , s 2 i / h s 2 i = { e, s 1 , s 2 s 1 } , e =  1 2 3 1 2 3  s 1 − →  1 2 3 2 1 3  s 2 − →  1 2 3 2 3 1  , and the Sch ub ert cells W π are given by W e =  0 1 0 0 0 1  , W s 1 =  1 ∗ 0 0 0 1  , W s 2 s 1 =  1 0 ∗ 0 1 ∗  . Note in particular that the last elemen ts in the ab o ve examples ha ve no ﬁxed p oin ts, and they are called derangements. As w e will show that each derangemen t of S M parametrizes a unique line- soliton solution generated b y the τ -function of the form (3.10). It is imp ortant for our purp oses to remark that each p erm utation π ∈ S M with mark ed piv ot positions can be uniquely expressed by the chor d diagr am . This p erm utation is the decorated p erm utation deﬁned in [38] for a parametrization of the totally non-negative Grassmann cells. Deﬁnition 4.2. A c hord diagram asso ciated with π ∈ S M is deﬁned as follows: Consider a line segmen t with M marked p oints b y the num b ers { 1 , 2 , . . . , M } in the increasing order from the left. (a) If i < π ( i ) (excedance), then draw a chord joining i and π ( i ) on the upp er part of the line. (b) If j < π ( j ) (deﬁciency), then draw a chord joining j and π ( j ) on the low er part of the line. (c) If l = π ( l ) (ﬁxed p oint), then (i) if l is a pivot, then draw a lo op on the upp er part of the line at this p oin t. (ii) if l is a non-pivot, then draw a lo op on the lo wer part of the line at this p oint. The dimension of each Sch ub ert cell of Gr( N , M ) can b e also found from the chord diagram, and it is given by dim W π = N + { # of crossings } + { # of cusps in the low er part } − { # of lo ops in the upp er part } . Here w e sa y that the p oin t mark ed by j is a “cusp”, if π ( j ) < j = π ( k ) < k or k < π ( k ) = j < π ( j ) for some k . In particular, the p oint j is a cusp in the low er part of the diagram, if π ( j ) < j = π ( k ) < k (see [10, 45]). KP SOLITONS IN SHALLOW W A TER 13 Example 4.3. Consider the case of Gr(2 , 4). The Sc hubert cells W { i,j } are mark ed b y the piv ots { i, j } with 1 ≤ i < j ≤ 4, and the p erm utation representations are given by  1 2 3 4 1 2 3 4  s 2 − →  1 2 3 4 1 3 2 4  s 1 − →  1 2 3 4 3 1 2 4  s 3 ↓ s 3 ↓  1 2 3 4 1 3 4 2  s 1 − →  1 2 3 4 3 1 4 2  s 2 ↓  1 2 3 4 3 4 1 2  The chord diagrams are sho wn b elow, and the p oints with ﬁlled circle indicate the pivots for those cells: One should note here that each ﬁxed p oin t corresp onds to a lo op of the diagrams, and the k 1 k 2 k 3 k 4 diagrams without lo ops are asso ciated with the derangements of the p ermutation group. As w e will show, eac h chord (not lo op) iden tiﬁes a line-soliton for y  0 (or y  0) corresp onding to the lo cation of the chord in the upp er (or low er) part of the chord diagram. F or example, in the case π =  1 2 3 4 3 1 4 2  , we hav e [1 , 3]- and [3 , 4]-solitons in y  0 and [1 , 2]- and [2 , 4]-solitons in y  0. 4.2. The Pl ¨ uc ker coordinates and total non-negativit y. W e here describ e the totally non- negativ e (TNN) Grassmannian Gr + ( N , M ) as a subspace of Gr( N , M ). Then we will show that the τ -function asso ciated with Gr + ( N , M ) is necessary and suﬃcient conditions for the solution generated by the τ -function to b e regular. W e ﬁrst note that the co ordinates of Gr( N , M ) is giv en b y the Pl ¨ uck er embedding into the pro jectivization of the wedge pro duct space ∧ N R M , i.e. Gr( N , M )  → P ( ∧ N R M ) , whic h maps eac h frame given by [ f 1 , . . . , f N ] ∈ Gr( N , M ) to the p oint on P ( ∧ N R M ), i.e. (4.2) f 1 ∧ · · · ∧ f N = P 1 ≤ j 1 <... 0 for i 1 < · · · < i N . Then τ ∈ Gr + ( N , M ) implies that τ -function is p ositive deﬁnite and the solution u ( x, y , t ) = 2 ∂ 2 x (ln τ ) is regular for all ( x, y, t ) ∈ R 3 . In order to prov e a con verse of this statemen t, w e ﬁrst show the follo wing: Let ( t 1 , t 2 , . . . , t M ) b e the higer times for the KP equation (see Remark 3.2, and here the ﬁrst three times t 1 = x, t 2 = y and t 3 = − t give the KP v ariables). KP SOLITONS IN SHALLOW W A TER 15 Prop osition 4.1. Supp ose that the τ -function is r e gular for al l ( t 1 , t 2 , . . . , t M ) . Then τ ∈ Gr + ( N , M ) . Pr o of . Let us ﬁrst write the exp onen tial terms, E j = exp  M P n =1 k n j t n + θ 0 j  =: ˆ E j e θ 0 j for j = 1 , . . . , M , with θ 0 j ∈ R , i.e. the shifts of t n ’s in the exp onen tial functions. Because the k -parameters are all distinct, one can take the co ordinates ( θ 0 1 , . . . , θ 0 M ) instead of ( t 1 , . . . , t M ). The τ -function is then giv en by τ = P 1 ≤ j 1 < ··· 0 with the piv ot set { e 1 , . . . , e N } and the ordering k 1 < · · · < k M , so that τ ≈ E ( e 1 , . . . , e N ) > 0. This implies that the τ -function v anishes at some p oin t in ( x, y )-plane, and therefore the solution u ( x, y , t ) is not regular. It is then clear from the pro of that the total non-negativity is not only suﬃcien t but necessary for the regularity of the solution. Namely we hav e the following. Corollary 4.1. The solution of the KP e quation gener ate d by the τ -function in the form (3.10) with (3.11) is non-singular for any initial data if and only if τ ∈ Gr + ( N , M ) . Th us the classiﬁcation of the regular soliton solutions is equiv alent to a study of the totally non-negativ e Grassmannian. R emark 4.5 . Since each τ -function can b e identiﬁed as a p oin t on Gr( N , M ), one can deﬁne a momen t map, µ : Gr( N , M ) → h ∗ R [24], µ ( τ ) = P 1 ≤ j 1 < ··· c > k 1 + k 2 , and E 1 b ecomes dominant for c < k 1 + k 2 . Hence, we hav e for y  0 w = ∂ x ln f − →    k 1 as x → −∞ , k 2 for − ( k 1 + k 2 ) y < x < − ( k 2 + k 3 ) y , k 3 as x → ∞ . In the neighborho o d of the line x + ( k 1 + k 2 ) y =constant, f ≈ E 1 + aE 2 , whic h corresp onds to a [1 , 2]-soliton and its lo cation is ﬁxed by the constant a . The solution in y  0 also consists of a [2 , 3]-soliton in the neighborho o d of the line x + ( k 2 + k 3 ) y =constant, and whose lo cation is determined by the lo cations of other line-solitons. Therefore, we need only tw o parameters a, b (b esides the k -parameters) to sp ecify the solution uniquely , and those parameters ﬁx the lo cations of line x + y tan Ψ [ i,j ] − C [ i,j ] t = x 0 [ i,j ] for [ i, j ]-soliton. F or [ i, j ] = [1 , 3] and [2 , 3] in x  0, we hav e x [1 , 3] = − 1 k 3 − k 1 ln b, x [2 , 3] = − 1 k 3 − k 2 ln b a . The shap e of solution generated by f = E 1 + aE 2 + bE 3 with a = b = 1 (i.e. at t = 0 three line-solitons meet at the origin) is illustrated via the contour plot in Figure 5.2. In this Figure, one can see that the line-soliton in y  0 lab eled by [1 , 3], is lo calized along the line θ 1 = θ 3 with direction parameter c = k 1 + k 3 ; tw o other line-solitons in y  0 lab eled by [1 , 2] and [2 , 3] are lo calized resp ectiv ely , along the phase transition lines with c = k 1 + k 2 and c = k 2 + k 3 . This solution represents a resonan t solution of three line-solitons. The resonan t condition among those three line-solitons is given by K [1 , 3] = K [1 , 2] + K [2 , 3] , Ω [1 , 3] = Ω [1 , 2] + Ω [2 , 3] , whic h are trivially satisﬁed with K [ i,j ] = ( k j − k i , k 2 j − k 2 i ) and Ω [ i,j ] = k 3 j − k 3 i . The resonant condition may b e symbolically written as [1 , 3] = [1 , 2] + [2 , 3] . One can also represent this line-soliton solution by a permutation of three indices: { 1 , 2 , 3 } whic h is illustrated b y a (linear) chor d diagr am shown b elo w. Here, the upp er c hord represen ts the [1 , 3]- soliton in y  0 and the low er tw o chords represent [1 , 2] and [2 , 3]-solitons in y  0. F ollowing the arro ws in the c hord diagram, one recov ers the p erm utation, π =  1 2 3 3 1 2  or simply π = (312) . In general, eac h line-soliton solution of the KP equation can b e parametrized by a unique permutation corresp onding to a chord diagram (see the next subsection). 18 YUJI K ODAMA Figure 5.2. Example of (2 , 1)-soliton solution and the chord diagram. The k - parameters are chosen as ( k 1 , k 2 , k 3 ) = ( − 5 4 , − 1 4 , 3 4 ). The right panel is the cor- resp onding chord diagrams. W e take the A -matrix A = (1 1 1) so that at t = 0 three line-solitons meet at the origin. Each E ( j ) with j = 1 , 2 or 3 indicates the dominan t exp onen tial term E j in that region. The b oundaries of any tw o adjacent regions give the line-solitons indicating the transition of the dominant terms E j . The k -parameters are the same as those in Figure 5.1, and the line-solitons are determined from the intersection p oin ts of the η j ( c )’s in Figure 5.1. Here a = b = 1 (i.e. τ = E 1 + E 2 + E 3 ) so that the three solitons meet at the origin at t = 0. The results describ ed in this example can b e easily extended to the general case where f has arbitrary num b er of exp onen tial terms (see also [26, 4]). Prop osition 5.1. If f = a 1 E 1 + a 2 E 2 + · · · + a M E M with a j > 0 for j = 1 , 2 , . . . , M , then the solution u c onsists of M − 1 line-solitons for y  0 and one line-soliton for y  0 . Suc h solutions are referred to as the ( M − 1 , 1)-soliton solutions; meaning that ( M − 1) line- solitons for y  0 and one line-soliton for y  0. Note that the line-soliton for y  0 is lab eled b y [1 , M ], whereas the other line-solitons in y  0 are labeled b y [ k , k + 1] for k = 1 , 2 , . . . , M − 1, coun terclo c kwise from the negative to the p ositiv e x -axis, i.e. increasing Ψ from − π 2 to π 2 . As in the previous examples one can set a 1 = 1 without any loss of generality , then the remaining M − 1 parameters a 2 , . . . , a M determine the lo cations of the M line-solitons. Also note that the xy -plane is divided in to M sectors for the asymptotic region with x 2 + y 2  0, and the b oundaries of those sectors are given by the asymptotic line-solitons. This feature is common even for the general case. Figure 5.3 illustrates the case for a (3 , 1)-soliton solution with f = E 1 + E 2 + E 3 + E 4 . The chord diagram for this solution represents the p ermutation π = (4123) = s 4 s 3 s 2 ∈ S (1) 4 . Example 5.2. Let us no w consider the case with N = 2 and M = 3: W e take the A -matrix in (3.11) of the form, A =  1 0 − b 0 1 a  . where a and b are p ositive c onstan ts, that is, A marks a p oin t on Gr + (2 , 3). Then the τ -function is giv en by τ = E (1 , 2) + aE (1 , 3) + bE (2 , 3) . In order to carry out the asymptotic analysis in this case one needs to consider the sum of tw o η j ( c ), i.e. η i,j = η i + η j for 1 ≤ i < j ≤ 3. This can still b e done using Figure 5.1, but a more eﬀective w ay is describ ed b elo w (see the graph of η ( k , c ) in Figure 5.5). F or y  0, the transitions of the dominant exp onen tials are given by following scheme: E (1 , 2) − → E (1 , 3) − → E (2 , 3) , KP SOLITONS IN SHALLOW W A TER 19 Figure 5.3. The time evolution of a (3 , 1)-soliton solution and the corresp onding c hord diagram. The upp er chord represen ts the [1 , 4]-soliton, and the lo wer ones represen t [1 , 2]-, [2 , 3]- and [3 , 4]-solitons. The chord diagram shows π = (4123). The intermediate solitons are [1 , 3]-soliton at t = − 8 and [2,4]-soliton at t = 8, resp ectiv ely . These solitons app ear as the resonant (2 , 1)-type solutions. as c v aries from large positive (i.e. x → −∞ ) to large negative v alues (i.e. x → ∞ ). The b oundary b et ween the regions with the dominan t exp onentials E (1 , 2) and E (1 , 3) deﬁnes the [2 , 3]-soliton solution since here the τ -function can b e approximated as τ ≈ E (1 , 2) + aE (1 , 3) = 2( k 2 − k 1 ) e θ 1 + 1 2 ( θ 2 + θ 3 − θ 23 ) cosh 1 2 ( θ 2 − θ 3 + θ 23 ) , so that we hav e u = 2 ∂ 2 x ln τ ≈ 1 2 ( k 2 − k 3 ) 2 sec h 2 1 2 ( θ 2 − θ 3 + θ 23 ) , where θ 23 is related to the parameter of the A -matrix (see b elo w). A similar computation as ab ov e near the transition b oundary of the dominant exp onen tials E (1 , 3) and E (2 , 3) yields τ ≈ 2( k 3 − k 1 ) ae θ 3 + 1 2 ( θ 1 + θ 2 − θ 12 ) cosh 1 2 ( θ 1 − θ 2 + θ 12 ) . The phases θ 12 and θ 23 are related to the parameters of the A -matrix, a = k 2 − k 1 k 3 − k 1 e − θ 23 , b = k 2 − k 1 k 3 − k 2 e − θ 12 . F or y  0, there is only one transition, namely E (2 , 3) − → E (1 , 2) , as c v aries from large p ositiv e v alue (i.e. x → ∞ ) to large negative v alue (i.e. x → −∞ ). In this case, a [1 , 3]-soliton is formed for y  0 at the b oundary of the dominant exp onentials E (2 , 3) and E (1 , 2). The contour plot of the line-soliton solution is shown in Figure 5.4. Notice that this ﬁgure can be obtained from Figure 5.2 b y changing ( x, y ) → ( − x, − y ). This solution can be represented b y the chord diagram corresp onding to the p ermutation π = (231) shown b elo w. Note that this diagram is the π -r otation of the chord diagram in Example 5.1 whose p erm utation π = (312) is the in verse of π = (231). As shown in those examples, it is now clear that each line-soliton appears as a boundary of tw o dominan t exp onen tials, and with the condition that k i + k j are all distinct for i 6 = j , we ha ve the follo wing Prop osition: 20 YUJI K ODAMA Figure 5.4. Example of (1 , 2)-soliton solution and the chord diagram. The k - parameters are the same as the (2 , 1)-soliton in the previous ﬁgure. The right panels are the corresp onding chord diagrams. The parameters in the A -matrices are chosen as a = 1 2 and b = 1 so that at t = 0 three line-solitons meet at the origin. Prop osition 5.2. Two dominant exp onentials of the τ -function in adjac ent r e gions of the xy -plane ar e of the form E ( i, m 2 , . . . , m N ) and E ( j, m 2 , . . . , m N ) for some N − 1 c ommon indic es m 2 , . . . , m N . As a consequence of Prop osition 5.2, the KP solution b eha ves asymptotically like a single line- soliton (5.3) u ( x, y, t ) ' 1 2 ( k j − k i ) 2 sec h 2 1 2 ( θ j − θ i + θ ij ) , in the neighborho o d of the line x + ( k i + k j ) y = constant, whic h forms the boundary b et ween the regions of dominant exp onen tials E ( i, m 2 , . . . , m N ) and E ( j , m 2 , . . . , m N ). Equation (5.3) deﬁnes an asymptotic line-soliton, i.e. [ i, j ]-soliton, as a result of those tw o dominant exp onen tials. In order to identify the set of asymptotic line-solitons asso ciated with a given solution, w e need to determine which exp onential terms E ( m 1 , m 2 , . . . , m N ) are actually dominant along each line [ i, j ] : x = − ( k i + k j ) y as | y | → ∞ . F or this purp ose, ﬁrst note that along a line x = − cy each exp onen tial term E ( m 1 , m 2 , . . . , m N ) has the form, E ( m 1 , m 2 , . . . , m N ) ∝ exp  N P n =1 η m n ( c ) y  , η m ( c ) = k m ( k m − c ) . Th us for y  0 (or  0), the dominant exp onen tial corresp onds to the largest (or least) v alue of the sum of η m n ( c ) for each c . When tw o dominant exponentials E ( i, m 2 , . . . , m N ) and E ( j , m 2 , . . . , m N ) are in balance along the direction of the [ i, j ]-soliton, we hav e η i ( c ) = η j ( c ) whic h implies that c = k i + k j . Since η m ( c ) − η i ( c ) = ( k m − k i )( k m + k i − c ) and the k -parameters are ordered as k 1 < k 2 < · · · < k M , we ha ve the following order relations among the other η m ( c )’s along c = k i + k j , (5.4) ( η i = η j < η m if m < i or j < m, η i = η j > η m if i < m < j. The relations among the phases η j ( c ) can b e seen easily from the plots of η j ( c ) versus c as w ell as η ( k , c ) = k ( k − c ) versus k for a ﬁxed v alue of c illustrated b y Figure 5.5. Prop osition 5.2 and the relations (5.4) are particularly useful in order to ﬁnd the asymptotic line-solitons from a giv en KP τ -function as demonstrated by the example b elo w. Example 5.3. Let us consider the 2 × 4 matrix, A =  1 0 0 − a 0 1 b c  , KP SOLITONS IN SHALLOW W A TER 21 η 1 η 2 η 3 η 4 c [1,2] [1,3] [2,3] [1,4] [2,4] [3,4] k 1 k 2 k 3 k 4 c = k 2 + k 4 η (k,c) k Figure 5.5. The left ﬁgure shows η j ( c ) = k j ( k j − c ) for j = 1 , . . . , 4. Eac h [ i, j ] at the intersection p oin t of η i ( c ) = η j ( c ) corresp onds to c = k i + k j . W e assume that the parameters k j are generic so that there is at most one in tersection p oin t for each c . The right ﬁgure is a plot of η ( k , c ) as a function of k with c = k 2 + k 4 , that is, the η along the dotted line in the left ﬁgure passing through the intersection p oin t η 2 ( c ) = η 4 ( c ). This ﬁgure shows the order η 3 < η 2 = η 4 < η 1 . where a, b, c are p ositiv e real num b ers. In this case, there are six maximal minors, ﬁve of which are p ositiv e, namely , ξ (1 , 2) = 1 , ξ (1 , 3) = b, ξ (1 , 4) = c, ξ (2 , 4) = a, ξ (3 , 4) = ab , and ξ (2 , 3) = 0. Then from (5.1) the τ -function has the form, τ =( k 2 − k 1 ) E (1 , 2) + b ( k 3 − k 1 ) E (1 , 3) + c ( k 4 − k 1 ) E (1 , 4) + a ( k 2 − k 4 ) E (2 , 4) + ab ( k 4 − k 3 ) E (3 , 4) . Prop osition 5.2 implies that the line solitons are lo calized along the lines x + cy =constant with c = k i + k j = tan Ψ [ i,j ] . Hence, we lo ok for dominant exp onential terms in the τ -function along those directions. F or y  0, the c -v alues decrease as w e sweep clo c kwise from negative to p ositiv e x -axis starting with the largest v alue c = k 3 + k 4 . W e hav e η 1 , η 2 > η 3 = η 4 from the order relations (5.4) for c = k 3 + k 4 . This means that η 1 + η 2 is the dominan t phase combination along this direction. Since ξ (1 , 2) 6 = 0, τ ( x, y , t ) ≈ ( k 2 − k 1 ) E (1 , 2) implying that u ≈ 0 along the line [3 , 4], so there is no [3 , 4] line-soliton. By similar reasoning one can verify that the [1 , 4]- and [1 , 2]-solitons are also imp ossible. Let us consider the direction c = k 2 + k 4 to chec k for the [2 , 4]-soliton. F rom (5.4) (see also Figure 5.5), η 3 < η 2 = η 4 < η 1 , and since both ξ (1 , 2) and ξ (1 , 4) = a are nonzero, the τ -function in (5.1) corresp onds to a dominant balance of exp onentials: τ ≈ ( k 2 − k 1 ) E (1 , 2) + c ( k 4 − k 1 ) E (1 , 4) along the line [2 , 4]. Therefore [2 , 4] corresp onds to an asymptotic line-soliton as y → ∞ . The [1 , 3]-soliton also exists by a similar argumen t. Thus, we hav e tw o asymptotic line-solitons [1 , 3]- and [2 , 4]-types for y  0. W e next lo ok for the asymptotic solitons for y  0 by sweeping from the negative x -axis to p ositiv e x -axis. Recall that in this case the dominant exp onen tial E ( i, j ) corresp onds to the least v alue of the sum η i ( c ) + η j ( c ). It is easy to see that [1 , 2] and [3 , 4]-solitons are imp ossible since E (3 , 4) and E (1 , 2) are resp ectiv ely , the only dominant exponentials along those directions. Then consider the [1 , 3]-soliton. Along c = k 1 + k 3 , (5.4) implies that η 2 < η 1 = η 3 < η 4 , and so the exp onen tials E (1 , 2) and E (2 , 3) would give the dominant balance. But E (2 , 3) is not present in the ab o ve τ -function b ecause ξ (2 , 3) = 0. So we conclude that [1 , 3]-soliton do es not exist as y  0, and for similar reasons, [2 , 4]-soliton is also imp ossible. Next, chec king for the [1 , 4]-soliton, we hav e η 2 , η 3 < η 1 = η 4 from (5.4). But as seen earlier, the dominant exp onential E (2 , 3) is not presen t in the τ -function. Ho wev er, there do es exist a balance b etw een the next dominan t exp onen tial pairs { E (1 , 3) , E (3 , 4) } or { E (1 , 2) , E (2 , 4) } dep ending on whether η 2 > η 3 or η 2 < η 3 . In either case, there exists an asymptotic line-soliton along [1 , 4]. A similar argumen t applies along the line [2 , 3] whic h corresp onds the other asymptotic line-soliton as y  0. 22 YUJI K ODAMA Figure 5.6. An example of (2 , 2)-soliton solution. The left left ﬁgure is at t = − 16 and the right one at t = 16. The parameters ( k 1 , . . . , k 4 ) are given b y ( − 1 , − 0 . 5 , 0 . 5 , 2). The intermediate solitons are [3 , 4]-type for t = − 16 and [1 , 2]- t yp e for t = 16. Note that the middle triangular section at t = − 16 corresp onds to the region with the dominant exp onential E (2 , 4), and that all ﬁve non-zero exp onen tial terms in the τ -function app ear in the ﬁgure at t = 16. In summary , the τ -function corresp onding to the A -matrix given ab o ve, generates a KP solution with asymptotic line-solitons [1 , 3] and [2 , 4] as y  0, and asymptotic line-solitons [1 , 4] and [2 , 3] as y  0. This line-soliton solution with the parameters a = b = c = 1 in the A -matrix is shown in Figure 5.6. W e note that the line-solitons asso ciated with the resonant (1 , 2)- and (2 , 1)-soliton solutions can b e determined in the same w ay as the ab ov e example b y applying for the dominant balance conditions giv en by Proposition 5.2 and (5.4). W e no w pro ceed to discuss a more general characterization of all line-soliton solutions of the KP equation whose τ -functions are given in the W ronskian form (3.10). 5.2. Characterization of the line-solitons. It should b e clear from the ab o ve examples that a dominant exp onen tial term determined b y the relations (5.4) is actually pr esent in the given τ - function if its co eﬃcient term given b y a maximal minor of the A -matrix is non-zero. Th us, in order to obtain a complete characterization of the asymptotic line solitons, it is necessary to consider the structure of the N × M co eﬃcien t A -matrix in some detail. W e consider the matrix A to b e in RREF, and we will also assume that A is irr e ducible as deﬁned b elo w: Deﬁnition 5.4. An N × M matrix A is irreducible if eac h column of A contains at least one nonzero elemen t, or each row contains at least one nonzero element other than the pivot once A is in RREF. If an N × M matrix A is not irreducible, then the corresponding τ -function gives the same KP solution u whic h is obtained from another τ -function asso ciated with a smaller size matrix ˜ A deriv ed from A . One can notice from the determinant expansion in (5.1) that (a) if the m -th column of A has only zero elements, then ξ ( m 1 , . . . , m N ) = 0 if m k = m for some k , that is, the exp onential E m will never app ear in the τ -function; in terms of the c hord diagram, this corresp onds to a lo op in the low er part of the diagram ( m is a non-piv ot index), (b) if the n -th row of A has the piv ot as the only non-zero element, then all ξ ( m 1 , . . . , m N ) 6 = 0 con tains the index n , that is, the exp onen tial E n can be factored out from the τ -function; in terms of the chord diagram, this corresp onds to a lo op in the upp er part of the diagram. So the irreducibility implies that we consider only derangements (i.e. no ﬁxed p oints) of the p ermu- tation. W e now present a classiﬁcation scheme of the line-soliton solutions by identifying the asymptotic line-solitons as y → ±∞ . W e denote a line-soliton solution by ( N − , N + )-soliton whose asymptotic KP SOLITONS IN SHALLOW W A TER 23 Interac tio n R egi on [e N , j N ] [e 1 , j 1 ] [i M-N , g M-N ] [i 1 , g 1 ] E(e 1 , e 2 , .. . , e N ) y = ∞ x = - ∞ y = - ∞ x = ∞ E( j 1 , j 2 , .. . , j N ) Figure 5.7. ( N − , N + )-soliton solution. The asymptotic line-solitons are de- noted b y their index pairs [ e n , j n ] and [ i m , g m ]. The sets { e 1 , e 2 , . . . , e N } and { g 1 , g 2 , . . . , g M − N } indicate pivot and non-pivot indices, resp ectiv ely . Here N − = M − N and N + = N for the τ -function on Gr( N , M ), and E ( · , . . . , · ) represents the dominan t exp onen tial in that region. form consists of N − line-solitons as y → −∞ and N + line-solitons for y → ∞ in the xy -plane as sho wn in Figure 5.7. The next Prop osition provides a general result characterizing the asymptotic line-solitons of the ( N − , N + )-soliton solutions (the pro of can b e found in [8]): Prop osition 5.3. L et { e 1 , e 2 , . . . , e N } and { g 1 , g 2 , . . . , g M − N } denote r esp e ctively, the pivot and non-pivot indic es asso ciate d with an irr e ducible, N × M , TNN A -matrix. Then the soliton solution obtaine d fr om the τ -function in (5.1) with this A -matrix has the fol lowing structur e: (a) F or y  0 , ther e ar e N asymptotic line-solitons of [ e n , j n ] -typ e for some j n . (b) F or y  0 , ther e ar e ( M − N ) asymptotic line-solitons of [ i m , g m ] -typ e for some i m . An imp ortant consequence of Prop osition 5.3 is that it deﬁnes the p airing map π : [ M ] → [ M ] on the integer set [ M ] := { 1 , 2 , . . . , M } according to (5.5) ( π ( e n ) = j n , n = 1 , 2 , . . . , N , π ( g m ) = i m , m = 1 , 2 , . . . , M − N . Recall that { e n } N n =1 and { g m } M − N m =1 are resp ectiv ely , the pivot and non-piv ot indices of the A -matrix and form a disjoin t partition of [ M ]. Then the unique index pairings in Prop osition 5.3 imply that the map π is a p ermutation of M indices. More precisely , π ∈ S M where S M is the group of p erm utations of the index set [ M ]. F urthermore, since π ( e n ) = j n > e n , n = 1 , . . . , N and π ( g m ) = i m < g m , m = 1 , . . . , M − N , π deﬁned by (5.5) is a p erm utation with no ﬁxed p oin t, i.e. der angements . Y et another feature of π is that it has exactly N exc e danc es deﬁned as follows: an element l ∈ [ M ] is an exc e danc e of π if π ( l ) > l . The excedance set of π in (5.5) is the set of pivot indices { e 1 , e 2 , . . . , e N } . The ab ov e results can b e summarized to deduce the following c haracterization for the line-soliton solution of the KP equation [8]. Theorem 5.5. L et A b e an N × M , TNN, irr e ducible matrix which c orr esp onds to a p oint in the non-ne gative Gr assmannian Gr + ( N , M ) ⊂ Gr ( N , M ) . Then the τ -function (5.1) asso ciate d with this A -matrix gener ates an ( M − N , N ) -soliton solutions. The M asymptotic line-solitons asso ciate d with e ach of these solutions c an b e identiﬁe d via a p airing map π deﬁne d by (5.5) . The map π ∈ S M is a der angement of the index set [ M ] with N exc e danc es given by the pivot indic es { e 1 , e 2 , . . . , e N } of the A -matrix in RREF. 24 YUJI K ODAMA Figure 5.8. An example of (3 , 3)-soliton solution. The p ermutation of this solution is π = (451263). The k -parameters are chosen as ( k 1 , k 2 , . . . , k 6 ) = ( − 1 , − 1 2 , 0 , 1 2 , 1 , 3 2 ). The dominan t exp onential for x  0 is E (1 , 2 , 5), and each dominan t exp onential is obtained through the derangemen t representing the so- lution, e.g. after crossing [5 , 6]-soliton in the clockwise direction, the dominan t exp onen tial b ecomes E (1 , 2 , 6). That is, [5 , 6]-soliton is given b y the balance of those tw o exp onen tials. As explained in Section 4, the derangements π ∈ S M are represen ted by the chord diagrams with the arrows ab ov e the line p ointing from e n to j n for n = 1 , 2 , . . . , N , while arrows b elo w the line p oin t from g m to i m for m = 1 , 2 , . . . , M − N . Figure 5.8 illustrates the time evolution of an example of (3 , 3)-soliton solution. The c hord diagram shows all asymptotic line-solitons for y → ±∞ . Theorem 5.5 pro vides a unique parametrization of each TNN Grassmannian cell in terms of the derangemen t of S M . This agree with the result obtained by Postnik ov et al in [38, 45]. One should, ho wev er, note that Theorem 5.5 does not giv e us the indices j n and i m in the [ e n , j n ] and [ i m , g m ] line-solitons. The sp eciﬁc conditions that an index pair [ i, j ] identiﬁes an asymptotic line-soliton are obtained b y identifying the dominant exp onen tial in each domain in the xy -plane. The example b elo w illustrates how to apply Theorem 5.5, and identify all the asymptotic line-solitons for a given irreducible TNN A -matrix. Example 5.6. Let us consider the 3 × 5 matrix, A =   1 0 − a 0 b 0 1 c 0 − d 0 0 0 1 e   with ad − bc = 0 , where a, b, c, d and e are p ositive constants, that is, the A -matrix marks a p oin t on Gr + (3 , 5). Then the purp ose is to ﬁnd asymptotic line-solitons generated by the τ -function (3.10) asso ciated with this A -matrix. F rom Prop osition 5.3, one can see that the τ -function with this matrix will pro duce a (3 , 2)-soliton solution since N = 3 and M = 5. Moreov er, the asymptotic line-solitons for this solution are lab eled by [1 , j 1 ] , [2 , j 2 ] and [4 , j 3 ] for y  0 for some j 1 > 1 , j 2 > 2 and j 3 > 4. Similarly , the line-solitons for y  0 are lab eled by [ i 1 , 3] and [ i 2 , 5] for some i 1 < 3 and i 2 < 5. The basic idea to determine those indices j 1 , j 2 , j 3 and i 1 , i 2 is to apply Prop osition 5.3 and the dominant relations (5.4). KP SOLITONS IN SHALLOW W A TER 25 Figure 5.9. Example of shallow w ater wa ves. The upp er photos are taken at a b eac h in Mexico. The lo wer ﬁgures show the ev olution of the corresp onding exact (3 , 2)-soliton solution of (24153)-type shown in the chord diagram (see Example 5.6): the left one at t = 1 . 5 and the right one at t = 10. (Photographs b y courtesy of Mark J. Ablowitz.) Let us ﬁrst consider the case for y  0. Starting with the last pivot e 3 = 4, it is immediate to ﬁnd j 3 = 5, b ecause of j 3 > 4 (just Prop osition 5.3). W e no w take the next piv ot e 2 = 2 and ﬁnd the index j 2 . Since the index 5 is already tak en as the pair index of e 3 = 4, we need to chec k only the cases [2 , 4] and [2 , 3]. F or the existence of [2 , 4]-soliton, the dominant relation (5.4) requires that b oth ξ (1 , 2 , 5) and ξ (1 , 4 , 5) are not zero. Calculating those minors for our A -matrix, we hav e ξ (1 , 2 , 5) = e 6 = 0 , ξ (1 , 4 , 5) = d 6 = 0 , and hence [2 , 4]-soliton exists. Now w e consider the case with e 1 = 1, that is, we ha ve only [1 , 2] and [1 , 3] p ossibility . In the case of [1 , 3], we use again the dominant relation (5.4), and c heck the minors ξ (1 , 4 , 5) and ξ (3 , 4 , 5) which corresp onds to the dominan t exp onentials. W e then ﬁnd ξ (1 , 4 , 5) = d 6 = 0 but ξ (3 , 4 , 5) = bc − ad = 0. This implies that [1 , 3]-soliton is imp ossible for y  0. So the last one is [1 , 2]-t yp e, whic h can b e conﬁrmed by the condition ξ (1 , 4 , 5) = d 6 = 0 and ξ (2 , 4 , 5) = b 6 = 0. No w w e consider the case for y  0. Theorem 5.5 tells us that for the non-piv ot index g 1 = 3, only the pair [1 , 3] is possible (the index 2 is already taken because [1,2]-soliton exists, i.e. π (1) = 2). Then the ﬁnal soliton must b e [3 , 5]-type from the non-pivot index g 2 = 5. The last one can b e conﬁrmed by the least condition in (5.4) with ξ (2 , 3 , 4) = a 6 = 0 and ξ (2 , 4 , 5) = b 6 = 0. Th us w e ha ve a (2 , 3)-soliton solution of π = (24153)-t yp e for the τ -function (3.10) with the A -matrix considered. The photos in Figure 5.9 show some interacting shallo w water w av es, whic h w e think a realization of this example. W e demonstrate an exact solution whose parameters are giv en by ( k 1 , k 2 , . . . , k 5 ) = ( − 2 , − 1 , 0 , 0 . 5 , 2) and the A -matrix with ( a, b, c, d, e ) = (1 , 2 , 1 , 2 , 1). 6. (2 , 2) -soliton solutions Here w e giv e a summary of all soliton solutions of the KP equation generated by the 2 × 4 irreducible, TNN A -matrices. Prop osition 5.3 implies that eac h of the soliton solutions consists of 26 YUJI K ODAMA (3412) (4321) (2143) (4312) (3421) (3142) (2413) Figure 6.1. The c hord diagrams for seven diﬀerent types of (2 , 2)-soliton solutions. Eac h diagram corresp onds to a totally non-negative Grassmannian cell in Gr(2 , 4). t wo asymptotic line-solitons as y → ±∞ . That is, they are (2 , 2)-soliton solutions. W e outline b elow the classiﬁcation sc heme for the (2 , 2)-soliton solutions, and discuss some of the exact solutions in details for the applications discussed in the following sections. First note that for 2 × 4 matrices, there are only tw o types given by  1 0 − c − d 0 1 a b  and  1 a 0 − c 0 0 1 b  , The fact that A is TNN implies that the constan ts a, b, c and d must b e non-negative. F or the ﬁrst t yp e, one can easily see that ad = 0 is imp ossible b ecause then either ξ (3 , 4) < 0 or A is not irreducible. Then there are 5 p ossible cases with ad 6 = 0, namely , (1) ad − bc > 0 , (2) ad − bc = 0 , (3) b = 0 , c 6 = 0 , (4) c = 0 , b 6 = 0 , (5) b = c = 0 . F or the second type, ab 6 = 0 due to irreducibility . Hence, we hav e only tw o cases: (6) c 6 = 0 , (7) c = 0 . Th us w e ha v e total sev en diﬀeren t t yp es of A -matrices, and using Theorem 5.5, we can show that eac h A -matrix gives a diﬀerent (2 , 2)-soliton solution which can b e enumerated according to the sev en derangements of the index set [4] = { 1 , 2 , 3 , 4 } with tw o excedances. Namely , for those cases from (1) to (7) we hav e (1) π = (3412) , (2) π = (2413) , (3) π = (4312) , (4) π = (3421) , (5) π = (4321) , (6) π = (3142) , (7) π = (2143) . In Figure 6.1, w e show the c hord diagrams for all those seven cases. One should note that an y derangemen t of S 4 with exactly tw o excedances should b e one of the graphs. This uniqueness in the general case has b een used to count the num b er of totally non-negative Grassmann cells [38, 45]. Let us now summarize the results for all those seven cases of the (2 , 2)-soliton solutions: (1) π = (3412): This case corresp onds to the T-typ e 2-soliton solution which was ﬁrst obtained as the solution of the T o da lattice hierarc hy [4]. This is why w e call it “T-type” (see also [22]). The asymptotic line-solitons are [1 , 3]- and [2 , 4]-types for | y | → ∞ . The A -matrix is giv en by A =  1 0 − c − d 0 1 a b  , where a, b, c, d > 0 are free parameters with ad − bc > 0. This is the generic solution on the maximum dimensional cell of Gr + (2 , 4), and the corresp onding line-soliton has the most complicated pattern due to the fully resonant interactions among all line-solitons. KP SOLITONS IN SHALLOW W A TER 27 (2) π = (2413): The asymptotic line-solitons are given b y [1 , 2]- and [2 , 4]-solitons for y  0, and [1 , 3]- and [3 , 4]-solitons for y  0. The A -matrix is given by A =  1 0 − c − d 0 1 a b  , where a, b, c, d > 0 with ξ (3 , 4) = ad − bc = 0. Note the change of the solution structure by imp osing just one constraint ξ (3 , 4) = 0 to the previous case (1). (3) π = (4312): The asymptotic line-s olitons for this case are [1 , 4]- and [2 , 3]-solitons for y  0, and [1 , 3]- and [2 , 4]-solitons for y  0. The A -matrix is given by A =  1 0 − b − c 0 1 a 0  , where a, b, c > 0 are free parameters. Notice that t wo line-solitons for y  0 are the same as in the T-type solution (see the crossing in the low er chords in Figure 6.1). (4) π = (3421): The asymptotic line-solitons are given b y [1 , 3] and [2 , 4] for y  0, and for y  0, these are the [1 , 4]- and [2 , 3]-solitons. The A -matrix is given by A =  1 0 0 − c 0 1 a b  , where a, b, c > 0 are p ositive free parameters. This solution can b e considered as a dual of the previous case (3), that is, tw o sets of line-solitons for y  0 and y  0 are exc hanged (also notice the dualit y in the chord diagrams in Figure 6.1). The example discussed after Prop osition 5.3 corresp onds to this solution (see Figure 5.6). (5) π = (4321): The solution in this case is called the P-typ e 2-soliton solution which has asymptotic line-solitons of [1 , 4]- and [2 , 3]-types as | y | → ∞ . This type of solutions ﬁts b etter with the physic al assumption of quasi-tw o dimensionality with weak y -dep endence underlying the deriv ation of the KP equation. This is why we call it “P-type” (see [22]). The A -matrix is given by A =  1 0 0 − b 0 1 a 0  . The c hord diagram indicates that those t wo line-solitons m ust hav e the diﬀerent amplitudes, i.e. A [1 , 4] > A [2 , 3], but they can propagate in the same direction, which corresp ond to the t wo soliton solution of the KdV equation. (6) π = (3142): The asymptotic line-solitons are given b y [1 , 3]- and [3 , 4]-solitons for y  0, and [1 , 2]- and [2 , 4]-solitons for y  0. The A -matrix is given by A =  1 a 0 − c 0 0 1 b  , where a, b, c > 0. This solution is dual to the case (2) in the sense that the tw o sets of asymptotic line-solitons for y  0 and y  0 are switched, as well as the missing minors are switched by ξ (3 , 4) ↔ ξ (1 , 2). Also note the duality b et ween the corresp onding chord diagrams. (7) π = (2143): This case is called the O-typ e 2-soliton solution. The asymptotic line-solitons are of [1 , 2]- and [3 , 4]-t yp es as | y | → ∞ . The letter “O” for this type is due to the fact that this solution was original ly found to describ e the tw o-soliton solution of the KP equation (see for example [14]). The A -matrix for the O-type 2-soliton solution is given by A =  1 a 0 0 0 0 1 b  . Notice that this A -matrix is obtained as a limit c → 0 in the previous one of the case (6), i.e. (3142)-soliton solution. 28 YUJI K ODAMA Figure 6.2. The time evolution of an O-type soliton solution. Each E ( i, j ) in- dicates the dominan t exp onen tial in that region. The parameters are chosen as A [1 , 2] = A [3 , 4] = 0 . 1 and Ψ [3 , 4] = − Ψ [1 , 2] = 30 ◦ . No w let us describ e the details of some of the (2 , 2)-soliton solutions, which will b e imp ortan t for an application of those solutions to shallo w water problem discussed in the next Section. In particular, we explain how the A -matrix uniquely determines the structure of the corresp onding soliton solution such as the lo cation of the solitons and their phase shifts. 6.1. O-t yp e soliton solutions. This is the original tw o-soliton solution, and the solutions corre- sp ond to the chord diagram of π = (2143). A solution of this type consists of t wo full line-solitons of [1 , 2] and [3 , 4] (see Figure 6.2). Note here that they hav e phase shifts due to their collision. Let us describe explicitly the structure of the solution of this type: The τ -function deﬁned in (5.1) for this case is given by τ = E (1 , 3) + bE (1 , 4) + aE (2 , 3) + abE (2 , 4) , where a, b > 0 are the free parameters given in the A -matrix listed ab o ve. As w e will show that those tw o parameters can b e used to ﬁx the lo cations of those solitons, that is, they are determined b y the asymptotic data of the solution for large | y | . F or the later application of the solution, w e assume that [1 , 2]-soliton has a “negative” y -comp onen t in the wa ve-v ector (i.e. tan Ψ [1 , 2] < 0), and [3 , 4]-soliton has a “p ositiv e” y -comp onent, (i.e. tan Ψ [3 , 4] > 0, see Figure 3.1). Then for the region with large p ositiv e x , w e ha ve [1 , 2]-soliton in y > 0 and [3 , 4]-soliton in y < 0. F or [1 , 2]-soliton in x > 0 (and y  0), w e ha ve the dominan t balance b et ween E (1 , 4) and E (2 , 4). Then the τ -function can b e written in the following form, τ ≈ bE (1 , 4) + abE (2 , 4) = 2 be θ 4 + 1 2 ( θ 1 + θ 2 ) cosh 1 2  θ 1 − θ 2 + θ + 12  , whic h leads to the [1 , 2]-soliton solution in the region near θ 1 ≈ θ 2 for x  0, u = 2 ∂ 2 x ln τ ≈ 1 2 ( k 2 − k 1 ) 2 sec h 2 1 2  θ 1 − θ 2 + θ + 12  . Here the shift θ + 12 (+ indicates x > 0) is related to the parameter a in the A -matrix (see b elo w). F or [3 , 4]-soliton in x > 0 (and y  0), from the balance τ ≈ aE (2 , 3) + abE (2 , 4), we hav e u ≈ 1 2 ( k 4 − k 3 ) 2 sec h 2 1 2 ( θ 3 − θ 4 + θ + 34 ) . The shifts θ + 12 and θ + 34 are related to the parameters in the A -matrix, (6.1) a = k 4 − k 1 k 4 − k 2 e − θ + 12 , b = k 3 − k 2 k 4 − k 2 e − θ + 34 . KP SOLITONS IN SHALLOW W A TER 29 Figure 6.3. O-type interaction for tw o equal amplitude solitons. The parameters k i ’s are tak en to b e ( k 1 , k 2 , k 3 , k 4 ) = ( − 1 − 10 − 4 , − 10 − 4 , 10 − 4 , 1 + 10 − 4 ), which give A [1 , 2] = A [3 , 4] = 1 2 and tan Ψ [3 , 4] = − tan Ψ [1 , 2] = 1 + 2 × 10 − 4 (i.e. Ψ [3 , 4] ≈ 45 . 0057). The constan ts a, b in the A -matrix are c hosen so that the cen ter of in teraction p oin t is lo cated at the origin, and u max = u (0 , 0 , 0) ≈ 1 . 96 and the phase shift ∆ x [1 , 2] = ∆ [3 , 4] ≈ 7 . 8. Th us the parameters in the A -matrix can b e determined by the asymptotic data of the lo cations of those [1 , 2]- and [3 , 4]-solitons for x  0 and | y |  0. The most imp ortan t feature of the O-type solution is the phase shift due to the interaction of those tw o oblique line-solitons. The phase shift for [ i, j ]-soliton is deﬁned by θ ij = θ − ij − θ + ij where ± indicate the v alues for x → ±∞ . The v alues θ 12 and θ 34 turn out to be the same (see for example [16]), θ 12 = θ 34 = − ln ∆ O . where we note ∆ O := ( k 3 − k 2 )( k 4 − k 1 ) ( k 4 − k 2 )( k 3 − k 1 ) = 1 − ( k 2 − k 1 )( k 4 − k 3 ) ( k 4 − k 2 )( k 3 − k 1 ) < 1 . This implies that θ 12 = θ 34 > 0, and each [ i, j ]-soliton shifts in x with (6.2) ∆ x [ i,j ] = 1 k j − k i θ ij . The p ositiv e phase shifts ∆ x [1 , 2] > 0 and ∆ x [3 , 4] > 0 indicate an attr active force in the in teraction. Figure 6.3 illustrates an O-type in teraction of t wo solitons which hav e the same amplitude, A [1 , 2] = A [3 , 4] = 1 2 , and are symmetric with respect to the y -axis, Ψ [3 , 4] = − Ψ [1 , 2] ≈ 45 ◦ . Since the solution is close to the resonance, w e ha ve the large phase shifts ∆ x [1 , 2] = ∆ x [3 , 4] ≈ 7 . 8 and the maxim um v alue of the soltion u max ≈ 1 . 96 (almost four times larger than A [1 , 2] ). O-t yp e soliton solution has a steady X-shap e with phase shifts in b oth line-solitons. One can also ﬁnd the formula of the maxim um amplitude which o ccurs at the center of intersection p oin t (center of the X-shap e), which is given by (6.3) u max = A [1 , 2] + A [3 , 4] + 2 1 − √ ∆ O 1 + √ ∆ O q A [1 , 2] A [3 , 4] . (see for example [8, 12, 42].) Since 0 < ∆ O < 1, we hav e the b ound A [1 , 2] + A [3 , 4] < u max <  q A [1 , 2] + q A [3 , 4]  2 . It is also in teresting to note that the formula ∆ O has critical cases at the v alues k 1 = k 2 or k 2 = k 3 or k 3 = k 4 . F or the case with k 1 = k 2 or k 3 = k 4 (i.e. ∆ O = 1), one can see that one of the line-soliton b ecomes small, and the limit consists of just one-soliton solution. On the other hand, for 30 YUJI K ODAMA the case k 2 = k 3 (i.e. ∆ O = 0), the τ -function has only three terms, which corresp onds to a solution sho wing a Y-shap e in teraction (i.e. the phase shift becomes inﬁnity and the middle p ortion of the in teraction stretc hes to inﬁnit y). This limit has been discussed in [28, 32] as a resonant in teraction of three wa ves to make Y-shap e soliton. This limit giv es a critical angle b etw een those solitons whic h can be found as follo ws: First let us express eac h k j parameter in terms of the amplitude and the slop e, k 1 , 2 = 1 2  tan Ψ [1 , 2] ∓ q 2 A [1 , 2]  , k 3 , 4 = 1 2  tan Ψ [3 , 4] ∓ q 2 A [3 , 4]  , where the angle Ψ [ i,j ] is measured in the counterclockwise direction from the y -axis (see Figure 6.3). In particular, we hav e tan Ψ [1 , 2] = − q 2 A [1 , 2] + 2 k 2 , tan Ψ [3 , 4] = q 2 A [3 , 4] + 2 k 3 . F or simplicity , let us consider the sp ecial case when b oth solitons are of equal amplitude and symmetric with respect to the y -axis i.e., A [1 , 2] = A [3 , 4] = A 0 and Ψ [3 , 4] = − Ψ [1 , 2] = Ψ 0 > 0. This corresp onds to setting k 1 = − k 4 and k 2 = − k 3 . Then, for ﬁxed amplitude A 0 , the angle Ψ 0 has a lo wer b ound given by tan Ψ 0 = p 2 A 0 + 2 k 3 ≥ p 2 A 0 := tan Ψ c . The low er b ound is achiev ed in the limit k 2 = k 3 = 0, and the critical angle Ψ c is given by (6.4) Ψ c = tan − 1 p 2 A 0 . In [28], Miles in tro duced the follo wing parameter to describe the interaction prop erties for O-t yp e solution, (6.5) κ := tan Ψ 0 √ 2 A 0 = tan Ψ 0 tan Ψ c . With this parameter, the maximum amplitude of (6.3) for this symmetric case is given b y (6.6) u max = 4 A 0 1 + √ ∆ O , with ∆ O = 1 − 1 κ 2 . Th us, at the critical angle Ψ 0 = Ψ c (i.e., κ = 1), w e hav e u max = 4 A 0 and the phase shift θ [12] → ∞ , leading to the resonant Y-shap e interaction (see also [28, 12, 42]). One should note that if we use the form of the O-type solution even b ey ond the critical angle, i.e. k 3 < k 2 , then the solution b ecomes singular (note that the sign of E (2 , 3) c hanges). In earlier works, this w as considered to be an obstacle for using the KP equation to describe an in teraction of t w o line- solitons with a smaller angle. On the contrary , the KP equation should give a b etter approximation to describ e oblique interactions of solitons with smaller angles. Thus one should exp ect to ha ve explicit solutions of the KP equation describing such phenomena. It turns out that the new types of (2 , 2)-soliton solutions discussed ab ov e can indeed serve as go o d mo dels for describing line-soliton in teractions of solitons with small angles. W e will show in Section 8 how these solutions are related to the Mach r eﬂe ctions in shallow water wa ves. 6.2. (3142) -t yp e soliton solutions. W e consider a solution of this type which consists of t wo line- solitons for large positive x and t wo other line-solitons for large negative x . W e then assume that the slop es of tw o solitons in eac h region hav e opp osite signs, i.e. one in y > 0 and other in y < 0 (see Figure 6.4). The line-solitons for the (3142)-t yp e solution are determined from the balance b etw een t wo appropriate exp onen tial terms in its τ -function which has the form, τ = E (1 , 3) + bE (1 , 4) + aE (2 , 3) + abE (2 , 4) + cE (3 , 4) . KP SOLITONS IN SHALLOW W A TER 31 Figure 6.4. The time evolution of a (3142)-t yp e soliton solution. The line-solitons ha ve A [1 , 3] = A [2 , 4] = 0 . 5 and Ψ [2 , 4] = − Ψ [1 , 3] = 25 ◦ . The critical angle is giv en by Ψ c = tan − 1 √ 2 A 0 = 45 ◦ , which also gives Ψ c = Ψ [3 , 4] = − Ψ [1 , 2] . The parameters in the A -matrix are given by (6.9) with θ + 13 = θ + 24 = 0 and s = 1. The amplitude of the intermediate [1 , 4]-soliton approaches asymptotically to 1 . 075. The solution contains three free parameters a, b and c , which can b e used to determine the lo cations of three (out of four) asymptotic line-solitons (e.g. tw o in x  0 and one in x  0). Thus, the parameters are completely determined from the asymptotic data on large | y | . Let us ﬁrst consider the line-solitons in x  0: There are tw o line-solitons which are [1 , 3]-soliton in y  0 and [2 , 4]-soliton in y  0. The [1 , 3]-soliton is obtained by the balance b et ween the exp onen tial terms bE (1 , 4) and cE (3 , 4), and the [2 , 4]-soliton is by the balance b et ween aE (2 , 3) and cE (3 , 4). Consequently , the phase shifts of [1 , 3]- and [2 , 4]-solitons for x  0 are given b y (6.7) θ + 13 = ln k 4 − k 1 k 4 − k 3 + ln b c , θ + 24 = ln k 3 − k 2 k 4 − k 3 + ln a c . No w w e consider the line-solitons in x  0: They are [3 , 4]-soliton in y  0 and [1 , 2]-soliton in y  0. The phase shifts are given resp ectiv ely by θ − 34 = ln k 3 − k 1 k 4 − k 1 − ln b, θ − 12 = ln k 3 − k 1 k 3 − k 2 − ln a W e then deﬁne the parameter s (representing the total phase shifts θ + 13 + θ − 34 = θ + 24 + θ − 12 ), (6.8) s := exp  − θ + 13 − θ − 34  , whic h leads to (6.9) a = k 3 − k 1 k 3 − k 2 se θ + 24 , b = k 3 − k 1 k 4 − k 1 se θ + 13 , c = k 3 − k 1 k 4 − k 3 s. The s -parameter represents the relative lo cations of the intersection p oin t of the [1 , 3]- and [3 , 4]- solitons with the x -axis, in particular, θ + 13 + θ − 34 = 0 when s = 1 (see Figure 6.5). Thus the parameters a, b and c are related to the lo cations of [1 , 3]-soliton (with θ + 13 ), of [2 , 4]-soliton (with θ + 24 ), and the in tersection p oin t of [1 , 3]- and [3 , 4]-solitons (with s ). No w w e consider the case where the [1 , 3]- and [2 , 4]-solitons hav e the same amplitude ( A [1 , 3] = A [2 , 4] = A 0 ) and they are symmetric with resp ect to the x -axis (Ψ [2 , 4] = − Ψ [1 , 3] = Ψ 0 ). Then in terms of the k -parameters, we hav e k 3 − k 1 = k 4 − k 2 = p 2 A 0 . Also the symmetry of the wa v e-vectors, i.e. Ψ [2 , 4] = Ψ 0 = − Ψ [1 , 3] , gives k 2 + k 4 = − ( k 1 + k 3 ) = tan Ψ 0 . 32 YUJI K ODAMA Figure 6.5. (3142)-type soliton solution with the s -parameter. The line-solitons are giv en by A [1 , 3] = A [2 , 4] = 0 . 5 and Ψ [2 , 4] = − Ψ [1 , 3] = 25 ◦ (these then giv es the other t wo solitons uniquely). The parameters in the A -matrix are chosen as (6.9) with θ + [1 , 3] = θ + [2 , 4] = 0. Then at s = 1, all the solitons meet at the origin, i.e. the s -parameter shifts [1 , 2]- and [3 , 4]-solitons. This implies that we hav e (6.10) k 4 = − k 1 > 0 , k 3 = − k 2 > 0 . The angle Ψ 0 tak es the v alue in (0 , Ψ c ), where the critical angle is given by the condition k 2 = k 3 = 0, i.e. Ψ c = tan − 1 p 2 A 0 . Notice that this formula is the same as that of the O-t yp e soliton solution (see (6.4)), and the (3142)-t yp e exists when the κ -parameter is less than one, i.e. for (3142)-type, we ha ve κ = tan Ψ 0 √ 2 A 0 < 1 . F rom (6.10), one can easily deduce the following facts for [1 , 2]- and [3 , 4]-solitons in x < 0: (a) Those solitons ha ve the same amplitude, i.e. A [1 , 2] = A [3 , 4] = 1 2 ( k 4 − k 3 ) 2 = 1 2 ( k 4 + k 2 ) 2 = 1 2 tan 2 Ψ 0 = κ 2 A 0 . Th us, if the [1 , 3]- and [2 , 4]-solitons in x > 0 are close to the y -axis (i.e. a small Ψ 0 ), then the amplitudes of the solitons in x < 0 are small; whereas at the critical angle Ψ 0 = Ψ c , the solitons [1 , 2] and [3 , 4] in x < 0 take the maximum amplitude A [1 , 2] = A [3 , 4] = A 0 . (b) The directions of the w av e-vectors for the [1 , 2] and [3 , 4]-solitons are also symmetric, i.e. tan Ψ [3 , 4] = − tan Ψ [1 , 2] = k 3 + k 4 . Moreo ver, the symmetry (6.10) implies that tan Ψ [3 , 4] = k 4 − k 2 = p 2 A [2 , 4] = √ 2 A 0 , so Ψ [3 , 4] = Ψ c = tan − 1 p 2 A 0 . Th us the directions of the wa v e-v ectors for the [1 , 2] and [3 , 4]-solitons in x < 0 dep end only on the amplitude of the solitons in x > 0 but not on their directions (i.e., angle of their V-shap e). Let us choose the parameters in the A -matrix for the (3142)-soliton solution appropriately , so that at t = 0 all the solitons in tersect at the origin (see Figure 6.4). Then for t < 0, the resonan t in teraction b etw een [1 , 3]- and [3 , 4]-solitons (as well as [2 , 4]- and [1 , 2]-solitons) generates an inter- mediate line-soliton (called “stem” soliton) which is [1 , 4] soliton. The amplitude of this soliton is KP SOLITONS IN SHALLOW W A TER 33 Figure 6.6. T-t yp e in teraction with the s -parameter. The k -parameters are chosen as ( k 1 , k 2 , k 3 , k 4 ) = ( − 3 2 , − 1 2 , 1 2 , 3 2 ). The A -matrix is c hosen as (6.12) and (6.13) with θ + 13 = θ + 24 = 0 and r = 1. The s -parameter giv es the phase shift for the [1 , 2]- and [3 , 4]-solitons in x < 0. giv en by (6.11) A [1 , 4] = 1 2 ( k 4 − k 1 ) 2 = 1 2  p 2 A 0 + tan Ψ 0  2 = A 0 (1 + κ ) 2 . Note here that at the critical angle Ψ 0 = Ψ c , the amplitude takes the maximum A [1 , 4] = 4 A 0 (see [43, 37]). F or t > 0, the resonant interaction b et ween [1 , 3]- and [1 , 2]-solitons (as well as [2 , 4]- and [3 , 4]- solitons) generates an intermediate line-soliton of [2 , 3]-soliton. The amplitude of [2 , 3]-soliton is giv en by A [2 , 3] = 1 2 ( k 3 − k 2 ) 2 = 1 2  p 2 A 0 − tan Ψ 0  2 = A 0 (1 − κ ) 2 . Because of the symmetry (6.10), both [1 , 4]- and [2 , 3]-solitons are parallel to the y -axis, i.e. tan Ψ [1 , 4] = tan Ψ [2 , 3] = 0. 6.3. T-t yp e soliton solutions. There are four parameters in the A -matrix for T-type soliton solution. Here we explain that those parameters give the information of the lo cations of those line- solitons, the phase shift and on-set of the op ening of a b o x. Thus three of those four parameters are determined by the asymptotic data on large | y | , and we need an internal data for the other one. F ollowing the arguments in the previous section, one can ﬁnd the phase shifts of the line-solitons of [1 , 3] and [2 , 4]: F or [1 , 3]-soliton in x > 0 (and y  0), the phase shift is calculated as θ + 13 = ln k 4 − k 1 k 4 − k 3 − ln D b , where D = ad − bc = ξ (3 , 4). F or the same soliton in x < 0 (and y  0), we hav e θ − 13 = ln k 2 − k 1 k 3 − k 2 − ln c . So the total phase shift θ 13 := θ − 13 − θ + 13 dep ends on the A -matrix unlike the cases of O- and P-types, and it can take any v alue. F or [2 , 4]-soliton in x > 0 (and y  0), we hav e θ + 24 = ln k 3 − k 2 k 4 − k 3 − ln D c . and for the same one in x < 0 (and y  0), we hav e θ − 24 = ln k 2 − k 1 k 4 − k 1 − ln b. 34 YUJI K ODAMA Figure 6.7. T-type in teraction with the r -parameter. The k -parameters are the same as those in Figure 6.6. The A -matrix are chosen as (6.12) with θ + 13 = θ + 24 = 0 and s = 1. The r -parameter gives the on-set of the b ox, and it does not aﬀect the lo cations of all four line-solitons, that is, r is an internal parameter. Note that the total phase shift θ 24 = θ + 24 − θ − 24 is the same as that for [1 , 3]-soliton, i.e. the phase conserv ation along the y -axis θ + 13 + θ − 24 = θ − 13 + θ + 24 holds. Then as in (6.8) for the case of (3142)-t yp e, w e deﬁne the s -parameter, s := exp( − θ + 13 − θ − 24 ) , whic h represents the intersection p oint of [1 , 3]- and [2 , 4]-soliton. With the s -parameter, we hav e (6.12) b = k 2 − k 1 k 4 − k 1 se θ + 13 , c = k 2 − k 1 k 3 − k 2 se θ + 24 , D = k 2 − k 1 k 4 − k 3 s. Namely , the three parameters b, c and D = ad − bc determine the lo cations and the phase shift (i.e. the intersection p oint of [1 , 3]- and [2 , 4]-solitons). One other parameter is then related to an on-set of a b ox at the intersection p oin t (see Figure 6.7). In order to c haracterize this parameter, let us consider the in termediate solitons of [1 , 4] and [2 , 3]. First note that for t  0, [1 , 4]-soliton appears as the dominant balance betw een E (1 , 2) and E (2 , 4). Then one can ﬁnd the phase shift θ + 14 (here + indicates t > 0), θ + 14 = ln k 2 − k 1 k 4 − k 2 − ln d. Similarly one can get the phase shift θ − 14 for t  0 as θ − 14 = ln k 3 − k 1 k 4 − k 3 − ln D a . No w consider the sum of θ ± 14 , i.e. θ + 14 + θ − 14 = ln ( k 2 − k 1 )( k 3 − k 1 ) ( k 4 − k 2 )( k 4 − k 3 ) − ln dD a . Also, for the [2 , 3]-soliton, one can get θ + 23 + θ − 23 = ln ( k 2 − k 1 )( k 4 − k 2 ) ( k 3 − k 1 )( k 4 − k 3 ) − ln aD d . No w we in tro duce a parameter r in the form, (6.13) a d = r k 4 − k 2 k 3 − k 1 , so that we hav e θ + 14 + θ − 14 = ln r s , θ + 23 + θ − 23 = − ln ( r s ) . KP SOLITONS IN SHALLOW W A TER 35 Supp ose that at t = 0, [1 , 3]- and [2 , 4]-solitons in x > 0 are placed so that they meet at the origin, that is, w e c ho ose θ + 13 = θ + 24 = 0. Also if there is no phase shifts for those solitons, i.e. s = 1. then the sums b ecome θ + 14 + θ − 14 = ln r = − ( θ + 23 + θ − 23 ) . This implies that at t = 0 (and s = 1) if r = 1, then the T-type soliton solution has an exact shap e of “X” without an y opening of a b o x at the intersection p oin t on the origin. Moreov er, at t = 0 if r > 1, then [1 , 4]-soliton app ears in x > 0 and [2 , 3]-soliton in x < 0; whereas if 0 < r < 1, then [1 , 4]-soliton app ears in x < 0 and [2 , 3]-soliton in x > 0. Figure 6.7 illustrates those cases with s = 1. The parameter r determines the exp onential term that is dominant in the region inside the b o x. When r < 1, E (2 , 4) is the dominant exp onen tial term, and when r > 1 the dominant exp onen tial is E (1 , 3). One should note that the parameter r cannot be determined by the asymptotic data, that is, r is considered as an “internal” parameter. 7. Numerical simula tion and the st ability of the soliton solutions In this section, we present some numerical sim ulations of the KP equation with “V-shap e” initial w av e form related to a ph ysical situation (see for examples [37, 43, 15]). The main purp ose of the n umerical sim ulation is to study the interaction prop erties of line-solitons, and we will show that the solutions of the initial v alue problems with V-shap e incident w av es approach asymptotically to some of the exact soliton solutions of the KP equation discussed in the previous section. This implies a stability of those exact solutions under the inﬂuence of certain deformations (notice that the deformation in our cases are not so small). The initial v alue problem considered here is essen tially an inﬁnite energy problem in the sense that each line-soliton in the initial w av e is supported asymptotically in either y  0 or y  0, and the interactions o ccur only in a ﬁnite domain in the xy -plane. In the numerical scheme, we consider the rectangular domain D = { ( x, y ) : | x | ≤ L x , | y | ≤ L y } , and each line-soliton is matched with a KdV soliton at the b oundaries y = ± L y . The details of the numerical scheme and the results can b e found in [20]. W e consider the initial data given in the shap e of “V” with the amplitude A 0 and the oblique angle Ψ 0 > 0, (7.1) u ( x, y, 0) = A 0 sec h 2 r A 0 2 ( x − | y | tan Ψ 0 ) . Note here that t wo semi-inﬁnite line-solitons are propagating tow ard each other into the p ositive x -direction, so that they interact strongly at the corner of the V-shap e. At the b oundaries y = ± L y of the numerical domain, those line-solitons are patched to the KdV one-soliton solutions given by u ( x, ± L y , t ) = A 0 sec h 2 r A 0 2 ( x ∓ L y tan Ψ 0 − ν t ) , with ν = 3 4 tan 2 Ψ 0 + 1 2 A 0 . Note here that these solitons corresp ond to the exact one-soliton solution of the KdV equation with the velocity shift due to the oblique propagation of the line-soliton, i.e. ∂ 2 u/∂ y 2 = tan 2 Ψ 0 ∂ 2 u/∂ x 2 . The numerical sim ulations are based on a sp ectral metho d with windo w-technique similar to the metho d used in [43] (see [20] for the details). The V-shap e initial w av e was ﬁrst considered by Oik aw a and Tsuji (see for example [37, 43]) in order to study the generation of freak (or rogue) wa ves. They noticed generations of diﬀerent t yp es of asymptotic solutions depending on the initial oblique angle Ψ 0 , and found the resonan t in teractions whic h create lo calized high amplitude wa ves. In this section, we present the results for the cases corresponding to A 0 = 2 and t wo diﬀerent angles, Ψ 1 and Ψ 2 with Ψ 1 < Ψ c < Ψ 2 . where the critical angle is giv en by Ψ c = tan − 1 √ 2 A 0 ≈ 63 . 4 ◦ . Then we explain these results in terms of certain (2 , 2)-soliton solutions discussed in the previous section, and in particular, we describe the connection with the Mac h reﬂection (this will b e further discussed in Section 8). 36 YUJI K ODAMA Figure 7.1. Initial data with V-shap e wa ve. Each line of the V-shap e is lo cally a line-soliton solution. W e set those line-solitons to meet at the origin. The main idea here is to consider the V-shape initial wa v e as the part of some (2 , 2)-soliton solutions listed in the previous section. In order to identify those soliton s olutions from the V-shap e initial wa ve form, let us ﬁrst denote them as [ i 1 , j 1 ]-soliton for y  0 and [ i 2 , j 2 ]-soliton for y  0. Then using the relations, k j − k i = √ 2 A 0 = 2 and k j + k i = tan Ψ 0 , for [ i, j ]-soliton and the Miles parameter κ = tan Ψ 0 / √ 2 A 0 of (6.5), we hav e (7.2) ( k i 1 = − (1 + κ ) , k j 1 = 1 − κ, k i 2 = − (1 − κ ) , k j 2 = 1 + κ. Notice that k j 2 = − k i 1 and k i 2 = − k j 1 b ecause of the symmetry in the initial wa ve. Moreo ver, at the critical angle Ψ 0 = Ψ c (i.e. κ = 1), we hav e k i 2 = k j 1 = 0. W e also note k i 1 as the smallest parameter and k j 2 as the largest one, so that dep ending on the angle Ψ 0 , we obtain the follo wing ordering in the k -parameters: F or 0 < Ψ 0 < Ψ c (i.e. κ < 1), we hav e k i 1 < k i 2 < 0 < k j 1 < k j 2 , implying that the corresp onding chords of the [ i 1 , j 1 ]- and the [ i 2 , j 2 ]-solitons ov erlap. That is, [1 , 3] c hord app ears on the upp er side of the diagram, and [2 , 4] chord on the low er side. This means that the tw o solitons can b e identiﬁed as part of either the (3412)-type (T-type) or the (3142)-t yp e solution (see the chord diagrams in Figure 6.1). F or Ψ c < Ψ 0 < π 2 (i.e. κ > 1), we hav e k i 1 < k j 1 < 0 < k i 2 < k j 2 . In this case, the corresp onding chords are separated, and the tw o solitons form part of either (2413)- or (2143)-type (O-type) solution. Here [1 , 2]- and [3 , 4]-c hords app ear on the upp er and low er sides of the chord diagram, resp ectively . Then the n umerical simulations show that we hav e the follo wing types of the asymptotic solutions dep ending on the v alues Ψ 0 : (a) If the angle satisﬁes Ψ 0 < Ψ c (i.e. κ < 1), then the solution con verges asymptotically to (3142)-t yp e soliton solution (not T-type) (b) If the angle satisﬁes Ψ c < Ψ 0 (i.e. κ > 1), then the solution conv erges asymptotically to an O-t yp e soliton solution (not (2413)-type). The conv ergence here is in a lo cally deﬁned L 2 -sense with the usual norm, k f k L 2 ( D ) :=  Z Z D | f ( x, y ) | 2 dxdy  1 2 . KP SOLITONS IN SHALLOW W A TER 37 where D ⊂ R 2 is a compact set which cov ers the main structure of the in teractions in the solution. T o conﬁrm the conv ergence statements, we deﬁne the (relative) error function, (7.3) E ( t ) := k u t − u t exact k 2 L 2 ( D t r )  k u t exact k 2 L 2 ( D t r ) , with the solution u t ( x, y ) := u ( x, y , t ) and an exact solution u t exact ( x, y ), where D t r is the circular disc given by D t r :=  ( x, y ) ∈ R 2 : ( x − x 0 ( t )) 2 + ( y − y 0 ( t )) 2 ≤ r 2  . The center ( x 0 ( t ) , y 0 ( t )) of the circular domain D t r is chosen as the in tersection p oint of t wo lines de- termined from the corresp onding exact solution. W e ﬁnd the exact solution u t exact ( x, y ) by minimiz- ing E ( t ) at certain large time t = T 0 : In the minimization pro cess, we assume that the k -parameters remain the same as those given b y (7.2), and v ary the corresp onding A -matrix to adjust the solution pattern (recall that the A -matrix determines the lo cations of the line-solitons in the solution, see Section 5). After minimizing E ( t ), that is, ﬁnding the corresp onding exact solution, we chec k that E ( t ) further decreases for a larger time t > T 0 up to a time t = T 1 > T 0 , just before the eﬀects of the b oundary enter the disc D t r (those eﬀects include the p erio dic condition in x and a mismatch on the b oundary patching). W e take the radius r in D t r large enough so that the main in teraction area is cov ered for all t < T 0 , but D t r should b e kept a wa y from the b oundary to av oid any inﬂuence coming from the b oundaries. The time T 1 > T 0 giv es an optimal time to dev elop a pattern close to the corresp onding exact solution, but it is also limited to av oid an y disturbance from the b oundaries for t < T 1 . Thus, our con vergence implies the separation of the radiations from the soliton solution, just like the case of the KdV equation (see the end of Section 3). W e also note that the conv ergence here implies a c ompletion of the partial chord diagram consist- ing of only tw o chords whic h corresp onds to the semi-inﬁnite solitons in the initial V-shap e w a ve. Namely , the asymptotic solution of the initial v alue problem with V-shape initial w a ve is giv en b y an exact solution parametrized by a unique c hord diagram, and the initial (partial) chord diagram is completed by adding tw o other solitons (chords) generated by the in teraction. The completion ma y not b e unique, and in [23], we prop osed a concept of minimal completion in the sense that the completed diagram has the minimum total length of the chords and the corresp onding TNN Grassmannian cell has the minimum dimension. How ev er, this problem is still op en, and w e need to mak e a precise statement of the minimal completion of partial chord diagram given by the initial w av e proﬁle. 7.1. Regular reﬂection: κ > 1 . W e consider the V-shap e initial wa ve with A 0 = 2 and tan Ψ 0 = 12 5 , (Ψ 0 ≈ 67 . 3 ◦ ) which gives κ = 1 . 2. Here the critical angle is Ψ c = tan − 1 (2) ≈ 63 . 4 ◦ , and we exp ect asymptotically an O-type soliton solution. The corresp onding k -parameters are obtained from (7.2), i.e. ( k 1 , k 2 , k 3 , k 4 ) =  − 11 5 , − 1 5 , 1 5 , 11 5  . Figure 7.2 illustrates the result of the numerical simulation. The top ﬁgures show the direct simu- lation of the KP equation. The wak e b ehind the interaction p oin t has a large negative amplitude, and it disp erses and decays in the negative x -direction. This sho ws a separation of the radiations from the exact solution similar to the case of KdV soliton. The steady pattern left after shedding the radiations can b e identiﬁed as an O-t yp e solution. The middle ﬁgures sho w the corresponding O-t yp e exact solution whose A -matrix is determined by minimizing the error function E ( t ) at t = 6, A =  1 1 . 91 0 0 0 0 1 0 . 17  Using (6.1), we obtain the shift of the initial line-solitons, x [1 , 2] = x [3 , 4] = − 0 . 020 . (Note here that because of the symmetric proﬁle, the shifts for initial solitons are the same.) The negativ e shifts imply the slow-do wn of the incidence wa ves due to the generation of the solitons 38 YUJI K ODAMA 0 2 4 6 0 0.1 0.2 0.3 0.4 t E(t) Figure 7.2. Numerical simulation of V-shap e initial wa ve for κ > 1 (regular reﬂection). The initial wa ve consists of [1 , 2]-soliton in y > 0 and [3 , 4]-soliton in y < 0, with A [1 , 2] = A [3 , 4] = 2 and Ψ 0 ≈ 67 . 3 ◦ (Ψ c = 63 . 4 ◦ . The upp er ﬁgures sho w the result of the direct simulation. Notice a large w ake behind the interaction p oin t whic h extends the initial solitons. The middle ﬁgures show the corresp onding exact solution of O-type. The circle in these ﬁgures show the domain D t r with r = 12. Ab out t = 3, the wak es seem to b e out of the domain. The b ottom graph shows the error function E ( t ) which is minimized at t = 6. The solid chords in the diagram indicates the inciden t solitons, and the dotted ones show the reﬂected solitons (i.e. a completion of the chord diagram [23]). extending the initial solitons in the negative x -direction. The phase shifts ∆ x [ i,j ] for the O-type exact solution are calculated from (6.2), and they are ∆ x [1 , 2] = ∆ x [3 , 4] = 0 . 593 . The p ositivit y of the phase shifts is due to the attractive force b et ween the line-solitons, and this explains the slo w-down of the initial solitons, i.e. the small negative shifts of x [ i,j ] . The b ottom graph in Figure 7.2 sho ws E ( t ) of (7.3), where w e tak e r = 12 for the domain D t r . One can see a rapid con vergence of the solution to the O-t yp e exact solution with those parameters. One should how ever remark that when Ψ 0 is close to the critical one, i.e. k 2 ≈ k 3 , there exists a large phase shift in the soliton solution, and the con vergence is v ery slo w. Note that in the limit k 2 = k 3 the amplitude of the intermediate soliton generated at the intersection p oin t reaches four times larger than the initial solitons. This large amplitude wa ve generation has b een considered as the Mach reﬂection problem of shallow w ater w av e [28, 37, 8, 23] (see also Section 8). The chord diagram in Figure 7.2 sho ws a c ompletion of the (partial) chord diagram: The solid chords indicate the initial solitons forming V-shap e, and the dotted chords corresp onds to the solitons generated by the interaction (see [23] for further discussion). KP SOLITONS IN SHALLOW W A TER 39 0 5 10 0 0.02 0.04 0.06 t E(t) Figure 7.3. Numerical simulation of V-shap e initial wa v e for κ < 1 (Mach reﬂec- tion): The initial w av e consists of [1 , 3]-soliton in y > 0 and [2 , 4]-soliton in y < 0 with A 0 = 2 and Ψ 0 = 45 ◦ . The top ﬁgures show the result of the direct simula- tion, and the middle ﬁgures show the corresponding exact solution of (3142)-type. Notice that a large amplitude intermediate soliton is generated at the intersection p oin t, and it corresponds to the [1 , 4]-soliton with the amplitude A [1 , 4] = 4 . 5. The circles in the middle ﬁgures are D t r with r = 12, which cov er w ell the main part of the interaction regions up to t = 12. The b ottom graph of the error function E ( t ) whic h is minimized at t = 10. The solid chords in the diagram indicate the incident solitons, and the dotted ones show the reﬂected solitons. 7.2. The Mac h reﬂection: κ < 1 . W e consider the initial V-shap e w av e with A 0 = 2 and Ψ 0 = 45 ◦ (i.e. κ = 0 . 5). The angle Ψ 0 is now less than the critical angle Ψ c ≈ 63 . 4 ◦ . The asymptotic solution is exp ected to b e of (3142)-type whose k -parameters are obtained from (7.2), i.e. ( k 1 , k 2 , k 3 , k 4 ) =  − 3 2 , − 1 2 , 1 2 , 3 2  . Figure 7.3 illustrates the result of the n umerical simulation. The top ﬁgures sho w the direct sim ulation of the KP equation. W e again observe a b ow-shape wak e b ehind the interaction point. The wak e expands and decays, and then we see the app earance of new solitons which form resonant in teractions with the initial solitons. One should note that the solution generates a large amplitude in termediate soliton at the in teraction p oin t, and this soliton is iden tiﬁed as [1 , 4]-soliton with the amplitude A [1 , 4] = 4 . 5 . This [1 , 4]-soliton is called the Mach stem in the Mach reﬂection [8, 23] (see also Section 8). The middle ﬁgures in Figure 7.3 sho w the corresp onding exact solution of (3142)-type whose A -matrix is found by minimizing E ( t ) at t = 10, A =  1 1 . 92 0 − 1 . 96 0 0 1 0 . 64  40 YUJI K ODAMA 0 2 4 6 0 0.02 0.04 0.06 0.08 t E(t) Figure 7.4. Numerical sim ulation for X-shape initial w av e with A 0 = 2 and Ψ 0 = 45 ◦ (i.e. κ = 0 . 5 < 1): The initial wa ve is the sum of [1 , 3]- and [2 , 4]-solitons. The top ﬁgures show the numerical simulation, and the middle ﬁgures show the corresp onding exact solution of (3412)-t yp e, i.e. T-t yp e. Notice that the circle sho wing D t r with r = 22 co v ers well the box generated by the resonan t in teraction up to t = 7. The b ottom graph shows the error function E ( t ) of (7.3) which is minimized at t = 6. The four solid chords in the diagram show the asymptotic solitons in the incident wa ve, and they form a T-type soliton solution. Using (6.9), we obtain the phase shifts x [ i,j ] for the initial solitons of [1 , 3]- and [2 , 4]-type in x > 0, and the s -parameter, x [1 , 3] = x [2 , 4] = − 0 . 01 , s = 0 . 980 . Those v alues indicate that the solution is very close to the exact solution for all the time. The negativ e v alue of the shifts x [ i,j ] is due to the generation of a large amplitude soliton [1 , 4]-t yp e (i.e. the initial solitons slow do wn), and s < 1 implies that the [1 , 4]-soliton is generated after t = 0. Also note that the [1 , 4]-soliton now resonan tly in teract with [1 , 3]- and [2 , 4]-solitons to create new solitons [1 , 2]- and [3 , 4]-solitons (called the reﬂected wa ves in the Mach reﬂection problem [28, 8, 23]). This pro cess then seems to comp ensate the shifts of incident wa ves, even though we observe a large wak e b ehind the interaction p oin t. The b ottom graph in Figure 7.3 shows a rapid conv ergence of the initial wa ve to (3142)-t yp e soliton solution with those parameters of the A -matrix and k v alues giv en abov e, and for the error function E ( t ), we minimize it at t = 10 for D t r with r = 12. 7.3. T-t yp e interaction with X-shap e initial w a v e. In this example, we consider an X-shape initial wa ve given by the sum of tw o line-solitons. F or simplicity , we consider a symmetric initial w av e with A 0 = 2 and Ψ 0 = 45 ◦ (i.e. extend the initial wa ve in Figure 7.3 into the negative x - region). Since κ = 0 . 5 < 1, the corresp onding chord diagram sho ws the T-t yp e with the k -parameters KP SOLITONS IN SHALLOW W A TER 41 ( k 1 , k 2 , k 3 , k 4 ) = ( − 3 2 , − 1 2 , 1 2 , 3 2 ). Although T-t yp e soliton app ears for smaller angle Ψ 0 , one should not take so small v alue. F or an e xample of the symme tric case, if we take Ψ 0 = 0 giving twice higher amplitude than one-soliton case, we obtain KdV 2-soliton solution with diﬀerent amplitudes (as can b e shown b y the metho d of IST). So for the case with v ery small angle Ψ 0 , we expect to see those KdV solitons near the intersection p oin t. How ever, the solitons exp ected from the c hord diagram ha ve almost the same amplitude as the incidence solitons for the case with a small angle. The detailed study also shows that near the intersection p oint for T-type solution at the time when all four solitons meet at this point (i.e X-shape), the solution at the in tersection p oin t has a small amplitude due to the repulsive force similar to the KdV solitons. Then the initial X-shap e wa ve with small angle generates a large soliton at the intersection point. This then implies that our initial w av e given by the sum of tw o line-solitons creates a large disp ersiv e p erturbation at the intersection p oin t, and one may need to wait a long time to see the conv ergence. The top ﬁgures in Figure 7.4 illustrate the n umerical simulation, whic h clearly shows an opening of a resonant b o x as exp ected by the feature of T-type. The corresp onding exact solution is illustrated in the middle ﬁgures, where the A -matrix of the solution is obtained b y minimizing the error function E ( t ) of (7.3) at t = 6, A =  1 0 − 0 . 368 − 0 . 330 0 1 1 . 198 0 . 123  In the minimization pro cess, we tak e x [1 , 3] = x [2 , 4] due to the symmetric proﬁle of the solution, and adjust the on-set of the b o x (see subsection 6.3). Note that the symmetry reduces the num b er of free parameters to three. W e obtain x [1 , 3] = x [2 , 4] = 0 . 025 , r = 3 . 63 , s = 0 . 350 . The p ositiv e shifts of those [1 , 3]- and [2 , 4]-solitons in the wa vefron t indicate also the p ositive shift of the newly generated soliton of [1 , 4]-t yp e at the front. This is due to the repulsive force which exists in the KdV type interaction as explained ab o ve, that is, the in teraction part in the initial w av e has a larger amplitude than that of the exact solution, so that this part of the solution mo ves faster than that in the exact solution. This diﬀerence may result in a shift of the lo cation of the [1 , 4]-soliton. The relatively large v alue r > 1 indicates that the onset of the box is actually muc h earlier than t = 0, and s < 1 shows the p ositiv e phase shifts as calculated from (6.12). The b ottom graph in Figure 7.4 sho ws the evolution of the error function E ( t ) of (7.3) whic h is minimized at t = 6. Note here that the circular domain D t r with r = 22 cov ers well the main feature of the interaction patterns for all the time computed for t ≤ 7. The chord diagram in the ﬁgure sho ws four asymptotic solitons in the initial wa ve which form a T-type soliton solution (see [20] for further discussion). 8. Shallow w a ter w a ves: The Mach reflection In this last section, we discuss a real application of the exact soliton solutions of the KP equation describ ed in the previous sections to the Mach reﬂection phenomena in shallow water. In [28], J. Miles considered an oblique interaction of t w o line-solitons using O-type solutions. He observed that resonance o ccurs at the critical angle Ψ c , and when the initial oblique angle Ψ 0 is smaller than Ψ c , the O-type solution becomes singular (recall that at the critical angle Ψ c , one of the exponential term in the τ -function v anishes, see subsection 6.1). He also noticed a similarit y b et w een this resonant in teraction and the Mach reﬂection found in sho ck w av e interaction (see for example CF:48,Wh:64). This is illustrated by the left ﬁgure of Figure 8.1, where an incidence w av e shown b y the vertical line is propagating to the righ t, and it hits a rigid wall with the angle − Ψ 0 measured counterclockwise from the axis p erpendicular to the wall (see also [15]). If the angle Ψ 0 (equiv alen tly the inclination angle of the wall) is large, the reﬂected w av e b ehind the incidence w av e has the same angle Ψ 0 , i.e. a regular reﬂection o ccurs. Ho wev er, if the angle is small, then an intermediate wa ve called the Mach stem app ears as illustrated in Figure 8.1. The Mach stem, the incident wa ve and the reﬂected wa v e in teract resonantly , and those three wa ves form a resonant triplet. The right panel in Figure 8.1 42 YUJI K ODAMA Ψ 0 O O −Ψ 0 −Ψ 0 Φ Ψ 0 Figure 8.1. The Mach reﬂection. The left panel illustrates a semi-inﬁnite line- soliton (incidence wa ve) propagating parallel to the wall with the mirror image. The right panel is an equiv alen t system to the left one when w e ignore the viscous eﬀect on the w all. The incident w av e then forms a V-shap e wa v e at t = 0 as discussed in Section 7. The resulting wa ve pattern sho wn here is a (3142)-soliton solution. illustrates the wa ve propagation which is equiv alent to that in the left panel, if one ignores the eﬀect of viscosity on the wall (i.e. no b oundary lay er). At the p oint O , the initial wa ve has V-shap e with the angle Ψ 0 , which forms the initial data for our simulation discussed in the previous section. Then as we presented, the n umerical simulation describ es the reﬂection of line-soliton with an inclined w all, and these results explain well the Mach reﬂection phenomena in terms of the exact soliton solutions of the KP equation. 8.1. Previous n umerical results of the Boussinesq-type equations. One of the most in- teresting things of the Mac h reﬂection is that the KP theory predicts an extraordinary four-fold ampliﬁcation of the stem wa ve at the critical angle [28]. W e recall the formulae of the maximum amplitudes which are given by (6.6) for the O-type solution ( κ > 1) and (6.11) for (3142)-type solution ( κ < 1). Let α denote the ampliﬁcation factor in terms of the Miles parameter κ of (6.5), i.e. (8.1) α =    (1 + κ ) 2 , for κ < 1 , 4 1 + √ 1 − κ − 2 , for κ > 1 . Sev eral laboratory and n umerical experiments tried to conﬁrm the form ula (8.1), in particular, the four-fold ampliﬁcation at the critical v alue κ = 1 (see for example [36, 27, 15, 41, 46]). In [15], F unak oshi made a numerical simulation of the Mach reﬂection problem using the system of equations,        η t + ∆ ψ + α ∇ · ( η ∇ ψ ) − β 6 ∆ 2 ψ = 0 ,  ψ − β 2 ∆ ψ  t + η + α 2 |∇ ψ | 2 = 0 , whic h is equiv alent to the Boussinesq-type equation (2.4) up to this order. He considered the initial w av e to b e the KdV soliton with higher order corrections up to O (  ). In his paper, he mainly presen ted the results for the incidence w av es with the amplitude a i = 0 . 05 = ˆ a 0 /h 0 and the angles π 40 ≤ Ψ 0 ≤ π 3 . He concluded that his results agree very well with the resonantly interacting solitary w av e solution predicted by Miles. How ever his results on the ampliﬁcation parameter α are slightly shifted to the lo wer v alues of the Miles κ -parameter. T anak a in [41] then re-examined F unakoshi’s KP SOLITONS IN SHALLOW W A TER 43 0 1 2 3 4 1 2 3 4 FUN AKO SHI TANA KA YEH et al MILES (19 77 ) (19 80) (19 93) (20 10) κ α Figure 8.2. Numerical results of the ampliﬁcation factor α versus the κ -parameter. The circles sho w F unak oshi’s result [15], the squares show T anak a’s result [41]. The blac k dots sho ws the exp erimen tal results by Y eh et al [46]. results for higher amplitude incidence wa ves with a i = 0 . 3 using the high-order sp ectral metho d. He noted that the eﬀect of large amplitude tends to preven t the Mach reﬂection to o ccur, and all the parameters suc h as the critical angle Ψ c are shifted to ward the v alues corresp onding to the regular reﬂection (i.e. O-type). F or example, he obtained the maximum ampliﬁcation α = 2 . 897 at κ = 0 . 695. Ho wev er, we claim in our recent pap er [47] that those previous results did not prop erly interpret their comparisons with the theory , and in fact their results are in go o d agreement with the the predictions given by the KP theory except for the cases near κ = 1. One should emphasize that the KP equation is deriv ed under the assumptions of quasi-tw o dimensionalit y and weak nonlinearit y . Th us the key ingredien t is to include higher order corrections to those assumptions when w e compare the n umerical or experimental results with the theory . In particular, the quasi-t wo dimensionalit y can b e corrected by comparing the KP soliton with the KdV soliton in the propagation direction as men tioned in Section 3. More precisely , we hav e the amplitude correction (3.6), i.e. ˆ a 0 = a 0 cos 2 Ψ 0 = 2 h 0 A 0 3 cos 2 Ψ 0 , where ˆ a 0 is the amplitude observ ed in the n umerical computation of the Boussinesq-type equation (whic h has rotational symmetry in R 2 ). This then suggests that the κ -parameter should be ev aluated b y the follo wing formula using the exp erimen tal amplitude ˆ a 0 , (8.2) κ := tan Ψ 0 √ 2 A 0 = tan Ψ 0 p 3(ˆ a 0 /h 0 ) cos Ψ 0 . Because of the quasi-tw o dimensional approximation, i.e. | Ψ 0 |  1, Miles in his pap er [28] replaced tan Ψ 0 b y Ψ 0 , and then in [15, 41], the authors contin ued on to use this replacement. Then their computations with rather large v alues of Ψ 0 ga ve signiﬁcan t shifts of the κ -parameter. W e then re-ev aluate their results with our form ula (8.2), and the new results are sho wn in Figure 8.2. Since F unakoshi’s simulations are based on small amplitude incidence wa ves, his results agree quite well with the KP predictions. T anak a’s results are also in go o d agreement with the KP theory except for the cases near the critical angle (i.e. κ = 1), where the ampliﬁcation parameter α gets close to 3. This region clearly violates the assumption of the weak nonlinearity . Although one needs to make higher order corrections to weak nonlinearit y , the original plots of T anak a’s are signiﬁcan tly improv ed with the formula (8.2). The black dots in Figure 8.2 indicate the results of recent lab oratory exp erimen ts 44 YUJI K ODAMA T able 1. Ampliﬁcation factor α for diﬀerent v alues of κ = tan Ψ 0 / √ 2 A 0 : α ˜ x =71 . 1 (Exp.) are the laboratory data at x = 71 . 1 ( ˜ x = 4 . 27 m), α x =71 . 1 (KP) are calculated from the corresp onding KP exact solutions at t = 41 . 05, and α x = ∞ (KP) are from (6.6) and (6.11). In the row of A 0 = 0 . 413 with Ψ 0 = 30 ◦ , the v alues of α in the brack ets are obtained at x = 50 . 8, b ecause of the wa ve breaking immediately after this p oint; hence, the greater ampliﬁcation cannot b e realized [47]. κ A 0 Ψ 0 α x =71 . 1 (Exp . ) α x =71 . 1 (KP) α x = ∞ (KP) 1.392 0.086 30 ◦ 2.10 2.36 2.36 1.242 0.108 30 ◦ 2.13 2.51 2.51 1.017 0.161 30 ◦ 2.24 3.38 3.38 0.887 0.212 30 ◦ 2.33 2.43 3.56 0.731 0.312 30 ◦ 2.52 2.61 2.99 0.722 0.127 20 ◦ 1.89 1.84 2.96 0.635 0.413 30 ◦ (2.48) (2.54) 2.67 0.591 0.189 20 ◦ 1.95 1.93 2.53 0.516 0.249 20 ◦ 1.99 2.08 2.30 0.425 0.367 20 ◦ 2.01 1.99 2.03 done by Y eh and his colleagues [46]. W e will discuss their exp erimental results in the next section. Before closing this section, w e remark on the length of the Mac h stem (i.e. the intermediate soliton of [1 , 4]-t yp e). F rom (3142)-soliton solution, one can ﬁnd the p oint ( x ∗ , y ∗ ) of the interaction of the triplet with [1 , 3]-, [3 , 4]- and [1 , 4]-solitons [8] ,        x ∗ = 1 4 (tan Ψ c + tan Ψ 0 ) 2 t = A 0 2 (1 + κ ) 2 t, y ∗ = 1 2 (tan Ψ c − tan Ψ 0 ) t = r A 0 2 (1 − κ ) t, (see Figure 8.1). In the physical co ordinates with ( ˜ x ∗ , ˜ y ∗ , ˜ t ), we hav e ˜ x ∗ = c 0  1 + ˆ a 0 cos 2 Ψ 0 h 0 (1 + κ ) 2  ˜ t, ˜ y ∗ = c 0 s ˆ a 0 cos 2 Ψ 0 3 h 0 (1 − κ ) ˜ t. The angle Φ in Figure 8.1 is then given by tan Φ = ˜ y ∗ / ˜ x ∗ , which is approximated in [28, 15] by tan Φ ≈ r ˆ a 0 3 (1 − κ ) . Using the corrected formula tan Φ = ˜ y ∗ / ˜ x ∗ , one can see again a go o d agreemen t with the KP theory (see Figure 8 in [15]). 8.2. Exp erimen ts. Recently , Y eh and his colleagues [46] p erformed sev eral lab oratory exp erimen ts on the Mach reﬂection phenomena using 7.3 m long and 3.6 m wide wa ve tank with a water depth of 6.0 cm. Here we brieﬂy describ e their results and show that our KP theory can predict very well the evolution of the wa ves observed in the exp eriments. The wa ve tank is equipp ed with 16 axis directional-w av e mak er system along the 3.6 m long side w all mark ed b y x = 0. An oblique incident solitary w av e is created by driving those 16 paddles synchronously along the sidew all, and the w av e maker is designed to generate a KdV soliton with any heights b efore the breaking. The temp oral and spatial v ariations of water-surface proﬁles are measured b y the Laser Induced Fluorescen t (LIF) method (a highly accurate measuremen t tec hnique). The water dyed with ﬂuorescein (green) ﬂuoresces when excited by the laser sheet. The illuminated image of the water-proﬁles are recorded b y a high-sp eed and high-resolution video camera. KP SOLITONS IN SHALLOW W A TER 45 Figure 8.3. Two views of the temporal v ariation of the w ater-surface proﬁle in the y -direction (p erpendicular to the wall) at x = 71 . 1 [47]: The top panels show the exp erimen tal result, and the b ottom ones sho w the corresp onding (3142)-t yp e exact soliton solution of the KP equation. The incident wa ve amplitude A 0 = 0 . 212, and the angle Ψ 0 = 30 ◦ . The ampliﬁcation factors obtained from the exp erimen t and the exact solutions are close, and they are α x =71 . 1 (Exp . ) = 2 . 33 and α x =71 . 1 (KP) = 2 . 43 (see T able 1). 8.2.1. The Mach r eﬂe ction. In T able 1, their exp erimen tal results of the ampliﬁcation factors α are compared with those obtained from the exact solutions of the KP equation (i.e. O-t yp e for κ > 1 and (3142)-type for κ < 1). The wa ves were measured at ˜ x = 4 . 27 m ( x = ˜ x/h 0 = 71 . 1) which is the farthest measuring lo cation in the exp eriments, except for the case with A 0 = 0 . 413. In the later case, the α v alues in the brac kets are measured at x = 50 . 8 b ecause of the wa ve-breaking (notice that at this p oin t α = 2 . 48 implies a i = 1 . 02). W e calculate the corresp onding KP exact solution at t = 41 . 05 (recall here that the relations (3.8) gives ˜ x − c 0 ˜ t = h 0 x and c 0 ˜ t = 3 h 0 2 t ). The ampliﬁcation factor α is still growing along the propagation direction, and the v alues obtained from the exact solutions are in go od agreement with the measurements. W e note here that near the critical case (i.e. κ = 1) the growth of the stem amplitude is very slo w and at x = 71 . 1 ( ˜ x = 4 . 27 m) the ampliﬁcation factor is only achiev ed ab out 65%. Also for the cases with small oblique angle, i.e. Ψ 0 = 20 ◦ in T able 1, the ampliﬁcation factor α grows slow er when the incidence wa ve amplitude is smaller, that is, at x = 71 . 1, α is almost constant (slightly decreases) as κ increases. How ever the asymptotic v alue of α for large x increases as κ increases. This means that the observ ed w av es are still in the transient stage, and a longer tank is necessary to observe further growth of the ampliﬁcation factor (see [47] for the details). In Figure 8.3, we show the image of the wa ve-proﬁle at x = 71 . 1 for the case when the incident wa ve amplitude hav e A 0 = 0 . 212 and Ψ 0 = 30 ◦ . The 46 YUJI K ODAMA corresp onding exact solution with those parameters is of (3142)-type (i.e. κ = 0 . 887 < 1). The upp er panels show tw o views in diﬀeren t angles of the temporal v ariation of the w av e proﬁle of the exp erimen t at x = 71 . 1, which is made from 250 slices of the spatial proﬁles (100 slices p er second) with appro ximately 3000 pixel resolution n the y -direction. As exp ected form the (3142)-type exact solution, the stem-wa ve formation is realized, in which the incident and reﬂected w av es separate a wa y from the w all b y the stem-wa v e. The lo wer panels sho w the corresp onding (3142)-t yp e exact solution at t = 41 . 05 (the x -co ordinate is conv erted to the t -co ordinate, using (3.8)). Here the k -parameters are ( k 1 , k 2 , k 3 , k 4 ) = ( − 0 . 614 , − 0 . 037 , 0 . 037 , 0 . 614) from A 0 = 0 . 212 and Ψ 0 = 30 ◦ . Then we calculate the A -matrix using (6.9) with s = 1, and take A =  1 8 . 797 0 − 1 . 128 0 0 1 0 . 530  . This choice of the A -matrix places the incidence wa ve crossing at the origin at t = 0, i.e. θ + [1 , 3] = θ + [2 , 4] = 0 in (6.9). W e see a goo d agreement b et w een the experiment and the KP theory . At x = 71 . 1, the w av e-proﬁle observed in the exp erimen t is close to the corresp onding exact solution of (3142)-type, that is, the radiations generated at the b eginning stage disp ersed and well separated from the main part of the wa ve-proﬁle as predicted in the numerical simulation (see subsection 7.2). 8.2.2. T-typ e inter action. Figure 8.4 shows a preliminary result for T-type interaction pattern gen- erated in the same tank by Y eh and his collab orators. The T-type solution is the most complex and in teresting soliton solution asso ciated with the τ -function on Gr + (2 , 4). The initial wa ve has the V-shap e (half of the X-shap e), then other half of the X-shap e is generated b y the line-soliton with op- p osite angle. The upp er panels of Figure 8.4 sho w the ev olution of the wa ve pattern with A 0 = 0 . 431 and Ψ 0 = 25 ◦ . Behind the crossing wa ve form (the right side in the ﬁgure), the large wak es are generated at the early stage of the ev olution, but they even tually separate from the main pattern of the T-type interaction, as we observed in the numerical simulation. The ﬁgures clearly show the formation of a b o x pattern as exp ected b y the KP theory . The low er panels show the corresp onding T-t yp e exact soliton solution whose parameters are ( k 1 , k 2 , k 3 , k 4 ) = ( − 0 . 697 , − 0 . 231 , 0 . 231 , 0 . 697) and the A -matrix given by (6.12) and (6.13) with θ + [1 , 3] = θ + [2 , 4] = 0 , s = 40 and r = 1, i.e. A =  1 0 − 40 . 34 − 24 . 07 0 1 24 . 07 13 . 37  The large v alue of the s -parameter indicates large phase shift of the inciden t line-solitons (see Figure 6.6), that is, the parts of line-solitons in the righ t side (i.e. b ehind the in teraction point) were created with some delay ed time. In ﬁnding those parameter v alues, w e did not make a precise minimization of certain error function, like the one given in (7.3). In a future communication, we hop e we will b e able to develop a metho d to determine the exact solution from the exp erimen tal data, that is, the in verse problem of the KP equation. Ac knowledgemen ts I would like to thank Harry Y eh for letting me to use his excellent and imp ortant exp erimen tal data b efore publication and for making extremely fruitful collab oration on the Mac h reﬂection problem. I am grateful to Sarbarish Chakrav art y for his many excellen t suggestions which made a tremendous impro vemen t on the pap er and for man y v aluable discussions throughout our in tensive but very pleasan t collab oration. I would also like to thank m y colleagues, Chiu-Y en Kao, Masayuki Oik a wa and Hidek azu Tsuji for many useful discussions related to the sub jects presented in this pap er. Sp ecial thanks to Mark J. 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KP solitons in shallow water

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