The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behavior, implicit solutions and wave breaking

The disp ersionless 2D T o da e quation: dressing, Cauc h y problem, longti me b eha viour, implicit solutions and w a v e breaking S. V. Mana ko v 1 , § and P . M. Santini 2 , § 1 L andau Institute for The or etic al Physics, Mosc ow, Russia 2 Dip artimento d i F i s ic a, Universit` a di R oma ”L a Sapien z a ”, and Istituto Nazionale di Fisic a Nucle ar e, Sezio ne di R oma 1 Piazz.le Aldo Mor o 2, I-0018 5 R oma, Italy § e-mail: manak ov@itp.ac.r u, paolo.santi ni@roma1.i nfn.it Septem b er 14, 2021 Abstract W e ha v e recen tly solved the inv erse s p ectral p r oblem for one- parameter families of v ector fi elds, and u s ed this resu lt to constru ct the formal solution of the Cauch y problem for a class of in tegrable non- linear partial differen tial equations in m u ltidimen sions, including the second hea venly equation of Plebanski and the disp ers ionless K adom t- sev - Pet viash vili (dKP) equation, arising as comm utation of vecto r fields. In this pap er w e make use of the ab ov e theory i) to construct the non lin ear Riemann-Hilb ert dr essing for the so-called tw o dimen- sional disp ersionless T od a equation ( e xp ( ϕ )) tt = ϕ ζ 1 ζ 2 , elucidating the sp ectral mechanism resp onsible for w a ve breaking; ii) we present the formal solution of the Cauch y problem for the wa v e form of it: ( exp ( ϕ )) tt = ϕ xx + ϕ y y ; iii) we obtain the longtime b ehavio ur of th e solutions of suc h a Cauc h y problem, sho wing that it is essen tially describ ed by the longtime b reaking form ulae of the d K P solutions, confirming the exp ected un iversal c h aracter of the dKP equation as protot yp e mo del in the description of the gradien t catastrophe of t w o- dimensional wa v es; iv) we finally c haracterize a class of sp ectral data allo wing one to linea rize the RH problem, corresp ond ing to a class of implicit solutions of the PDE. 1 1 In tr o d uction It w as observ ed long ago [1] that the comm utation of m ultidimensional v ector fields can generate integrable nonlinear pa r t ial differen tial equations (PDEs) in arbitrary dimensions. Some of t hese equations are disp ersionless (or quasi- classical) limits of integrable PDEs, ha ving the disp ersionless Kadom tsev - P etviash vili (dKP) equation u xt + u y y + ( u u x ) x = 0 [2],[3] as univ ersal protot yp e example, t hey arise in v ar ious problems of Mathematical Phys ics and are in tensiv ely studie d in the recen t literature (see, f.i., [4] - [18 ]) . In particular, an elegant in tegration sche me applicable, in general, to nonlinear PDEs a sso ciated with Hamiltonian v ector fields, w as presen ted in [7] and a nonlinear ¯ ∂ - dressing was dev elop ed in [13]. Sp ecial classes of nontrivial solutions were also deriv ed (see, f .i., [12], [15]). The Inv erse Sp ectral T ra nsfor m (IST) for 1-pa r a meter fa milies of mul- tidimensional vec tor fields has b een deve lop ed in [19] (see also [20]). This theory , intro ducing in teresting no v elties with r esp ect to the classical IST for soliton equations [21, 2 2], has allo w ed one to construct the f o rmal solution of the Cauc hy problems for the sec ond heav enly equation [23 ] in [19] and for the nov el system of PD Es u xt + u y y + ( uu x ) x + v x u xy − v y u xx = 0 , v xt + v y y + uv xx + v x v xy − v y v xx = 0 , (1) in [24]. The Cauch y problem for the v = 0 reduction of (1), the dKP equation, w a s also pres en ted in [24], while the Cauc hy problem for the u = 0 reduction of (1), an in tegrable system introduced in [16], w as giv en in [25]. This IST and its ass o ciated nonlinear Riemann - Hilb ert (RH) dress ing turn out to be efficien t to ols to study sev eral prop erties of the solution space, suc h as: i) the c haracterization of a distinguished class of sp ectral dat a for whic h the asso ciated nonlinear RH problem is linearized and solv ed, corresponding to a class of implicit solutions of the PDE (as it w as done for the dKP equation in [26] and for the D una jski generalization [27] of the sec ond hea ve nly equation in [28]); ii) the construction of the longtime b eha viour of the solutions o f the Cauc hy problem [26]; iii) the p ossibilit y to establish whether or not t he lac k of disp ersiv e terms in the nonlinear PDE causes the breaking of lo calized initial profiles and, if y es, to in v estigate in a surprisingly explicit w a y the analytic asp ects of suc h a w a ve breaking (as it w a s recen tly done fo r the (2+1)-dimensional dKP mo del in [26]). 2 In this pap er we make use of this theory to study another distinguished mo del a r ising as the comm utation of ve ctor fields, the so- called 2 dimensional disp ersionless T o da (2ddT) equation φ ζ 1 ζ 2 =  e φ t  t , φ = φ ( ζ 1 , ζ 2 , t ) , (2) or ϕ ζ 1 ζ 2 = ( e ϕ ) tt , ϕ = φ t , (3) also called Boy er-Finley [29 ] equation o r SU( ∞ ) T o da equation [30], the natural contin uous limit of the 2 dimensional T o da lattice [34 , 35] φ n ζ 1 ζ 2 = c  e φ n +1 − φ n − e φ n − φ n − 1  , φ = φ n ( ζ 1 , ζ 2 ) . (4) The 2ddT equation w as probably first deriv ed in [3 1] as an exact reduction of t he second heav enly equation; then in [3 2 ] as a distinguished example of an integrable system in multidim ensions. Some of its in tegra bilit y properties ha v e b een inv estigated in [33] and the in tegration metho d presen ted in [7] is applicable to it. Both elliptic and h yp erb olic v ersions of (2) are relev an t, describing, for instance, in tegrable H - spaces (heav ens) [29, 36] and in tegrable Einstein - W eyl geometries [37]-[38],[30]. String equations solutions [9] of it are relev ant in the ideal Hele-Sha w pro blem [39, 40, 4 1, 42, 43]. The in tegrabilit y o f (2 ) follows f rom the fact that (2) is the condition of comm uta t ion [ ˆ L 1 , ˆ L 2 ] = 0 for the follo wing pair of one-parameter families of v ector fields [8]: ˆ L 1 = ∂ ζ 1 + λv ∂ t +  − λv t + φ ζ 1 t 2  λ∂ λ , ˆ L 2 = ∂ ζ 2 + λ − 1 v ∂ t +  λ − 1 v t − φ ζ 2 t 2  λ∂ λ , (5) where v = e φ t 2 (6) and λ ∈ C is the sp ectral parameter, implying the existenc e of common eigenfunctions of b oth v ector fields; i.e., the existen ce o f the Lax pair: ψ ζ 1 = − λv ψ t +  λv t − φ ζ 1 t 2  λψ λ , ψ ζ 2 = − λ − 1 v ψ t −  λ − 1 v t − φ ζ 2 t 2  λψ λ . (7) 3 Equations (7) and (2) can b e written in the follo wing ”Hamiltonian” form [8]: ψ ζ 1 + {H 1 , ψ } ( λ,t ) = 0 , ψ ζ 2 + {H 2 , ψ } ( λ,t ) = 0 , H 1 ζ 2 − H 2 ζ 1 − {H 1 , H 2 } ( λ,t ) = 0 , (8) where H 1 = λv − φ ζ 1 2 , H 2 = − λ − 1 v + φ ζ 2 2 (9) and { f , g } ( λ,t ) := λ ( f λ g t − f t g λ ) . (10) If, in par t icular, ζ 1 = z = x + iy 2 , ζ 2 = ¯ z = x − iy 2 , x, y ∈ R , (11) equation ( 2) b ecomes the following nonlinear w a v e equation φ xx + φ y y =  e φ t  t , (12) or ϕ xx + ϕ y y = ( e ϕ ) tt . (13) In a ddition, if | λ | = 1 and φ ∈ R , equations (9) giv e the “real” Hamiltonian form ulation [32 ] ψ x + { H 1 , ψ } ( θ, t ) = 0 , ψ y + { H 2 , ψ } ( θ, t ) = 0 , H 1 y − H 2 x − { H 1 , H 2 } ( θ, t ) = 0 , (14) of (7 ) and (12), for the real Hamiltonians H 1 , H 2 : H 1 = i 2 ( H 1 + H 2 ) = sin θe φ t 2 − 1 2 φ y , H 2 = 1 2 ( H 2 − H 1 ) = − cos θ e φ t 2 + 1 2 φ x (15) and the P oisson brac ket { f , g } ( θ, t ) := f θ g t − f t g θ , (16) ha ving introduced the parametrization λ = e − iθ , θ ∈ R . (17) The pap er is o rganized as follows . In § 2 we presen t the dressing sch eme for equations (2) and (1 2 ), giv en in terms of a v ector nonlinear RH problem. 4 As for the dressing of dKP presen ted in [26], since the normalization of the eigenfunctions turns out to dep end on t he unkno wn solution of 2ddT, a clo- sure condition is necessary , allowing one to construct the solution o f 2ddT through an implicit system of algebraic equ ations, whose inv ersion is resp on- sible for the w av e breaking of an initial lo calized profile. I n § 3 we presen t the IST f o r the 2 ddT equation (12) and use it to obtain the formal solution of the Cauc h y for suc h equation. In § 4 w e obtain the longtime b ehaviour of the solutions of suc h a Cauc hy problem, sho wing that the solutions break also in the longtime regime, and tha t suc h regime is essen tially desc rib ed b y the longtime breaking fo rm ulae of the dKP solutions [26 ], confirming the exp ected univ ersal c haracter of the dKP equation as pro t o t ype mo del in the description of the gra dien t catastrophe of t wo-dimensional w a v es. In § 5 w e c ha r acterize a class of RH sp ectral data allowing one to decouple and linearize the R H problem, generating a class of implicit solution of 2ddT para metrized b y an arbitrary real function o f one v ariable. 2 Nonlinear RH d r e ssing In this section w e in tro duce the v ector nonlinear RH problem enabling one to construct larg e classes of solutions of the Lax pair (7) and of the 2ddT equations (2) and (12). Pr op osition . Consider the follow ing ve ctor nonlinear RH problem ξ + j ( λ ) = ξ − j ( λ ) + R j ( ξ − 1 ( λ ) + ν 1 ( λ ) , ξ − 2 ( λ ) + ν 2 ( λ )) , λ ∈ Γ , j = 1 , 2 (18) on an arbitrary closed contour Γ of the complex λ plane, where ~ R ( ~ s ) = ( R 1 ( s 1 , s 2 ) , R 2 ( s 1 , s 2 )) T are giv en differen tia ble sp ectral data dep ending on the second arg umen t s 2 through exp ( is 2 ) and satisfying the constrain t {R 1 ( s 1 , s 2 ) , R 2 ( s 1 , s 2 ) } ( s 1 ,s 2 ) = 1 , R j ( s 1 , s 2 ) := s j + R j ( s 1 , s 2 ) , j = 1 , 2 , (19) with { f , g } ( s 1 ,s 2 ) := f s 1 g s 2 − f s 2 g s 1 ; (20) where ν j , j = 1 , 2 are the explicit functions ~ ν =  ν 1 ν 2  =  ( ζ 1 λ + ζ 2 λ − 1 ) v − t − ζ 1 φ ζ 1 i ln λ + i φ t 2  (21) 5 and ~ ξ + = ( ξ + 1 , ξ + 2 ) T and ~ ξ − = ( ξ − 1 , ξ − 2 ) T are the unkno wn ve ctor solutions of the RH problem (18), analytic risp ectiv ely inside and outside the contour Γ and suc h t hat ~ ξ − → ~ 0 as λ → ∞ . Then, assuming that the ab ov e RH problem a nd its linearized f orm are uniquely solv able, w e ha v e the following results. 1) If lim λ →∞ ( iλξ − 2 ) = φ ζ 1 e − φ t 2 , iξ + 2 (0) = φ t , (22) it follows that ~ π ± = ~ ξ ± + ~ ν are common eigenfunctions of ˆ L 1 , 2 : ˆ L 1 ~ π ± = ˆ L 2 ~ π ± = ~ 0 satisfying the relations { π ± 1 , π ± 2 } ( λ,t ) = λ ( π ± 1 λ π ± 2 t − π ± 1 t π ± 1 λ ) = i (23) and the p oten tials φ ζ 1 , φ t , reconstructed through (22), solv e the 2ddT equa- tion (2) . 2) In addition, if the v aria bles ζ 1 , ζ 2 are sp ecified a s in (11), if the RH data satisfy the additional reality constraint ~ R  ~ R ( ~ s )  = ~ s, ∀ ~ s ∈ C (24) and Γ is the unit circle, then the eigenfunctions satisfy the follo wing symme- try relation: ~ π − ( λ ) = ~ π + (1 / ¯ λ ) (25) and φ ∈ R . Remark 1 The RH problem (18) can o b viously b e f o rm ulated directly in terms of the eigenfunctions ~ π ± as follo ws: π + j ( λ ) = R j ( π − 1 ( λ ) , π − 2 ( λ )) = π − j ( λ ) + R j ( π − 1 ( λ ) , π − 2 ( λ )) , λ ∈ Γ , j = 1 , 2 , (26) with the normalization ~ π − ( λ ) = ~ ν ( λ ) + O ( λ − 1 ) , | λ | >> 1 . (27) Remark 2 The dep endence of ~ R on s 2 through exp ( is 2 ) ensures that the solutions ~ ξ ± of the RH problem do not exhibit the ln λ singularit y con tained in the norma lizat io n (21). It follo ws that the eigenfunctions π ± 2 con tain the 6 ln λ singularity only as an additive singularity , while π ± 1 do not exhibit such singularit y . Remark 3 Before adding t he closure conditions ( 22), the solutions ~ ξ ± of the RH problem dep end, via the normalization (21), on the undefined fields φ t and φ ζ 1 , through the com bination ( t + ζ 1 φ ζ 1 ); then the t w o closure condi- tions (22) mus t b e view ed as a nonlinear system of t w o algebraic equations for φ t and φ ζ 1 defining implicitly the solution φ ζ 1 , φ t of the 2ddT equation. Therefore, as in the dKP case [2 6], w e exp ect that this sp ectral features b e resp onsible for the w a ve breaking of lo calized initial data ev olving according to the nonlinear w av e equation (3). Details on ho w tw o-dimensional w a v es ev olving a ccording to the 2ddT equation break will b e presen ted elsew here. An a lternativ e closure, perhaps useful in the realit y reduc tion case desc rib ed in part 2) of t he a b ov e Prop osition, is giv en b y the equations − i ξ + 1 (0) 2 = I m ( t + z φ z ) , iξ + 2 (0) = φ t . (28) With this closure, indeed, w e obtain a system of algebraic equations inv olving t + z φ z , its imagina r y part and φ t . Remark 4 The symmetry relations (25) are a distinguished example of the follo wing symmetry of the common eigenfunctions of t he La x pair (7), when ζ 1 = z , ζ 2 = ¯ z as in (11) and φ ∈ R : if ψ ( λ ) is a solution o f (7), then ψ (1 / ¯ λ ) is a solution to o . Remark 5 F or part 2) of the a b ov e Prop osition, when the con tour Γ is the unit circle, the RH problem is c haracterized b y the f o llo wing system of nonlinear integral equations for ξ ± j ( λ ) , | λ | = 1 (ha ving para metrized λ as in (17)): ξ ± j ( e − iθ ) − 1 2 π 2 π R 0 dθ ′ 1 − (1 ∓ ǫ ) e i ( θ ′ − θ ) R j  ( z e − iθ ′ + ¯ z e iθ ′ ) v − t − z φ z + ξ − 1 ( e − iθ ′ ) , θ ′ + i φ t 2 + ξ − 2 ( e − iθ ′ )  = 0 , j = 1 , 2 , (29) and the closure conditions (22) read φ z e − φ t 2 = 1 2 π i 2 π R 0 dθ e − iθ R 2  ( z e − iθ + ¯ z e iθ ) v − t − z φ z + ξ − 1 ( e − iθ ) , θ + i φ t 2 + ξ − 2 ( e − iθ )  = 0 , (30) 7 φ t = − 1 2 π i 2 π R 0 dθ R 2  ( z e − iθ + ¯ z e iθ ) v − t − z φ z + ξ − 1 ( e − iθ ) , θ + i φ t 2 + ξ − 2 ( e − iθ )  = 0 . (31) Pr o of . F or part 1), w e first apply the op erato rs ˆ L 1 , 2 to the RH problem (26), obtaining the linearized RH problem ˆ L j ~ π + = A ˆ L j ~ π − , where A is the Jacobian matrix of the transformation (26): A ij = ∂ R i /∂ s j , i, j = 1 , 2. Since, due to the normalization (27), ˆ L j ~ π − → ~ 0 as λ → ∞ , it fo llo ws that, b y uniqueness, ~ π ± are common eigenfunctions of the v ector fields ˆ L 1 , 2 : ˆ L 1 , 2 ~ π ± = ~ 0 and, consequen tly , that φ t , φ z are solutions of the 2ddT equation (2). Then the eigenfunctions exhibit the following asymptotics: π − 1 = λz v − t − z φ z + λ − 1 ( ¯ z v + a − v − 1 ) + O ( λ − 2 ) , | λ | >> 1 , π + 1 = λ − 1 ¯ z v − t − ¯ z φ ¯ z + λ ( z v + a + v − 1 ) + O ( λ 2 ) , | λ | << 1 , π − 2 = i ln λ + i φ t 2 − iλ − 1 φ z v − 1 + O ( λ − 2 ) , | λ | >> 1 , π + 2 = i ln λ − i φ t 2 + iλφ ¯ z v − 1 + O ( λ 2 ) , | λ | << 1 , (32) where a − ¯ z = − a + z = ( z e φ t ) z − ( ¯ z e φ t ) ¯ z . (33) implying the closure conditions (22) ,( 2 8). Since, from (26), { π + 1 , π + 2 } ( t,λ ) = {R 1 , R 2 } ~ π − { π − 1 , π − 2 } ( t,λ ) , λ ∈ Γ, equation (19) implies that { π + 1 , π + 2 } ( t,λ ) = { π − 1 , π − 2 } ( t,λ ) , λ ∈ Γ; i.e., the Poiss on brac k ets of the ± eigenfunctions are analytic in the whole complex λ plane. Since { π − 1 , π − 2 } → i as λ → ∞ , it follo ws that { π + 1 , π + 2 } = { π − 1 , π − 2 } = i . F or part 2), applying ~ R ( · ) to the complex conjugate o f the RH problem (26) and using the reality condition (24) it f o llo ws that ~ π + ( λ ) = ~ π − ( λ ) , | λ | = 1. By the Sc hw artz reflection principle, it fo llo ws the symm etry (25) and, using the Lax pair (7), the reality condition φ ∈ R . ✷ 3 The Cauc h y probl em for the 2ddT equation In this section we prese n t the formal solution of the Cauc h y problem for the w av e form of the 2ddT equation:  e φ t  t = φ xx + φ y y , x, y ∈ R , t > 0 , φ ( x, y , t ) ∈ R , φ ( x, y , 0) = A ( x, y ) , φ t ( x, y , 0 ) = B ( x, y ) . (34) 8 where t he assigned initial conditions A ( x, y ) , B ( x, y ) a re lo calized in the ( x, y ) plane f or x 2 + y 2 → ∞ . T o do it, w e use the IST for vec tor fields dev elop ed in [19, 20, 24, 25]. In this respect, w e recall t w o ba sic facts: since t he Lax pair of the 2ddT is made o f v ector fields, i) the sp ac e of eigenfunction s is a ring (if f 1 and f 2 are eigenfunctions, an y differen t iable function F ( f 1 , f 2 ) is an eigenfunction); ii) since the v ector fields a re also Hamiltonian, the sp ac e of eige n functions is also a Lie algebr a, who se Lie br acket is the Poisson br acket (10) (if f 1 and f 2 are eigenfunctions, a lso { f 1 , f 2 } ( λ,t ) is a n eigenfunction). Multiplying the first and second equations o f (7 ) (with ζ 1 = z , ζ 2 = ¯ z as in (11)) b y λ − 1 and λ respectiv ely , then adding and subtracting the resulting equations, one o btains the equiv alen t and more conv enien t La x pair: ˆ L 1 ψ := λψ ¯ z − λ − 1 ψ z −  − 2 v t + λ φ ¯ z t 2 + λ − 1 φ z t 2  λψ λ = 0 , (35) ˆ L 2 ψ := ψ t + v − 1 2 ( λψ ¯ z + λ − 1 ψ z ) − v − 1 4 ( λφ ¯ z t − λ − 1 φ z t ) λψ λ = 0 , (36) where the first equation m ust b e view ed as the sp ectral problem (in whic h v t shall b e replaced, in the direct problem, b y ( φ z ¯ z / 2) exp ( − φ t / 2), due to (1 2)) and the second equation as t -evolution of the eigenfunction. Eigenfunctions and sp ectral data . Now w e in tro duce the Jo st and an- alytic eigenfunctions for the spectral problem (35). Since the asso ciat ed undressed op erator: λ∂ ¯ z − λ − 1 ∂ z coincides with the undressed op erator of the sp ectral problem for the (2 + 1)-dimensional self-dual Y ang-Mills equa- tion [44], the construction of the Jost and analytic Green’s functions is tak en from there. W e define Jost eigenfunctions of the sp ectral problem ( 35) o n the unit circle of the complex λ plane, using the parametrization (1 7). Introducing the conv enien t real v ariables ξ , η , θ ′ as follows: ξ = cos θ x + sin θ y , η = − sin θ x + cos θ y , θ ′ = θ , (37) the L a x pa ir (35 ),(36) b ecomes ˆ L 1 ψ := ψ η − 1 2 [ − ( φ ξ ξ + φ ηη ) v − 1 + φ ξ t ] ( η ψ ξ − ξ ψ η + ψ θ ′ ) = 0 , (38) ˆ L 2 ψ := ψ t + v − 1 ψ ξ − ( v − 1 ) η ( η ψ ξ − ξ ψ η + ψ θ ′ ) = 0 . (39) 9 A con v enient basis of Jost eigenfunctions are the solutions f 1 and f 2 of equa- tion (38 ) satisfying the b oundar y conditions ~ f ( ξ , η , θ ′ ) :=  f 1 ( ξ , η , θ ′ ) f 2 ( ξ , η , θ ′ )  →  ξ θ ′  , as η → −∞ ; (40) they are ch aracterized b y the linear inte gral equation ~ f =  ξ θ ′  + 1 2 η R −∞ dη ′ [ − ( φ ξ ξ + φ η ′ η ′ ) v − 1 + φ ξ t ] ( η ′ ~ f ξ − ξ ~ f η ′ + ~ f θ ′ ) . (41) It follows that f 1 ( ξ , η , θ ) and f 2 ( ξ , η , θ ) − θ are 2 π - p erio dic in θ . The η → ∞ limit of ~ f defines the scattering v ector ~ σ ( ξ , θ ) = ( σ 1 ( ξ , θ ) , σ 2 ( ξ , θ )) T as follo ws ~ f ( ξ , η , θ ) → ~ S ( ξ , θ ) =  ξ θ  + ~ σ ( ξ , θ ) , as η → ∞ ; (42) namely: ~ σ ( ξ , θ ) = 1 2 Z R dη  − ( φ ξ ξ + φ ηη ) v − 1 + φ ξ t  ( η ~ f ξ − ξ ~ f η + ~ f θ ) . (43) Also t he scattering vector is 2 π -p erio dic in θ : ~ σ ( ξ , θ + 2 π ) = ~ σ ( ξ , θ ); i.e., its dep endence on the second argumen t θ is through exp ( iθ ). The analytic eigenfunctions of (35) are defined instead via t he integral equations: ~ ψ ± ( z , ¯ z , λ ) =  ψ ± 1 ( z , ¯ z , λ ) ψ ± 2 ( z , ¯ z , λ )  =  λz + λ − 1 ¯ z i ln λ  + i 4 R C dz ′ ∧ d ¯ z ′ G ± ( z − z ′ , ¯ z − ¯ z ′ , λ ) h − φ z ¯ z v − 1 + λ φ ¯ z ′ t 2 + λ − 1 φ z ′ t 2 i λ ~ ψ ± λ ( z ′ , ¯ z ′ , λ ) , (44) where G ± are the ana lytic Green’s functions G ± ( z , ¯ z , λ ) = ∓ 1 π 1 ( λz + λ − 1 ¯ z ) , sgn(1 − | λ | ) = ± 1 (45) suc h tha t λG ± ¯ z − λ − 1 G ± z = δ ( z ), reducing, on the unit circle | λ | = 1, to G ± ( z , ¯ z , λ ) = ∓ 1 π 1 ξ ∓ iǫη , 0 < ǫ << 1 . (46) 10 Since G + and G − are analytic resp ectiv ely inside and outside the unit circle of the complex λ plane, t hen ( ψ + 1 , ψ + 2 ) and ( ψ − 1 , ψ − 2 ) are also analytic, resp ectiv ely , inside and outside the unit circle of the complex λ plane, af- ter subtracting their singular parts, giv en resp ective ly b y ( λ − 1 ¯ z v , i ln λ ) and ( λz v , i ln λ ), as it can b e seen b y solving the inte gral equations (44) b y iter- ation o r fro m the followin g λ - asymptotics: ψ − 1 = λz v − z φ z + λ − 1 ( ¯ z v + a − v − 1 ) + O ( λ − 2 ) , | λ | >> 1 , ψ + 1 = λ − 1 ¯ z v − ¯ z φ ¯ z + λ ( z v + a + v − 1 ) + O ( λ 2 ) , | λ | << 1 , ψ − 2 = i ln λ + i φ t 2 − iλ − 1 φ z v − 1 + O ( λ − 2 ) , | λ | >> 1 , ψ + 2 = i ln λ − i φ t 2 + iλφ ¯ z v − 1 + O ( λ 2 ) , | λ | << 1 , (47) where a ± are defined in (33). In a ddition, equations (46) imply the limits G + ( z − z ′ , ¯ z − ¯ z ′ , λ ) → − 1 π 1 ξ − ξ ′ ± iε , as η → ∓∞ , G − ( z − z ′ , ¯ z − ¯ z ′ , λ ) → 1 π 1 ξ − ξ ′ ∓ iε , as η → ∓∞ . (48) Therefore, on the unit circle | λ | = 1, the η → −∞ limit of ( ψ + 1 , ψ + 2 ) and ( ψ − 1 , ψ − 2 ) ar e analytic resp ectiv ely in the upp er and low er parts of the com- plex ξ plane, while the η → ∞ limit of ( ψ + 1 , ψ + 2 ) and ( ψ − 1 , ψ − 2 ) are analytic resp ectiv ely in the low er and upp er parts of the complex ξ plane. This me c h- anism, first o bserve d in [44], play s an imp o rtan t role in the IST for v ector fields (see [19],[20],[2 4],[25]). Since the Jo st eigenfunctions ~ f = ( f 1 , f 2 ) T are a go o d basis in the space of eigenfunctions of the sp ectral problem (38 ) for | λ | = 1, one can express the analytic eigenfunctions in terms of them through the follo wing formulae, v alid for | λ | = 1: ~ ψ ± = ~ K ± ( ~ f ) = ~ f + ~ χ ± ( f 1 , f 2 ) , (49) defining the sp ectral data ~ χ ± as differen tiable functions of t w o arguments . In t he η → −∞ limit, equations (49) reduce to lim η →− ∞ ~ ψ ± −  ξ θ  = ~ χ ± ( ξ , θ ) , (50) implying t ha t i) ~ χ + ( ξ , θ ) and ~ χ − ( ξ , θ ) are analytic in the first v ariable ξ resp ectiv ely in the upper and low er half parts of the complex ξ plane, and ii) ~ χ ± ( ξ , θ ) are 2 π -p erio dic in θ : ~ χ ± ( ξ , θ + 2 π ) = ~ χ ± ( ξ , θ ) (their dep endence on the second arg umen t θ is through exp ( iθ )). 11 A t η → ∞ , equations (49) reduce to lim η →∞ ~ ψ ± −  ξ θ  = ~ σ + ~ χ ± ( ξ + σ 1 , θ + σ 2 ) . (51) Applying the op erato r R R dξ 2 π R 0 dθ 2 π e − i ( ωξ + nθ ) · , n ∈ Z to equations (51) and using the ab ov e established analiticit y prop erties in ξ and the 2 π -p erio dicit y in θ , we obtain the follo wing linear in tegral equations connecting the (F ourier transforms of the) scattering data ~ σ to the (F ourier transforms of the) sp ectral data ~ χ ± : ˜ ~ χ ± ( ω , n ) + H ( ± ω )  ˜ ~ σ ( ω , n ) + R R dω ′ ∞ P n ′ = −∞ ˜ ~ χ ± ( ω ′ , n ′ ) Q ( ω ′ , n ′ , ω , n )  = ~ 0 , (52) where H is the Hea viside step function and Q ( ω ′ , n ′ , ω , n ) = R R dξ 2 π 2 π R 0 dθ 2 π e i ( ξ ( ω ′ − ω )+( n ′ − n ) θ )  e i ( ω ′ σ 1 ( ξ ,θ )+ n ′ σ 2 ( ξ ,θ )) − 1  , ˜ ~ χ ± ( ω , n ) = R R dξ 2 π R 0 dθ 2 π e − i ( ωξ + nθ ) ~ χ + ( ξ , θ ) , ˜ ~ σ ( ω , n ) = R R dξ 2 π R 0 dθ 2 π e − i ( ωξ + nθ ) ~ σ + ( ξ , θ ) . (53) A t last, eliminating, fro m equations (49), the Jost eigenfunctions ~ f , one obtains, through algebraic ma nipulat io n, the fo llowing v ector no nlinear RH problem o n the unit circle of the complex λ plane: ψ + 1 = R 1 ( ψ − 1 , ψ − 2 ) = ψ − 1 + R 1 ( ψ − 1 , ψ − 2 ) , | λ | = 1 , ψ + 2 = R 2 ( ψ − 1 , ψ − 2 ) = ψ − 2 + R 2 ( ψ − 1 , ψ − 2 ) . (54) W e remark that t he 2 π - p erio dicity properties o f the scattering data ~ χ ± ( ξ , θ ) in the v ariable θ imply that the dep endence o f ~ R on the second arg umen t s 2 is also through exp ( is 2 ), to guaran t y that the ln λ singularit y is just an additiv e o ne f or ψ ± 2 , and is absen t for ψ ± 1 . Recapitulating, in the direct pro blem, at t = 0, w e go from the initial conditions φ , φ t of the 2ddT equation to the initial scattering v ector ~ σ ( ξ , θ ); from it w e construct, through the linear in tegral equations (5 2 ), the scat- tering data ~ χ ± ( ξ , θ ) and, through a lgebraic manipulation, the RH sp ectral 12 data ~ R ( ~ s ) = ( R 1 ( s 1 , s 2 ) , R 2 ( s 1 , s 2 )). In the inv erse problem, o ne giv es the RH sp ectral data ~ R ( ~ s ) and reconstructs t he v ector solutio ns ~ ψ ± of the RH problem ( 5 4), defined by the normalization: ~ ψ − = ( z λ + ¯ z λ − 1 ) e φ t 2 − z φ z i ln λ + i φ t 2 ! + ~ O ( λ − 1 ) , | λ | >> 1 . (55) A t last, the closure conditio ns lim λ →∞ λ ( iψ − 2 + ln λ ) = φ z e − φ t 2 , lim λ → 0 ( iψ + 2 + ln λ ) = φ t 2 , (56) consequenc es of the asymptotics (47), allo w one to reconstruct the solution of the 2 ddT equation through the solution of a system of t w o algebraic equations for φ t and φ z . Time ev olut ion of the sp ectral data . T o construct the t - evolution of the sp ectral data w e observ e that ~ f and ~ ψ ± , eigenfunctions of the sp ectral problem (35): ˆ L 1 ~ f = ˆ L 1 ~ ψ ± = ~ 0, are solutions of the following equations in v olving the second Lax op erator ˆ L 2 ~ f = ˆ L 2 ~ ψ ± = (1 , 0) T , implying the follo wing elemen tary time ev o lutio ns of the data: ~ σ ( ξ , θ , t ) = ~ σ ( ξ − t, θ , 0) , ~ χ ± ( ξ , θ , t ) = ~ χ ± ( ξ − t, θ , 0) , ~ R ( ξ , θ , t ) = ~ R ( ξ − t, θ , 0 ) . (57) In addition, it follows that the common Jost eigenfunctions ~ J and the com- mon analytic eigenfunctions ~ π ± of the Lax pair (35),(36) are obtained from ~ f and ~ ψ ± simply as follows: ~ J := ~ f − t (1 , 0) T , ~ π ± := ~ ψ ± − t (1 , 0 ) T . (58) It is easy to v erify that the analytic eigenfunctions ~ π ± , the RH data ~ R ( ~ s ) and the asso ciated RH pr o blem of this section coincide with those app earing in the dressing construction of § 2. Hamiltonian constraints on the data . The Ha milto nian c haracter of the 2ddT dynamics implies the following fo r mulae f or the P oisson brac k ets of the relev an t eigenfunctions: { J 1 , J 2 } ( λ,t ) = { π ± 1 , π ± 2 } ( λ,t ) = i (59) 13 whic h, in turn, imply that the transformations ~ s → ~ K ± ( ~ s ) and ~ s → ~ R ( ~ s ) are canonical: {K ± 1 , K ± 2 } ( s 1 ,s 2 ) = {R 1 , R 2 } ( s 1 ,s 2 ) = 1 . (60) T o pro v e (59), one first shows that J 3 := { J 1 , J 2 } ( λ,t ) → i as η → −∞ , π − 3 := { π − 1 , π − 2 } ( λ,t ) → i as λ → ∞ , π + 3 := { π + 1 , π + 2 } ( λ,t ) → i as λ → 0. Since the vec tor fields a re Hamiltonian, J 3 , π ± 3 are also common eigenfunctions, and equations (59) hold, b y uniquenes s. Equations (60) are consequences of (59) and of the relations ~ π ± = ~ K ±  ~ J  , ~ π + = ~ R  ~ π −  . (61) Realit y constrain ts . The definition (5 8b) and the condition φ ∈ R imply the symmetry relat io ns ~ π − ( λ ) = ~ π + (1 / ¯ λ ); (62) consequen tly , from (18 ), the reality constraint (24) on the RH data holds true. Small field limit and Radon T ransform . As for the IST of the hea v enly [20] and dKP [24] equations, in t he small field limit | φ | , | φ t | << 1, the direct and in v erse sp ectral transforms presen ted in this sec tion reduce to the direct and in v erse Radon transform [45]. Indeed, the mapping fro m the init ia l data { A ( x, y ) , B ( x, y ) } to the scattering v ector ~ σ reduces to the direct Radon transform: ~ σ ( ξ , θ ) ∼ 1 2 R R  η 1  h − ( ∂ 2 ξ + ∂ 2 η ) A ( x ( ξ , η , θ ) , y ( ξ , η , θ ))+ ∂ ξ B ( x ( ξ , η , θ ) , y ( ξ , η , θ )) i dη , x ( ξ , η , θ ) = ξ cos θ − η sin θ , y ( ξ , η , θ ) = ξ sin θ + η cos θ , (63) while the sp ectral data ~ χ ± and ~ R are constructed from ~ σ as follo ws: ~ χ ± ( ξ , θ ) ∼ − ˆ P ± ξ ~ σ ( ξ , θ ) , ~ R ( ξ , θ ) ∼ − i ˆ H ξ ~ σ ( ξ , θ ) , (64) where ˆ P ± ξ and ˆ H ξ are r isp ectiv ely the ( ± ) a na lyticit y pro jectors and the Hilb ert transform in the v ariable ξ : ˆ P ± ξ g ( ξ ) := ± 1 2 π i Z R dξ ′ ξ ′ − ( ξ ± i 0) g ( ξ ′ ) , ˆ H ξ g ( ξ ) := 1 π P Z R dξ ′ ξ − ξ ′ g ( ξ ′ ) . (65) 14 A t last, the first of the closure conditions (56) of the in v erse problem reduce s to the in v erse Radon transform φ t ( x, y , t ) ∼ − 1 2 π i 2 π R 0 dθ R 2 ( ξ − t, θ ) ∼ − 1 2 π 2 2 π R 0 dθ P R R dξ ′ ξ ′ − ( ξ − t ) σ 2 ( ξ ′ , θ ) , ξ = x cos θ + y sin θ , (66) that can b e sho wn to b e equiv alen t to the w ell-known P oisson fo rm ula φ ( x, y , t ) = ∂ t R R 2 dx ′ dy ′ 2 π L ( x − x ′ , y − y ′ , t ) A ( x ′ , y ′ ) + R R 2 dx ′ dy ′ 2 π L ( x − x ′ , y − y ′ , t ) B ( x ′ , y ′ ) , (67) where L ( x, y , t ) := H ( t 2 − x 2 − y 2 ) p t 2 − x 2 − y 2 (68) and H ( · ) is the Hea viside step function, describing the solution of the Cauc h y problem φ tt = φ xx + φ y y , x, y ∈ R , t > 0 , φ ( x, y , t ) ∈ R , φ ( x, y , 0) = A ( x, y ) , φ t ( x, y , 0) = B ( x , y ) . (69) for t he linear w a v e equation in 2+1 dimensions. 4 The longti me b e ha viou r of the solu tions In this section w e sho w, as it w as done in the dKP case [26], that t he sp ectral mec hanism causing the breaking of a lo calized initial conditio n ev o lving ac- cording to the 2ddT equation is presen t also in the longtime regime. W e will actually sho w that the longtime br e aki n g of the 2ddT solutions is essential ly describ e d b y the lon g time br e aking formulae of the dKP solutions f ound in [26]; this is an imp ortant confirmation of the exp ected universal char a cter of the dKP e quation as pr ototyp e m o del i n the description of the gr adie n t c atastr ophe of two-dim e nsional waves . W e remark that it is clearly meaningful to study the long time b eha viour of the solutions of the 2ddT equation only if no breaking take s place b efore, at finite time. In this section we a ssume that the initial conditio n b e small, then the nonlinearit y b ecomes import a n t only in the long time regime and no breaking t ak es place b efore. 15 Motiv ated by the longtime b eha viour of the solutions of the linear wa v e equation u tt = u xx + u y y , lo calized, with amplitude O ( t − 1 2 ), in the region p x 2 + y 2 − t = O (1), we study the longtime b eha viour of the solutions of the 2 ddT equation in the space-time region z = t + r 2 e iα , α, r ∈ R , α = O (1) , t >> 1 , (70) implying that r = p x 2 + y 2 − t, α = arctan y x . (71) Substituting ( 70) into the in tegral equations (2 9) a nd ke eping in mind that , in the longtime regime, φ t is small, so that, f. i., v ∼ 1 + φ t / 2 + φ 2 t / 8, we obtain ξ ± j ( λ ) − 1 2 π 2 π R 0 dθ ′ 1 − (1 ∓ ǫ ) e i ( θ ′ − θ ) R j  − 2 t sin 2  θ ′ − α 2  + r cos( θ ′ − α )+ t + r 2 cos( θ ′ − α ) φ t (1 + φ t 4 ) − z φ z + ξ − 1 ( e − iθ ′ ) , θ ′ + ξ − 2 ( e − iθ ′ )  ∼ 0 , j = 1 , 2 . (72) Since the main contribution to these in tegrals o ccurs when sin(( θ ′ − α ) / 2 ) ∼ 0, w e mak e the c hange of v aria ble θ ′ = α − µ ′ / √ t , obtaining ξ ± j ( λ ) − 1 2 π √ t R R dµ ′ 1 − (1 ∓ ǫ ) e i ( α − θ − µ ′ √ t ) R j  − µ ′ 2 2 + X + ξ − 1  e − i ( α − µ ′ √ t )  , α + ξ − 2  e − i ( α − µ ′ √ t )   ∼ 0 , j = 1 , 2 , (73) where X := r + t + r 2 φ t + t 8 φ 2 t − z φ z . (74) If | θ − α | >> t − 1 / 2 , equations (73) imply that ξ ± j ( λ ) = O ( t − 1 / 2 ): ξ ± j ( λ ) ∼ 1 2 π √ t ( 1 − (1 ∓ ǫ ) e i ( α − θ ) ) R R dµ ′ R j  − µ ′ 2 2 + X + ξ − 1  e − i ( α − µ ′ √ t )  , α + ξ − 2  e − i ( α − µ ′ √ t )   , j = 1 , 2 . (75) If, instead, θ − α = − µt − 1 / 2 , | µ | = O (1), then ξ ± j ( λ ) = O (1): ξ ± j  e − i ( α − µ √ t )  ∼ 1 2 π i R R dµ ′ µ ′ − ( µ ± iǫ ) R j  − µ ′ 2 2 + X + ξ − 1  e − i ( α − µ ′ √ t )  , α + ξ − 2  e − i ( α − µ ′ √ t )   , j = 1 , 2 . (76) 16 Therefore it is not p ossible to neglect, in the ab ov e in tegral equations, ξ − j , j = 1 , 2 in the arguments of R j , j = 1 , 2; it follow s t ha t these in tegral equations remain nonlinear eve n in the lo ngtime regime. A t last, using equations (75), the asymptotic form of the closure condi- tions read, fo r t >> 1: φ t ∼ − 1 2 π i √ t R R dµ ′ R 2  − µ ′ 2 2 + X + ξ − 1  e − i ( α − µ ′ √ t )  , α + ξ − 2  e − i ( α − µ ′ √ t )   , (77) φ z = − e − iα φ t (1 + φ t 2 ) (1 + O ( t − 1 )) . (78) Comparing (77) and (78), a nd using (70), w e infer that z φ z ∼ − t + r 2 φ t (1 + φ t 2 ). Using this asymptotic relation in (77), we finally obtain the follo wing result. In t he space-time region z = t + r 2 e iα , α, r ∈ R , t >> 1 , X := r + ( t + r ) φ t + 3 t 8 φ 2 t = p x 2 + y 2 − t + p x 2 + y 2 φ t + 3 t 8 φ 2 t = O (1 ) , (79) the longtime t >> 1 b ehaviour of the solutio ns of the 2ddT equation ( expφ t ) t = φ xx + φ y y is describ ed b y the following implicit (scalar) equation: φ t = 1 √ t F  p x 2 + y 2 − t + p x 2 + y 2 φ t + 3 t 8 φ 2 t , arctan y x  + o  1 √ t  , (80) where F is g iven by F ( X, α ) = − 1 2 π i R R dµ ′ R 2  − µ ′ 2 2 + X + a 1 ( µ ′ ; X , α ) , α + a 2 ( µ ′ ; X , α )  (81) and a j ( µ ; X , α ) , j = 1 , 2 ar e t he solutions of the in tegr a l equations a j ( µ ; X , α ) = 1 2 π i R R dµ ′ µ ′ − ( µ − iǫ ) R j  − µ ′ 2 2 + X + a 1 ( µ ′ ; X , α ) , α + a 2 ( µ ′ ; X , α )  , j = 1 , 2 . (82) Outside the asymptotic region (79) the solution decay s faster. W e first remark that, since φ t = O ( t − 1 2 ), the condition X = O (1) implies that r = p x 2 + y 2 − t = O ( √ t ); it fo llows that, in the longtime regime t >> 17 1, the solution of the Cauc h y problem fo r the 2ddT eq uation is conce n t rated, with amplitude O ( t − 1 2 ), in the asymptotic region p x 2 + y 2 − t = O ( √ t ). W e also remark that the asymptotic solution (80)- ( 82) is connected to the initial conditions o f the Chauc h y problem through the direct problem presen ted in the previous section. 5 A dist inguis h ed class of i mpl i c it s olution s In this sec tion, in analogy with the results of [26],[28], we construct a class of explicit solutions of the vector nonlinear RH problem (18) and, corresp ond- ingly , a class of implicit solutions of the 2ddT equation parametrized by an arbitrary real sp ectral function of one v ariable. Supp ose that the t w o comp onen ts of the RH sp ectral data ~ R in (18) are giv en by : R j ( s 1 , s 2 ) = ( − 1) j +1 if  e s 1 + s 2  , j = 1 , 2 , (83) in terms of the single real spectral function f of a single argumen t, dep ending on s 1 and s 2 only through their sum. Then the RH problem (26) b ecomes π + 1 = π − 1 + if  e π − 1 + π − 2  , | λ | = 1 , π + 2 = π − 2 − if  e π − 1 + π − 2  (84) and the following prop erties hold. i) The reality and Hamiltonian constrain ts (24) and (19) are satisfied. ii) π + 1 + π + 2 = π − 1 + π − 2 . Consequen tly , using the analyticit y prop erties of the eigenfunctions, it follo ws that the functions ∆ + and ∆ − , defined b y ∆ ± := π ± 1 + π ± 2 − ( z λ + ¯ z λ − 1 ) v − i ln λ, (85) are analytic resp ectiv ely inside a nd outside the unit circle of the λ -plane and satisfy the equation ∆ + = ∆ − ; therefore they are equal to a constan t in λ . Ev aluating suc h a constan t at λ = 0 and at λ → ∞ , w e obtain the follo wing equalities ∆ + = ∆ − = − t − ¯ z φ ¯ z − i φ t 2 = − t − z φ z + i φ t 2 . (86) 18 This implies tha t i) the solutions of the 2ddT equation generated b y the a b ov e RH problem satisfy the linear (2 + 1)-dimensional PDE φ t = i ( ¯ z φ ¯ z − z φ z ) (87) and, substituting in (12) the ex pression of φ t in terms of φ z , φ ¯ z giv en in (87), one obtains the following nonlinear t w o dimensional constraint: i ( ¯ z ∂ ¯ z − z ∂ z )  e i ( ¯ z φ ¯ z − z φ z )  = φ z ¯ z (88) on the solutions of 2ddT constructed by the ab ov e RH problem. ii) π + 1 + π + 2 = π − 1 + π − 2 is the follo wing explicit and elemen tary function of λ : w ( λ ) := π + 1 + π + 2 = π − 1 + π − 2 = ( z λ + ¯ z λ − 1 ) e − φ t 2 + i ln λ − t − z φ z + i φ t 2 . (89) iii) Since, from (89), π − 1 + π − 2 = w ( λ ) is an explicit function of λ , the vector nonlinear RH problem (84) decouples in to tw o scalar, linear RH problems: π + 1 = π − 1 + if  e w ( λ )  , π + 2 = π − 2 − if  e w ( λ )  , (90) whose explicit solutions are giv en b y ξ ± j ( λ ) = ( − 1) j +1 1 2 π i H | λ | =1 dλ ′ λ ′ − (1 ∓ ǫ ) e ar g λ f  e w ( λ ′ )  , j = 1 , 2 , (91) where ξ ± j = π ± j − ν j , and the closure conditions ( 2 2) read φ z e − φ t 2 = − 1 2 π i I | λ | =1 dλf  e w ( λ )  , φ t = 1 2 π i I | λ | =1 dλ λ f  e w ( λ )  . (92) Although the R H problem (90) is linear, since w ( λ ) in (89) dep ends on the unkno wns φ t , φ z , the closure conditions (92) are a nonlinear algebraic system of tw o equations for the t w o unknowns φ t , φ z , defining implicitly a class of solutions of the 2ddT equation parametrized by the arbitr a ry real sp ectral function f ( · ) of a single v ariable. Ac kno wledgemen ts . 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