A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation

Contemporary Mathematics A generating function for the N -soliton solutions of the Kadomtsev-Pe tviashvili II equation Sarbarish Chakrav arty and Y uji K odama A BS T RA C T . This work describe s a classificat ion of the N -soliton solutions of the Kadomtse v-Petvia shvili II equation in terms of chord diagrams of N chords joining pairs of 2 N points. The dif ferent classes of N -solito ns are enumera ted by the distrib ution of crossings of the chords. The generat ing function of the chord diagrams is expre ssed as a continue d fracti on, special cases of which are moment generat ing functions for certain kinds of q -orth ogonal polynomials. 1. Introduction The classical theory of enumerative combinato rics has indeed a far -reaching scope, encompassing disparate areas in math ematical, physical, biolog ical and social sciences. Comb inatorial entities such as per mutations, partitions, tree s, lattice paths, graphs and th eir v arious enumerations fin d applications ranging from eco nometrics, DN A structures, and statistical mech anics to cod ing theory , kno ts and enu merati ve algebraic geom etry . The pur pose of th e present n ote is to elaborate on a some wh at unexpected relationship b etween a classical combinatorial p roblem stud ied by T ouchar d in the 1950s and the classification of a special class of solitary wa ve solutions ( solitons ) of an e xactly solv able nonlinear partial differential equation disco vered s ome 20 years later . This nonlinear wa ve equation , k nown after its discoverers as the Kadomtsev-Petviashvili (KP) equation, (1.1) ∂ ∂ x  − 4 ∂ u ∂ t + ∂ 3 u ∂ x 3 + 6 u ∂ u ∂ x  + 3 σ ∂ 2 u ∂ y 2 = 0 , describes the ev olu tion of small-amplitud e, weakly two-dimen sional solitary wav es in a weakly d ispersi ve medium [ 18 ]. Dependin g on the sign of σ , th ere are two versions of the KP equatio n namely , KPI and KPII. Thro ughout this a rticle we con sider Eq.( 1.1 ) with σ = + 1, wh ich is the KPII eq uation. T he f unction u = u ( x , y , t ) is the rescaled amplitude of the wa ve-form. The KP equ ation arises in many physical setting s includin g water wa ves and plasmas (see e.g. [ 15 ] for a revie w). It is a completely integrable system with remarkab ly rich math ematical structure which is well-docum ented in several monograp hs [ 1, 14, 22 ]. In th is article, we consider a family of re al, non- singular solitary wave solution s of the KPII equatio n, known as the N -solito n solutions. At any given time t , these wav e -forms are localized along certain lin es in the xy -plane, an d decay e xp onentially everywhere else. In th e gener ic case, they form a pattern of N in tersecting straight l ines a s | y | → ∞ in the xy -plane, whereas in the near - field region the N lin es interact to form intermediate lines and web-like s tructures as sho wn in Fig. 1.1. T he simplest kind of such solution is the 1-soliton, which is a constant amplitude w ave localized along a line in the xy - plane, an d trav eling with unifor m velocity perpendicu lar to the line. For N > 1, the asympto tic form of the N -soliton solution coincid es w ith N 1-so litons alo ng d if f erent directio ns, as | y | → ∞ a nd unifo rmly in t . For this reason, these solutions are often referred to as the line-solitons . Sev er al researchers [ 12, 13, 2 3, 24, 31 ] as well as the au thors [ 5, 20, 3, 4, 6 ] h a ve stud ied the soliton solutions of KPII. The g eneral line-soliton con figurations called the ( N − , N + ) -solitons con sist of N − line solito ns as y → − ∞ an d 1991 Mathemati cs Subject Classificat ion. 37K10i,33D45 i,05A15. Parti ally supported by National Science Foundati on Grant No. DMS-0307181. Parti ally supported by National Science Foundati on Grant No. DMS-0404931. c  2002 American Mathematical Society 1 N + line soliton s as y → ∞ [ 3, 6 ]. T he N -so litons co rrespond to the special case wh en N − = N + = N , and when the N line solitons as y → ∞ are pair-wise identical (as w ave-forms) to the N line soliton s as y → − ∞ . An interesting f eature of the N -solito n solutions is th e fact that th ese solutions can be essentially (i.e., u p to space -time translations) recon structed from the asym ptotic d ata alon e, comprising N pairs of am plitudes and d irections associated with the N line solitons as | y | → ∞ [ 4, 6, 7 ]. As a direct consequence of this result, it is po ssible to classify all the N -soliton solutions into ( 2 N − 1 ) !! d istinct equivalence classes, correspo nding to the ways of partitio ning the integer set { 1 , 2 , . . . , 2 N } into N distinct p airs. The purp ose of th is paper is to extend our stud ies to a char acterization of th e N -soliton solu tions accordin g to soliton interactio n patter ns, and give a classification of such inter actions in ter ms of certain partitions (perfect matchings) of the inte ger set { 1 , 2 , . . . , 2 N } . W e highlight some i nteresting connections between N -solitons of KPII on one hand, and combinato rics o f chord diagrams and q -Hermite polynomials on the other . (a) -100 -50 0 50 100 -100 -50 0 50 100 (b) -100 -50 0 50 100 -75 -50 -25 0 25 50 75 (c) -50 0 50 100 -50 0 50 100 Figure 1.1: N-soliton solutions of the KPII equation illustrating different spatial i nteraction patterns: (a) 2-soliton solution, (b) resonant 3-soliton solution, (c) partially resonant 4-soliton solution. H ere and in all following fi gures, t he horizontal axis is x , vertical axis is y , and the grap hs show contour lines of ln u ( x , y , t ) for fixed t . 2. Background In this section we gi ve a brief ov er vie w o f th e chord d iagrams with 2 N p oints, as well as the line-soliton solutions of the KPII equation. T he aim is to und erscore the connection b etween these two seemingly disjoint mathematical objects. In particular, we illustrate how a KPII line-so liton can be rep resented by a set of index p airs, which leads naturally to the con struction of a chord d iagram on an integer set. W e shall use this constructio n in Section 3, to identify e ach line soliton of an N -solito n solution b y using a chord jo ining two specific points amo ng 2 N points, and study the soliton interaction patterns in terms of such chord diagrams. 2.1. Chord diagra ms. Let u s first describ e a cho rd diagram con sisting of N ch ords. Consider a p artition o f the integer set [ 2 N ] : = { 1 , 2 , . . . , 2 N } into N distinct 2-elemen t b locks or pairings p n : = [ i n , j n ] , 1 ≤ i n < j n ≤ 2 N , n = 1 , 2 , . . . , N , such that [ 2 N ] is a union of the block s p 1 , p 2 , . . . , p N . I n combin atorics, such a partition is r eferred to as a (perfe ct) matching o f [ 2 N ] . W e will denote the set of all matchings o f [ 2 N ] by M N . The to tal n umber of match ings in M N is giv en by | M N | = 1 · 3 · 5 . . . · ( 2 N − 1 ) = : ( 2 N − 1 ) !! . A standar d way to rep resent a m atching X of M N , is to m ark 2 N points o n a lin e from left to rig ht labeled b y 1 , 2 , . . . , 2 N , and jo in th e two points of eac h p airing p by a semicircular arc a bove the line. The resulting d iagram (see e.g. Fig. 2.1) is called a linear chord diagram , wherea s a chord diagram would co rrespond to labeling the 2 N points in a clockwise manner on a circle, and joining the two points of each pairing by a chord. W ithou t lo ss of generality , the smallest in teger i n from each pair ing o f X ∈ M N can be arranged in a strictly increasing order 1 = i 1 < i 2 < . . . i N ≤ 2 N − 1. Howe ver the j n ’ s are not ordered in general. Definition 2.1. Let p r and p s with r < s be distinct pairings (equi valently , a pair of chords). Then, 2 Figure 2.1: A linear Chord diagram (a) p r and p s form an alignment or a O-type configuration if i r < j r < i s < j s . That is, the pairs do not o verlap . (b) p r and p s form a cr ossing or a T -ty pe configuration if i r < i s < j r < j s . That is, the pairs partially overlap. (c) p r and p s form a nesting or a P-type configuration if i r < i s < j s < j r . That is the pairs complete ly ov erlap. The m eanings of O-, T -, an d P-type of configura tions will be clear later in Section 3 whe n we discu ss th e N - solitons of K PII. In Fig. 2.1, the p airing [ 4 , 6 ] for ms an alig nment (O- type con figuration) with [ 1 , 3 ] ; a cr ossing with [ 5 , 8 ] ; and a nesting with th e pairing [ 2 , 7 ] . F urtherm ore, the total n umber of (pairwise) crossings in Fig. 2.1 is 3; they occur between the pairs [ 1 , 3 ] and [ 2 , 7 ] ; [ 2 , 7 ] and [ 5 , 8 ] ; and [ 4 , 6 ] a nd [ 5 , 8 ] . Similarly , there is o ne nesting, { [ 2 , 7 ] , [ 4 , 6 ] } , and two alignments, { [ 1 , 3 ] , [ 4 , 6 ] } and { [ 1 , 3 ] , [ 5 , 8 ] } . It sho uld be clear that the of align ments, crossings and nestings fo r any X ∈ M N must add up to th e total number of pairwise c hord con figurations, i.e., N ( N − 1 ) / 2 . One o f the earliest results [ 10, 34 ] in the e numeration of cho rd diagrams is that the number of m atchings in M N with no crossings is gi ven by th e N th Catalan number C N = 1 N + 1  2 N N  , which ap pears in many comb inatorial problem s ( see, e. g., [ 32 ]). Similarly , the number o f diag rams in M N with n o nestings is also g i ven by C N . The p roblem of countin g the elemen ts of M N accordin g to the num ber of p airwise crossings of cho rds was con sidered by T ouchard [ 33 ], who ga ve an implicit form ula fo r th e e numerating g enerating function in terms of con tinued fractio ns. Su bsequently , Riordan [ 28 ] derived a remar kable explicit formula for the generating function based on T oucha rd’ s work. If cr ( X ) denotes the number of crossings of the element X ∈ M N , then the generating function by the numbe r o f crossings is defined via the polynomial F N ( q ) : = ∑ X ∈ M N q cr ( X ) , 0 ≤ cr ( X ) ≤ 1 2 N ( N − 1 ) , in the variable q wit h positiv e integer coef ficien ts. T he T ouch ard-Riordan formula for F N ( q ) is (2.1) F N ( q ) = 1 ( 1 − q ) N N ∑ n = 0 ( − 1 ) n  2 N N − n  −  2 N N − n − 1  q n ( n + 1 ) / 2 . The first few polynomials are F 1 ( q ) = 1 , F 2 ( q ) = q + 2 , F 3 ( q ) = q 3 + 3 q 2 + 6 q + 5 , and it easily follows from Eq. (2.1 ) that the numbe r o f non-crossing diagrams is gi ven by F N ( 0 ) =  2 N N  −  2 N N − 1  = 1 N + 1  2 N N  , which is the Catalan number C N mentioned earlier . Howe ver , the T oucha rd-Riordan formula is some wha t mysterious, in that it is not obvious from Eq. (2. 1 ) th at F N ( q ) is in fact a po lynomial in q of d egree N ( N − 1 ) / 2 as implied by its com binatorial o rigin, or that F N ( 1 ) = | M N | = ( 2 N − 1 ) !!. The se assertion s f ollo w o nly after detailed analysis of Eq. (2 .1) [ 28 ] (see also [ 11 ]). A purely combin atorial proof of the T ouchar d-Riordan f ormula also appeare d in Ref. [ 25 ] (see also [ 19 ]), and its relation to q -Hermite polyno mials was inves tigated in ref. [ 17 ]. 2.2. The τ -functio n and line so litons of KPII. It is well kn o wn (see e .g. [ 30, 1 4 ]) that the solutio n u ( x , y , t ) of the KPII equation is giv en in terms of the τ -function τ ( x , y , t ) as (2.2) u ( x , y , t ) = 2 ∂ 2 ∂ x 2 ln τ ( x , y , t ) . 3 W e con sider the class of solutions whose τ -functio ns are gi ven by the Wronskian determinan t fo rm, i.e., (2.3) τ ( x , y , t ) = Wr ( f 1 , . . . , f N ) = det      f 1 f 2 . . . f N f ′ 1 f ′ 2 . . . f ′ N . . . . . . . . . f ( N − 1 ) 1 f ( N − 1 ) 2 f ( N − 1 ) N      . with f ( j ) n = ∂ j f n / ∂ x j , and wh ere the functions { f n } N n = 1 form a set of linearly independent solutions of th e linear system ∂ f ∂ y = ∂ 2 f ∂ x 2 , ∂ f ∂ t = ∂ 3 f ∂ x 3 . The soliton solutions of KPII can be constructed fro m Eq. (2. 3 ) b y cho osing a finite d imensional solutio n for each function f n ( x , y , t ) , na mely , (2.4) f n ( x , y , t ) = M ∑ m = 1 a nm e θ m , n = 1 , 2 , . . . , N , where θ m ( x , y , t ) = k m x + k 2 m y + k 3 m t + θ 0 m , m = 1 , . . . , M , are ph ases with real distinct p arameters k 1 , k 2 . . . k M , an d real constants θ 0 1 , . . . , θ 0 M . The co nstant coefficients a nm define the N × M coefficient ma trix A : = ( a nm ) . The simplest example is the 1-soliton solution with N = 1 , M = 2, and τ = f 1 = e θ 1 + e θ 2 , for which u ( x , y , t ) = 2 ∂ 2 ∂ x 2 ln τ = 1 2 ( k 1 − k 2 ) 2 sech 2 1 2 ( θ 1 − θ 2 ) . This 1-solito n solu tion describe s a plane traveling wave-form with constant am plitude ( k 1 − k 2 ) 2 / 2. For fixed t , the wa ve-fo rm is localized in the ( x , y ) -plan e along the line L : θ 1 = θ 2 whose norm al h as the slop e c = k 1 + k 2 . The solution is cha racterized b y two ph ysical p arameters, namely , the soliton a mplitude parameter a = | k 1 − k 2 | and the soliton dir ectio n parameter c = k 1 + k 2 . In the gen eral case, substitution of Eq . (2.4) into the Wronsk ian of Eq. ( 2.3), and subseq uent de velopment of th e resulting determinan t via B inet-Cauchy formula, yields the following e x plicit form of the τ -function: (2.5) τ ( x , y , t ) = ∑ 1 ≤ m 1 < ··· < m N ≤ M A ( m 1 , . . . , m N ) exp [ θ ( m 1 , . . . , m N ) ] ∏ 1 ≤ s < r ≤ N ( k m r − k m s ) , where A ( m 1 , . . . , m N ) is the N × N max imal minor of A obtained from the column s 1 ≤ m 1 < · · · < m N ≤ M , and θ ( m 1 , . . . , m N ) : = θ m 1 + . . . + θ m N is a p hase combination o f N (ou t of M ) distinct phases. N ote t hat the transformation, G : A → A ′ : = G A , G ∈ GL ( N , R ) , amoun ts to an overall rescaling o f the min ors A ( m 1 , . . . , m N ) , an d hence, o f th e τ -function in Eq . (2.5); i.e., τ → τ ′ = det ( G ) τ . Sin ce such a rescaling leaves the solution u ( x , y , t ) in Eq . ( 2.2) in vari- ant, it is po ssible to reduce the coefficient matrix A to redu ced r o w- echelon form (RREF) by Gaussian elimination. Throu ghout the rest of this article, the coeffi cient matrix A will be assumed to be in RREF . The solutions u ( x , y , t ) resulting from the τ -function in Eq. (2.5) ar e singular for ar bitrary choices of the p arameters { k n } M n = 1 and the matrix A . T o a void such s ingularities, which correspond to the zero-locus of th e τ -function, one needs to impose certain positivity conditions. Condition 2.2 (Positive d efiniteness of τ ). (a) The phase parame ters are distinct, and are ordered as k 1 < k 2 < . . . < k M . (b) The N × M co ef ficien t matrix A satisfies rank ( A ) = N , and M > N . (c) All non-z ero N × N minors of A are positive. Remark 2.3 The matric es satisfying Cond ition 2.2(c) ab ov e, are called totally non- negati ve (TNN) matrices. The classification of th e ( N − , N + ) -soliton solution s is thus given by the classification of the N × M TNN m atrices A in RREF . From a mo re geo metric persp ecti ve, each TNN matrix pa rametrizes a u nique cell in the TNN Grassmannian Gr + ( N , M ) (see e.g. [ 26 ]), and the classification of the soliton solutions c orresponds to a further refinemen t o f th e Schubert decomp osition o f Gr ( N , M ) in to TNN Grassmann cells (see [ 20 ] for the case M = 2 N ). The r efinement is given by a classification of the co ef ficient matrix A wh ose N × N m inors A ( m 1 , . . . , m N ) repr esent the Pl ¨ ucker coordin ates of Gr ( N , M ) . It should be noted that each ( N − , N + ) -soliton solution correspo nding to a TNN matrix can 4 be param etrized by a c hord diagram [ 6, 8 ]. Th e geometric structur e of this classification will b e discussed in a f uture commun ication [ 7 ]. In Cond ition 2.2, the distinctne ss assumptio n on the set o f pha se parame ters ensures that the set { e θ m } M m = 1 is linearly indep endent, while rank ( A ) = N imp lies that the set of fu nctions { f n } N n = 1 is linearly ind ependent. Also wh en M = N , the sum in Eq. (2 .5) reduces to a sin gle expon ential term with a phase co mbination that is linear in x . The logarithm of such a τ -func tion is annihilated by the second derivati ve in Eq. (2.2), lead ing to the trivial solutio n u ( x , y , t ) = 0. Thus the co ndition N < M guaran tees non-tr i vial so lutions o f KPI I. Con dition 2. 2 ( c) tog ether with the orderin g k 1 < k 2 < . . . < k M make the sum in E q. (2. 5) totally p ositi ve. As a result, τ ( x , y , t ) is a p ositi ve function on R 3 , a nd the r esulting solu tion u ( x , y , t ) of KPII is non- singular , bound ed an d p ositi ve definite. Furthermo re, the asymptotic an alysis of th e τ -fun ction in Eq . ( 2.5) reveals that for any given value of t , the re exist a set of lin es given by { L i j : θ i = θ j , i < j } in the ( x , y ) -p lane, such that (2.6) u ( x , y , t ) ∼ 1 2 ( k j − k i ) 2 sech 2 1 2 ( θ j − θ i + δ i j ) , along each L i j either as y → ∞ , o r as y → − ∞ . Equation (2. 6 ) , wh ich has the same form as th e 1 -soliton solu tion defines an a symptotic lin e soliton along L i j associated with th e solution u ( x , y , t ) . Each line soliton which is pa rallel to the line L i j has the para meters a i j = | k i − k j | fo r the amplitud e and c i j = k i + k j for the directio n normal to the line L i j . Hen ce, we den ote each line soliton by the index pa iring p = [ i , j ] labeling th e line L i j . No te fro m Eq . ( 2.4) tha t the ind ices i , j labeling th e ph ases θ i , θ j in Eq . (2 .6) also label a p air o f distinct c olumns of the co ef ficien t matrix A . Due to this conne ction, it tu rns out that th e pair ing: p = [ i , j ] , 1 ≤ i < j ≤ M , of the lin e soliton s can be determ ined from the structure of the coefficient matrix A which is in RREF , and satisfies the following irreducibility condition. Condition 2.4 (Irred ucibility) (a) Each column of A contains at least one nonze ro element. (b) Each row of A contains at least one nonzero element in addition to the pi vot. Recall that, for an N × M m atrix in RREF , the leftmost non-vanishing entry in each nonzero ro w is called a pi vot, which is normalized t o unity . Th e index pairs [ i , j ] of the asymptotic line solitons sho wn in Eq. (2.6) are then gi ven by the following technical result, which is proved in R e f. [ 3 ]. Proposition 2.5. Let the sub-ma trices X [ i j ] and Y [ i j ] of A be defin ed in terms of their column indices as X [ i j ] : = [ 1 , 2 , . . . , i − 1 , j + 1 , . . . , M ] Y [ i j ] : = [ i + 1 , . . . j − 1 ] . Then, necessary a nd sufficient co nditions for a n ind e x p air [ i , j ] to specify an asymp totic line soliton, as in Eq. (2.6) , ar e the fo llowing r an k conditions. (i) Each line soliton as y → ∞ is labeled by a u nique index pair [ e n , j n ] with e n < j n , where { e n } N n = 1 label the pivot columns of A. Moreover , if rank ( X [ e n j n ]) = : r n , then r n ≤ N − 1 and rank ( X [ e n j n ] | e n ) = ran k ( X [ e n j n ] | j n ) = rank ( X [ e n j n ] | e n , j n ) = r n + 1 . (ii) Each line soliton as y → − ∞ is labeled by a uniqu e index pa ir [ i n , g n ] with i n < g n , wher e { g n } M − N n = 1 label the non-p ivot c olumns of A. Mor eover , if rank ( Y [ i n g n ]) = : s n , then s n ≤ N − 1 and rank ( Y [ i n g n ] | i n ) = ran k ( Y [ i n g n ] | g n ) = rank ( Y [ i n g n ] | i n , g n ) = s n + 1 . Her e ( Z | m , n ) deno tes the s ub-matrix Z of A au gmented by the columns m and n of A. It should be clear from the above result that the ( N − , N + ) -soliton solution of KPII ge nerated from the τ -functio n in Eq. (2.5) has e xactly N + = N asy mptotic line-solitons as y → ∞ and N − = M − N asympto tic line -solitons as y → − ∞ . From the τ -function data consisting of M distinct phase parameters k 1 , . . . , k M and a matrix A satisfying C ondition 2.4, Proposition 2.5 p rovides an explicit way to identify all th e asymp totic line soliton s o f the correspon ding solution o f the KPII equation. W e illustrate this method with an e xample. Example 2.6 Con sider the solution u ( x , y , t ) generated by th e τ -f unction of Eq. (2.3) in the case N = 2 and M = 4, with 4 real parameters k 1 < k 2 < k 3 < k 4 , and f 1 = e θ 1 − e θ 4 , f 2 = e θ 2 + e θ 3 , A =  1 0 0 − 1 0 1 1 0  . 5 The piv o t co lumns of A are lab eled by th e indices { e 1 , e 2 } = { 1 , 2 } , and the non -piv ot columns by the indices { g 1 , g 2 } = { 3 , 4 } . According to Proposition 2.5, the nu mber of asympto tic line solitons is N + = N − = 2. They are identified by th e in dex pairs [ 1 , j 1 ] , [ 2 , j 2 ] as y → ∞ , for som e j 1 > 1 and j 2 > 2; and by the ind ex pairs [ i 1 , 3 ] , [ i 2 , 4 ] as y → − ∞ , for some i 1 < 3 and i 2 < 4. W e first determine th e asymptotic line-solitons as y → ∞ using the rank condition s p rescribed in Proposition 2.5(i). For the first pivot colu mn e 1 = 1; starting f rom j = 2 and then repe at- edly in crementing the value o f j by u nity , we check the ran k of each sub -matrix X [ 1 j ] . Pro ceeding in this way , we find that the rank cond itions are satisfied only when j = 4: X [ 14 ] = / 0 . So, r ank ( X [ 14 ]) = 0 < N − 1. Moreover , rank ( X [ 14 ] | 1 ) = rank ( X [ 14 ] | 4 ) = rank ( X [ 14 ] | 1 , 4 ) = 1 sin ce colu mns 1 and 4 are parallel. Thus, the first asymp totic line soliton as y → ∞ is identified by the ind ex p air [ 1 , 4 ] . For e 2 = 2 , proceeding in a similar manner we find tha t j = 3 does satis fy t he rank co nditions since X [ 23 ] =  1 − 1 0 0  is of r ank 1 = N − 1 , and rank ( X [ 23 ] | 2 ) = rank ( X [ 23 ] | 3 ) = rank ( X [ 23 ] | 2 , 3 ) = 2. Th erefore, th e asympto tic lin e solitons as y → ∞ are identified with the index pair s [ 1 , 4 ] and [ 2 , 3 ] . W e n ext consider the asymp totics for y → − ∞ . Starting with th e non -pi vot co lumn g 1 = 3 , we app ly the rank condition s in Proposition 2.5(ii) to the colu mn i = 2 . T hen, we have Y [ 23 ] = / 0 , a nd rank ( Y [ 23 ] | 2 ) = r ank ( Y [ 23 ] | 3 ) = rank ( Y [ 2 3 ] | 2 , 3 ) = 1. Hence , the p air [ 2 , 3 ] iden tifies an asym ptotic line-so liton as y → − ∞ . For g 2 = 4, we consider i = 1 , 2 , 3 and fin d th at the ran k cond itions are satisfied only for i = 1. In this case, Y [ 14 ] =  0 0 1 1  , so r ank ( Y [ 14 ]) = 1 = N − 1 an d ran k ( Y [ 14 ] | 1 ) = ran k ( Y [ 14 ] | 4 ) = ran k ( Y [ 14 ] | 1 , 4 ) = 2 . Thus, the index pair [ 1 , 4 ] identifies the o ther asymptotic lin e-soliton as y → − ∞ . In summ ary , both pairs of asy mptotic line soliton s as y → ± ∞ are lab eled by the index pairs [ 1 , 4 ] and [ 2 , 3 ] . This is an example of a P-type 2-soliton solution (see S ection 3), and is sho wn in Fig. 3.1. It shou ld be emph asized that in g eneral N − 6 = N + , and that even in th e case N − = N + , the line so litons as y → ∞ are in general distinct from the line solitons as y → − ∞ in both amplitude and direction. In this article, we restrict our discussions primarily to the N -soliton sub class of the line-soliton solutions of KPII. Definition 2.7 . Let S + : = { [ e n , j n ] } N n = 1 and S − : = { [ i n , g n ] } M − N n = 1 denote the index sets id entifying the lin e solitons a s y → ∞ and as y → − ∞ , resp ecti vely , according to Proposition 2.5. Then two ( N − , N + ) -soliton solution s o f KPII are said to be in the same equi valence class if their as ymptotic line-solitons are labeled by identical sets S ± of index pairs, where | S + | : = N + = N and | S − | : = N − = M − N . The set S + ∪ S − of unique index pairin gs in Defin ition 2.7 h as a co mbinatorial interpretation . Let [ M ] : = { 1 , 2 , . . . , M } be th e integer set with the p i vot and no n-piv ot indices { e 1 , . . . , e N } ∪ { g 1 , . . . , g M − N } fo rming a d isjoint partition of [ M ] . Define the pairin g map π : [ M ] → [ M ] accord ing to Proposition 2.5(i) & (ii) as (2.7) π ( e n ) = j n , n = 1 , 2 , . . . , N , π ( g n ) = i n , n = 1 , 2 , . . . , M − N . Then π : [ M ] → [ M ] is a b ijection, i.e., π ∈ S M , the p ermutation group of [ M ] [ 6 ]. In add ition, Proposition 2.5 im plies the following. Proposition 2.8. The pairing map π defined by Eq. (2.7) is a derangement of [ M ] with N excedances, which ar e given by the pivot indices { e 1 , . . . , e N } of th e coef ficient matrix A in RREF . Recall that a permu tation π with no fixed p oint is called a d erangement , and an eleme nt l ∈ [ M ] is called an excedance of π if π ( l ) > l . Each equivalence class o f ( N − , N + ) -soliton solution s o f KPII is un iquely d etermined b y a derangem ent π , a s in Proposition 2.8. Th ese derang ements also give a u nique param etrization of a TNN Gr assmann cell (see Remark 2. 3) whose associated TNN matrix A s atisfies the irreducibility C ondition 2.4. Recall that in Example 2.6, both sets of l ine solitons as y → ± ∞ are given by [ 1 , 4 ] , [ 2 , 3 ] , wher e 1,2 are the p i vot ind ices and 3,4 are the n on-piv ot indices of the associated coefficient matrix A . In this case, the pairing map is a derang ement of the set [ 4 ] , and is gi ven by π =  1 2 3 4 4 3 2 1  ∈ S 4 , in the bi-word notation of permu tations in S 4 . Note that the exced ance set of π is { 1 , 2 } . In a ddition, π is also an in volution of S 4 , i.e., π − 1 = π . Since the set of all in volutions of S 2 N is isomorph ic to the set of perfect matchings M N introdu ced in Section 2.1 , the inv olu tions can be also rep resented by the ch ord diagrams represen ting the elements of M N . In p articular , the ch ord diagram for the inv o lution π ∈ S 4 giv en above depicts a nesting of the chor ds p 1 = [ 1 , 4 ] and p 2 = [ 2 , 3 ] a s shown b elo w in Fig. 3.1. W e rema rk that it is p ossible to represen t d erangements that are not 6 in volutions by linear chor d diagrams, with dire cted chords both above and below the line. Th ese diagram s ha ve been used to study the more g eneral ( N − , N + ) -soliton solution s of KPII in Ref. [ 6 ]. But her e we fo cus our a ttention to th e N -solito n solutions, which will be our ne xt topic of discussion. 3. N -solito n solutions When M = 2 N , it follows fro m Proposition 2. 5 that N − = N + = N . If in ad dition, we c onsider S − = S + in Definition 2. 7, then we recover the intere sting subclass of the ( N , N ) -so liton solu tions mentioned in Section 1, called N -solito n solutio ns, which a re char acterized by iden tical sets of asympto tic line- solitons as | y | → ∞ . Th en the m ain features of the N -soliton solu tions f ollo w primarily from our d iscussion in Section 2, in particular from Prop ositions 2.5 and 2.8. Th ese are li sted below . Property 3.1 N -soliton solutions hav e the following properties. (i) The τ -function of an N -soliton solution is e xpressed in terms of 2 N d istinct phase parameters and an N × 2 N coefficient matrix A which satisfies Condition s 2.2 and 2.4. In addition, th e N × N min ors of A satisfy the duality conditions [ 6, 20 ]: A ( m 1 , . . . , m N ) = 0 ⇐ ⇒ A ( l 1 , . . . , l N ) = 0 , where the indices { m 1 , . . . , m N } and { l 1 , . . . , l N } for m a disjoint partition of integers { 1 , 2 , . . . , 2 N } . T hat is, the phase combinatio n θ ( m 1 , . . . , m N ) is pre sent in the τ -function of Eq. (2.5) if and only if θ ( l 1 , . . . , l N ) is. (ii) Each N -soliton solu tion has e x actly N asymp totic line s olitons as y → ± ∞ identified by the same index pairs [ e n , g n ] with e n < g n , n = 1 , . . . , N . The sets { e 1 , . . . , e N } and { g 1 , . . . , g N } label respe cti vely the pivot a nd non-p i vot columns of the coef ficient matrix A . Hence, they form a disjoint partition of the integer set [ 2 N ] . (iii) The amplitu de and direction par ameters of the n th asymptotic lin e soliton [ e n , g n ] are the same as y → ± ∞ , and are given in t erms of the phase parameter s as a n = k g n − k e n , c n = k g n + k e n . (iv) The pairing map associated to an N -soliton solution, namely π ( e n ) = g n , π ( g n ) = e n , n = 1 , 2 , . . . , N , corre - sponds to a partition of the integer set [ 2 N ] into N distinct pairs of integers ( e n , g n ) , as in Section 2.1. Each such map is an in volution in S 2 N with no fixed points, a member of I 2 N = { π ∈ S 2 N : π − 1 = π and π ( i ) 6 = i , ∀ i ∈ [ 2 N ] } . Such per mutations can b e expressed as pro ducts of N disjoint 2- cycles, and their ch ord diagrams are iden- tical to tho se of the p erfect matching s M N (see Fig. 2 .1). The total nu mber of such inv o lutions is giv en by | I 2 N | = | M N | = ( 2 N − 1 ) !!. H ence, there are ( 2 N − 1 ) !! distinct equiv alence classes of N -soliton solution s. 3.1. Equiv a lence classes of 2-soliton solutions. When N = 2, there are three types of 2- soliton solutions referr ed to as the O-, T - and P-typ es (following the termin ology introduced in Ref. [ 20 ]) . They are iden tified by the can onical coefficient matrices associated with τ -functions, namely A O =  1 1 0 0 0 0 1 1  , A T =  1 0 − 1 − 1 0 1 x 1 x 2  , A P =  1 0 0 − 1 0 1 1 0  , (3.1) with x 1 > x 2 > 0 i n A T . By applying th e ra nk cond itions of Proposition 2.5 to th e ab ove coefficient matrices, it is easily verified that the O-ty pe 2-solitons hav e asymptotic line-so litons [1,2] an d [3,4]; th e T -typ e resonant 2- solitons have asymptotic line-solitons [1,3] and [2,4]; and the P-type 2-solitons ha ve asymptotic line-solitons [1,4] and [2,3]. These are sho wn in Fig. 3.1 . No tice th at each of the O- and P-type solitons interact via an X-junction. Af ter in teraction, each line soliton undergoes a position shift in the xy -plane. Howe ver, it can be shown that the position shifts for t he O-type solitons ar e oppo site in sign to that o f the P-typ e solitons [ 6 ]. On the o ther hand, the T -type solitons inter act via fou r Y -junctions, connecting the f our asymptotic line-so litons to four interme diate segments. Each of these intermed iate segments is a lso a line solito n. For example, in the T -type soliton s in Fig. 3.1 th e asymp totic line soliton [ 1 , 3 ] (a s y → − ∞ ) form s the intermed iate line-solitons [ 1 , 2 ] an d [ 2 , 3 ] at the bottom left Y -junction. The line-soliton [ 2 , 3 ] connects with th e asymp totic line-soliton [ 2 , 4 ] (a s y → ∞ ) and the line-soliton [ 1 , 2 ] con nects with the asymptotic line soliton [ 2 , 4 ] (as y → − ∞ ). Similarly , the asymptotic line soliton [ 1 , 3 ] (as y → ∞ ) forms th e interm ediate line- solitons [ 1 , 4 ] and [ 3 , 4 ] at the top rig ht Y -junctio n. The line-soliton [ 3 , 4 ] conne cts with the asymptotic line-soliton 7 Figure 3.1: Three different two-soliton solutions of KPII with t he same phase parameters ( k 1 , . . . , k 4 ) = ( − 2 , − 1 2 , 0 , 1 ) , illustrating the three 2-soliton equi valence classes: O-type, T -type and P-type 2-soliton solutions. [ 2 , 4 ] ( as y → ∞ ) and the lin e-soliton [ 1 , 4 ] conn ects with the asymptotic line soliton [ 2 , 4 ] (as y → − ∞ ). Fig. 3.1 also sh o ws the chord diag rams for th e corr esponding p airing maps, which are inv o lutions o f the permu tation gro up S 4 . They correspo nd to the d isjoint partitions of [ 4 ] into 2 pa irs. In cycle notation , these inv olu tions are given by π O = ( 12 )( 34 ) , π T = ( 13 )( 24 ) , and π P = ( 14 )( 23 ) , for the O-, T - an d P-type 2-soliton equiv alence class es. Acco rding to the Definition 2.1, th e chor d diagram for π O forms an align ment; whereas the diagram f or π T has a cr ossing between the chords correspon ding to the line solitons [ 1 , 3 ] and [ 2 , 4 ] ; and the diagram for π P is a nesting. An impo rtant d istinction a mong the thr ee typ es o f 2-soliton solutions is that they belo ng to different regions of the soliton parameter space. Su ppose ( a 1 , c 1 ) and ( a 2 , c 2 ) are the so liton parameters of the asymptotic line-solitons of each type, with the same set of distinct phase param eters. Since the p hase parame ters are ord ered: k 1 < · · · < k 4 , the soliton parameter s satis f y the following relations, which can be easily verified using Eqs. (3.1). (i) For O-type 2-soliton solutions, c 2 > c 1 and c 2 − c 1 > a 1 + a 2 . (ii) For T -type 2-soliton solutions, c 2 > c 1 , and | a 1 − a 2 | < c 2 − c 1 < a 1 + a 2 . (iii) For P-type 2-soliton s olutions, a 2 > a 1 and | c 2 − c 1 | < a 2 − a 1 . (iv) ( c 2 − c 1 ) O > ( c 2 − c 1 ) T > | c 2 − c 1 | P , ( a 1 + a 2 ) O < ( a 1 + a 2 ) T = ( a 1 + a 2 ) P , and | a 2 − a 1 | O = | a 2 − a 1 | T < ( a 2 − a 1 ) P . Note that for O- an d T -typ e solutions the solito n direc tions are or dered, while for P-type solutions the amplitudes are ordered. Any cho ice of the soliton parameters a 1 , c 1 ; a 2 , c 2 with a 1 , a 2 > 0 would lead to one of the three types of 2-soliton so lutions, provided that { c 1 ± a 1 , c 2 ± a 2 } are distinc t real num bers. Thus, the three ty pes of 2-soliton solutions p artition the so liton parameter sp ace into disjoint secto rs, bou nded by th e hyperp lanes | c 2 − c 1 | = a 1 + a 2 and | c 2 − c 1 | = | a 1 − a 2 | . At each boundar y b etween two sectors, two of the phase para meters coincide. In such a situation, it can be shown (by taking suitable limits) that the 2-soliton solution degenerates into a Y -ju nction [ 5, 20 ]. It should be clear from the abov e th at the O-, T - and P-type 2-soliton s olutions exhibit distinct types of interaction patterns, and belong to different regions of the soliton parameter s pace. For N > 2, in addition to the non-resonant (O- and P-type) a nd f ully reso nant ( T -type) solutions, a large family o f partially reson ant solutio ns exists. For example, when N = 3, Prope rty 3.1(iv) implies that ther e are 1 5 distinct equ i valent classes of 3-soliton solutio ns (see Fig .3.2). Unlike the N = 2 case ab ove, it turns o ut to be a comp licated task to classify the N -solito n solutions accor ding to their c oef ficient m atrices A , wh en N > 2. This task was r ecently carried o ut by th e authors, a nd will b e rep orted in a future pub lication [ 7 ]. Here we consid er a more direct classification schem e for the N - soliton equiv a lence classes, by characterizin g the pairwise inter actions be tween the N lin e solito ns o f O-, T - and P -type, much as in the 2-soliton case. In other words, we r epresent the N -soliton solutio ns by the correspon ding in volutions in I 2 N ⊂ S 2 N (equiv alently , the matchings in M N ), an d enumerate the solution s according to the n umber of alignm ents, crossings and nesting s of the associated chord diagram . W e describe this classification scheme below . 8 3.2. Combinatorics of N -soliton solut ions. Th roughou t this subsection, we associate a n N - soliton equ i valence class de fined by th e set S = { [ e n , g n ] } N n = 1 of asy mptotic line solitons (see Definition 2.7) with the p artition X = { p 1 , p 2 , . . . , p N } ∈ M N , where p n : = [ e n , g n ] . Recall from Property 3.1(ii) th at the integer set [ 2 N ] is a disjoint union of the index sets E : = { e 1 , . . . , e N } and G : = { g 1 , . . . , g N } , with the following orderings among the indices: (i) 1 = e 1 < e 2 < . . . < e N < 2 N , (ii) e n < g n for all n = 1 , 2 , . . . , N . An immediate consequenc e o f the above o rderings is that (3.2) n ≤ e n ≤ 2 n − 1 , n = 1 , . . . , N , since t here are at least n − 1 indices to the left of e n , namely e 1 , e 2 , . . . , e n − 1 ; and at least 2 N − 2 n + 1 ind ices to the right of e n , namely g n , e r , g r , r > n . The N -soliton classification scheme is obtaine d by considerin g various statistics over the po ssible chord con figurations for the ch ord diagra ms of M N . For this p urpose, using Definition 2.1 we intro duce the following sets, which record the total number of alignments, crossings and nestings for a gi ven chord in any chord diagram of M N . Definition 3.2 . Let p n = [ e n , g n ] be a given cho rd of a p artition X ∈ M N , and let B n : = { p r = [ e r , g r ] : r < n } be the subset of chords originating from the left of p n in the linear chord diagram of X . (a) The set O n of alignments with the chord p n forming O- type configu rations, an d the a lignment number al ( X ) , are defined by O n : = { p r = [ e r , g r ] ∈ B n : g r < e n } , al ( X ) : = N ∑ n = 1 | O n | . (b) The set T n of cro ssings with the chord p n forming T - type co nfigurations, and the crossing numb er cr ( X ) , are defined by T n : = { p r = [ e r , g r ] ∈ B n : e n < g r < g n } , cr ( X ) : = N ∑ n = 1 | T n | . (c) The set P n of nesting s with the chor d p n forming P-type config urations, and the nesting numb er ne ( X ) , are defined by P n : = { p r = [ e r , g r ] ∈ B n : g r > g n } , ne ( X ) : = N ∑ n = 1 | P n | . It follows from the above defin itions tha t B n is the d isjoint un ion of the sets O n , T n and P n , so that | O n | + | T n | + | P n | = n − 1, and al ( X ) + cr ( X ) + n e ( X ) = N ( N − 1 ) / 2, which is a count of all possible pairwise chord configurations in the p artition X . Note that for O n , the indices g r lie in the intervals ( e r , e r + 1 ) , 1 ≤ r < n . Hence, | O n | = e n − n . So the nu mber o f cr ossings an d n estings with the chord p n sum to | T n | + | P n | = ( n − 1 ) − ( e n − n ) = 2 n − e n − 1, which depend s only on the pivot index e n ∈ E . This ob serv a tion leads t o the following. Lemma 3.3. If M ( E ) ⊆ M N denotes the set of all partitions which have the same (pivot) inde x set E , then the n umber of partitions of M ( E ) havin g r cr ossings and s nestings is the coefficient of p s q r in m E ( p , q ) = N ∏ n = 1 [ 2 n − e n ] p , q , [ n ] p , q : = p n − q n p − q = ∑ i + j = n − 1 p i q j . The de g r ee of both p a nd q in m E ( p , q ) is N 2 − ( e 1 + e 2 + . . . + e N ) . Proof. The distribution of crossing s and n estings is the sum o f p ne ( X ) q cr ( X ) over all p artitions X ∈ M ( E ) . Using Definition 3.2 for cr ( X ) an d ne ( X ) , this distribution can be expressed as ∑ X ∈ M ( E ) p ( | P ( 1 ) | + ... + | P ( N ) | ) q ( | T ( 1 ) | + ... + | T ( N ) | ) = N ∏ n = 1 2 n − e n − 1 ∑ l = 0 p l q 2 n − e n − 1 − l , after interchan ging the sum and product, and using the fact that | T n | + | P n | = 2 n − e n − 1 for n = 1 , 2 , . . . , N . Since the second sum is precisely [ 2 n − e n ] p , q , the formula for m E ( p , q ) f ollo w s. It is easy to verify from the pr oduct formula that m E ( p , q ) is sy mmetric in p and q . Consequen tly , the number of diagrams with r crossings and s nestings is the same as the the number o f diag rams with s crossing s and r nestings [ 19 ]. Note also that the enumeratin g p olynomial f or the cr ossings alone is given by m E ( 1 , q ) ; wh ile m E ( p , 1 ) enumera tes only the nestings fo r the chord d iagrams of M ( E ) . In order to extend the results of Le mma 3.3, to the entire set 9 M N , on e needs to sum m E ( p , q ) over all possible cho ices of the in teger set E , with e n ∈ E satisfying Eq . (3.2). Using Lemma 3.3, the expression for the required generating polynomial is gi ven by (3.3) F N ( p , q ) : = ∑ X ∈ M N p ne ( X ) q cr ( X ) = ∑ { E } m E ( p , q ) = ∑ 1 = e 1 < e 2 <...< e N , k ≤ e k ≤ 2 k − 1 N ∏ n = 1 [ 2 n − e n ] p , q . Since we ha ve from Eq. (3.2) that e n ≥ n , n = 1 , 2 , . . . , N , it follows from Lemma 3.3 that the degree of p and q in F N ( p , q ) is giv en by ne ( X ) m ax = cr ( X ) m ax = N 2 − ( 1 + 2 + . . . + N ) = N ( N − 1 ) 2 . Furthermo re, lik e m E ( p , q ) , F N ( p , q ) is symm etric in p an d q , i.e., F N ( p , q ) = N ( N − 1 ) / 2 ∑ r , s = 0 c rs q r p s , c rs = c sr . Some interesting consequen ces o f Eq. (3.3) for special cases of F N ( p , q ) ar e collected belo w . Corollary 3.4. The fun ction F N ( p , q ) h as the following pr operties: (i) F N ( 1 , 1 ) = | M N | = ( 2 N − 1 ) !! . (ii) When p = 1 , q = 0 , the total number of non-cr ossing (i.e., on ly alignments and nestings) c h or d diagrams of M N equals the N t h Catalan number [ 10 ] . That is , F ( 1 , 0 ) = C N which also counts the total possible c h oices for the or der ed integer set E [ 6 ] . S imilarly , F N ( 0 , 1 ) = C N gives the total n umber of no n-nesting (i.e., only alignments and cr ossings) chor d diagr ams of M N . (iii) F N ( 1 , q ) = : F N ( q ) gives th e gene r a ting po lynomial for the numbe r of cr ossings intr od uced in Section 2 .1, which is given e xp licitly by the T ouchar d -Rior dan formula Eq. (2.1) . The polynomials F N ( p , q ) can be determined fro m a generating f unction F ( p , q , x ) wh ich is a f ormal po wer series, and has the following representation. Proposition 3.5. Th e g enerating function for F N ( p , q ) is the Stieltjes-type continue d f raction, namely F ( p , q , x ) : = ∞ ∑ N = 0 F N ( p , q ) x N = 1 1 − x [ 1 ] p , q 1 − x [ 2 ] p , q 1 − x [ 3 ] p , q 1 − · · · , F 0 ( p , q ) : = 1 . Proof. First consider the set E : = { 1 = e 1 < · · · < e N : e k ≤ 2 k − 1 } . Note that E can be decompo sed into d istinct subsets when e n = 2 n − 1. One has E = N − 1 [ n = 0  E n ∪ ˆ E n  , where E n : = { 1 = e 1 < · · · < e n : e k ≤ 2 k − 1 } for n 6 = 0 c an be vie wed as the n -truncates of th e original s et E , E 0 = ∅ , and ˆ E n : = { 2 n + 1 = e n + 1 < · · · < e N : 2 n + k ≤ e n + k < 2 ( n + k ) − 1 } . The set ˆ E n can be re-expressed as E ′ N − n = { 1 = e ′ 1 < · · · < e ′ N − n : k ≤ e ′ k < 2 k − 1 } , by shifting and relabeling the indices as e n + k : = e ′ k + 2 n . Note howe ver th at E ′ N − n (with all e ′ k < 2 k − 1) is not the sam e as E N − n (with all e k ≤ 2 k − 1). From Eq. (3.3), (3.4) F N ( p , q ) = ∑ { E } N ∏ k = 1 [ 2 k − e k ] p , q = N − 1 ∑ n = 0 F n ( p , q ) C N − n ( p , q ) , 10 where C n ( p , q ) = ∑ { E ′ n } n ∏ k = 1 [ 2 k − e ′ k ] p , q . I ntroduce the po we r series F ( p , q , x ) = ∞ ∑ N = 0 F N ( p , q ) x N , F 0 ( p , q ) : = 1 , C ( p , q , x ) = ∞ ∑ N = 1 C N ( p , q ) x N . Using Eq. (3.4) in the power series, one finds that F ( p , q , x ) − 1 equals the prod uct F ( p , q , x ) C ( p , q , x ) , which imp lies (3.5) F ( p , q , x ) = 1 1 − C ( p , q , x ) . Next, define the associated polynomials F n ( p , q ; l ) : = ∑ { E n } n ∏ k = 1 [ 2 k − e k + l ] p , q , C n ( p , q ; l ) : = ∑ { E ′ n } n ∏ k = 1 [ 2 k − e ′ k + l ] p , q , so that F n ( p , q ; 0 ) = F n ( p , q ) and C n ( p , q ; 0 ) = C n ( p , q ) . The corr esponding power series F ( p , q ; l , x ) and C ( p , q ; l , x ) are defined similarly to F ( p , q , x ) an d C ( p , q , x ) above, an d they also satisfy Eq.( 3.5 ). Furth ermore, for n > 1 th e associated polyno mials satisfy the relation C n ( p , q ; l ) = ∑ { E ′ n } n ∏ k = 1 [ 2 k − e ′ k + l ] p , q = [ l + 1 ] p , q ∑ { E ′ n } n ∏ k = 2 [ 2 k − e ′ k + l ] p , q (3.6) = [ l + 1 ] p , q ∑ { E n − 1 } n − 1 ∏ j = 1 [ 2 j − e j + ( l + 1 )] p , q = [ l + 1 ] p , q F n − 1 ( p , q ; l + 1 ) , (3.7) after an appropriate index shift, k = j + 1, and relabelings e ′ j + 1 = e j + 1 so that j ≤ e j ≤ 2 j − 1 for j = 1 , . . . n − 1. As a result, the set E ′ n changed to the set E n − 1 . The for mal power series constructed from the first an d last expressions in Eq.(3. 7 ) satisfies C ( p , q ; l , x ) = x [ l + 1 ] p , q F ( p , q ; l + 1 , x ) . From th e analogue of E q. (3.5) for the associated fu nctions F ( p , q ; l , x ) a nd C ( p , q ; l , x ) , one ther efore obtains F ( p , q ; l , x ) = 1 1 − [ l + 1 ] p , q xF ( p , q ; l + 1 , x ) . This yields the continu ed fraction representation for F ( p , q , x ) = F ( p , q ; 0 , x ) . W e ca n gra phically illustrate th e results of Prop osition 3.5 for N = 2 and 3 in terms of the cor responding cho rd diagrams. F o r N = 2, F 2 ( p , q ) = 1 + p + q , which implies that there is one each o f th e O-, T -, and P-type diag rams. These wer e display ed in Fig. 3.1. For N = 3, ther e are 15 chord diagrams, which are displayed in Fig.3.2. They are characterized by F 3 ( p , q ) = ( 1 + 2 p + p 2 + p 3 ) + ( 2 + 2 p + 2 p 2 ) q + ( 1 + 2 p ) q 2 + q 3 . Note that the total numb er of pairwise chord configuratio ns f or each c ase fo r N = 3 is 3(3-1)/2 = 3. W e use t he ordering ( p 1 p 2 , p 2 p 3 , p 3 p 1 ) fo r the chord-pairs, and indicate the interaction type for each pair belo w: (a) Non-reso nant cases: 1 ( PPP)-type, 1 (PPO)-type, 2 (POO) -type, 1 (OOO)-ty pe. These are on the first r o w in Fig.3.2, from left to right. (b) One-resonan t cases: 2 (TPP)-ty pe, 2 (TPO)-type, and 2 (TOO)-type. T hese are on the second ro w . (c) T wo-re sonant cases: 2 (TTP)-type and 1 (TTO)-type. T hese are on the the third ro w . (d) Three- (i.e., fully-) resonant case: 1 (TTT)- type. 11 Figure 3.2: The closed chord diagrams for 3-soliton solutions. T he dots indicate the piv ots ( e 1 , e 2 , e 3 ) , and the ordered letters belo w each diagram indicate the type of in teractions in ( p 1 p 2 , p 2 p 3 , p 3 p 1 ) with the soliton pa iring p n = [ e n , g n ] . The number of the diagrams ha ving the same number of crossings comes from the generating function F 3 ( q ) = q 3 + 3 q 2 + 6 q + 5. E.g., 5, the Catalan number C 3 = F 3 ( 0 ) , counts the diagrams in the first row . 3.3. Generating functio n and q - orthogonal po lynomials. In the special case wh en p = 1, the formu la fo r F ( q , x ) : = F ( 1 , q , x ) in Prop osition 3.5 reduc es to similar con tinued fraction expression for F ( q , x ) : = F ( 1 , q , x ) , namely (3.8) F ( q , x ) = ∞ ∑ N = 0 F N ( q ) x N = 1 1 − x [ 1 ] q 1 − x [ 2 ] q 1 − x [ 3 ] q 1 − · · · , F 0 ( q ) : = 1 , with [ n ] q : = 1 + q + . . . + q n − 1 . From Corollary 3.4(iii), it follows that F ( q , x ) is the gen erating fun ction for the polyno mials F N ( q ) that enumera te the c rossings of th e c hords of M N , whose explicit for mula is given by Eq. (2. 1) . Here we show that th e continu ed fraction f or F ( q , x ) is r elated to the mo ment generatin g function for the con tinuous q -Hermite poly nomials. I t follows that the T o uchard-Riord an po lynomials F N ( q ) are simply the even momen ts of the weight fun ction with resp ect to which the q -Herm ite polyn omials are o rthogonal. The latter result was also found in Ref. [ 17 ]. W e first collect some facts (see e.g. [ 9, 21 ]) from the s pectral th eory of bo unded, real, semi-infinite Jacobi matrices on the Hilbert space l 2 ( C ) : = { u = ( u 0 , u 1 , u 2 , . . . ) : u i ∈ C , ∞ ∑ k = 0 | u k | 2 < ∞ } . 12 Define the following tri-diagonal matrices L : =       0 1 0 · · · a 1 0 1 · · · 0 a 2 0 . . . . . . . . . . . . . . .       L n : =        0 1 0 · · · 0 a 1 0 1 · · · 0 . . . . . . . . . . . . . . . 0 · · · a n − 2 0 1 0 · · · 0 a n − 1 0        b L n − 1 : =        0 1 0 · · · 0 a 2 0 1 · · · 0 . . . . . . . . . . . . . . . 0 · · · a n − 2 0 1 0 · · · 0 a n − 1 0        , with a i > 0 , i = 1 , 2 , . . . . Next, consider the linear system of equations ( λ I − L ) ϕ = e 0 , f or ϕ = ( ϕ 0 , ϕ 1 , ϕ 2 , . . . ) T , where I : = diag ( 1 , 1 , . . . ) is the semi-in finite identity matrix and e 0 = ( 1 , 0 , 0 , . . . ) ∈ l 2 ( C ) . Fact (a) ( Resolven t of Jacobi matrix ): T he ( 0 , 0 ) -element of the resolvent of L , i.e., ϕ 0 = h e 0 , ( λ I − L ) − 1 e 0 i , can be developed in a continued fraction by re writing the linear system of equation s as follo ws: ϕ 0 = 1 λ − ϕ 1 ϕ 0 and ϕ n ϕ n − 1 = a n λ − ϕ n + 1 ϕ n , n ≥ 1 . Thus one has (3.9) ϕ 0 = 1 λ − a 1 λ − a 2 λ − a 3 λ − · · · . For n ≥ 0, the n th conv e rgent of this co ntinued fr action is of the form R n = N n D n , wh ere N 0 = 0 , D 0 = 1 and N n ( λ ) , D n ( λ ) n ≥ 1, are polyno mials in λ of degrees n − 1 and n , resp ecti vely . These seque nces of polynomials satis fy the 3-term recur- rence relation (3.10) λ P n = a n P n − 1 + P n + 1 , P n : = ( N n , D n ) , with P 0 = ( 0 , 1 ) an d P 1 = ( 1 , λ ) as respectiv e initial cond itions. Furtherm ore, it can be shown by Cram ´ er’ s rule that for n ≥ 2, D n = det ( λ I n − L n ) , N n = det ( λ I n − 1 − b L n − 1 ) , such that N n D n = h e ′ 0 , ( λ I n − L n ) − 1 e ′ 0 i , where I n is the n × n identity matrix and e ′ 0 = ( 1 , 0 . . . , 0 ) ∈ R n . Fact (b ) ( S pectral theorem ): If { a n } , n ≥ 1, is a positive boun ded sequence such that the Jacob i matrix L , defin ed above, is bounded on l 2 ( C ) , then there exists a unique spectral measure µ with compact support Σ such that (3.11) ϕ 0 = lim n → ∞ N n D n = h e 0 , ( λ I − L ) − 1 e 0 i = Z Σ d µ ( s ) λ − s , f or λ / ∈ Σ . Furthermo re, the polynomials { N n , D n } , n ≥ 1, are orthogonal with respect to th e m easure µ . In particular { D n } , n ≥ 1 form a sequence of monic polyno mials satisfy ing the orthogonality relations Z Σ D m ( s ) D n ( s ) d µ ( s ) = α n δ mn , with α n = n ∏ j = 1 a j . If we now set a n = [ n ] q for n ≥ 1 in th e contin ued fraction representation for ϕ 0 in E q. (3.9), and compar e th e resulting expression with the generating function in Eq. (3.8), we find that F ( q , x = λ − 2 ) = ∞ ∑ N = 0 F N ( q ) λ 2 N = λϕ 0 ( λ ) = ∞ ∑ k = 0 1 λ k Z Σ s k d µ ( s ) , where the last eq uality fo llo ws fro m Eq. (3 .11) . Note also fro m Eq. (3 .11) that the mo ments R Σ s k d µ ( s ) = h e 0 , L k e 0 i , k = 0 , 1 , 2 , . . . , clearly v an ish fo r an y odd k because of th e structure of the Jacobi m atrix L . Th erefore, we conclude that the 13 generating p olynomials F N ( q ) are given by the even mo ments of th e measu re µ . Moreover, Eq. (3. 10 ) with a n = [ n ] q is the well known 3- term recu rrence relation for the q -Hermite polynomials H n ( s , q ) (see e.g., [ 16 ]). Ind eed, it follows from the initials condition s that th e denominator polynomials above satis fy D n ( s ) = H n ( s , q ) for n = 0 , 1 , 2 , . . . . These polyno mials satisfy the orth ogonality relations Z a − a H m ( s , q ) H n ( s , q ) d µ ( s ) = [ n ] q ! δ mn , d µ ( s , q ) : = ν ( s , q ) d s , s ∈ [ − a , a ] , a : = 2 ( 1 − q ) − 1 / 2 , where | q | < 1 , and the weight function ν ( s , q ) is given in t erms of the Jacobi theta fun ction Θ 1 ( θ , q ) b y ν ( s , q ) = q − 1 8 π a Θ 1 ( θ , q ) = 2 π a ∞ ∑ n = 0 ( − 1 ) n q n ( n + 1 ) / 2 sin ( 2 n + 1 ) θ , cos θ : = s a = 1 2 ( 1 − q ) 1 / 2 s . Note that ν ( s , q ) is an even fun ction in s (i.e ., it is stable und er θ → π − θ , wher e θ ∈ [ − π , π ] ), so the od d mo ments vanish as observed above. The even moments for the q -Hermite polynomials are gi ven by F N ( q ) = Z a − a x 2 N ν ( q , x ) d x = 2 2 N π ( 1 − q ) N ∞ ∑ n = 0 ( − 1 ) n q n ( n + 1 ) / 2 Z π 0 cos 2 N θ ( cos 2 n θ − cos 2 ( n + 1 ) θ ) d θ , which yields the T o uchard-Riord an formu la, after e valuating the last integral and rearranging the summation indices. Remark 3.6 It is in triguing to note that th e generating function F ( p , q , x ) that enu merates the interaction typ es of the N -solito n solution of the KPII eq uation has its orig in in the theo ry of q -orthog onal polynomials. W e mentio n another relation with orth ogonal polyno mials, withou t p resenting the details. Conside r the c ase p = 0. It turns out that th e generating fun ction F ( 0 , q , x ) for the no n-nesting cho rd d iagrams is related to the m oment generating function for a certain class of q -orthogo nal p olynomials studied in Ref. [ 2 ] (see also [ 16 ]). A particularly i nteresting consequence of this relation is that the function F ( 0 , q , − q ) has a Rogers-Ramanujan interpretation. L et ϕ 1 and ϕ 2 be certain modular forms of weight 1 5 for the lev el-5 principal modular group Γ ( 5 ) < PS L ( 2 , Z ) ; namely , (3.12) ϕ 1 ( q ) = 1 η ( q ) 3 / 5 ∑ n ∈ Z ( − 1 ) n q ( 10 n + 1 ) 2 / 40 , ϕ 2 ( q ) = 1 η ( q ) 3 / 5 ∑ n ∈ Z ( − 1 ) n q ( 10 n + 3 ) 2 / 40 , where η ( q ) = q 1 / 24 ∞ ∏ n = 1 ( 1 − q n ) is the Dedekin d η -function. I t is well-known that the modu lar forms ϕ 1 and ϕ 2 admit infinite pr oduct rep resentations, wh ich constitute the Roger s-Ramunajan id entities. Accordingly , F ( 0 , q , − q ) can be represented as a quotient: F ( 0 , q , − q ) = q − 1 / 5 ϕ 2 ϕ 1 = ∞ ∏ n = 0 ( 1 − q 5 n + 1 )( 1 − q 5 n + 4 ) ( 1 − q 5 n + 2 )( 1 − q 5 n + 3 ) = 1 1 + q 1 + q 2 1 + q 3 1 + · · · . This is the Rogers-Raman ujan co ntinued fraction [ 27, 29 ]. 4. Conclusion W e have presented a classification schem e for th e N -solito n solution s of th e KPII e quation, based on th e co mbi- natorics o f cho rd dia grams consisting of N cho rds connec ting d istinct pairs of 2 N po ints. W e have shown that it is possible to associate the O -, T -, and P-type of p airwise interactio n pattern s amon g the N asympto tic line solitons o f the N -solito n configuration with the alignments, crossings and nestings among pairs of chords in the chord diagram. As a result, th e equiv alence classes of N -solito n solutions can be en umerated by the same g enerating po lynomial F N ( p , q ) (Eq. (3 .3 ) ) of the distribution of n estings and cr ossings for th e set M N of all ch ord diagra ms of [ 2 N ] . It f ollo ws fro m 14 Proposition s 2.5 and 2.8 that ea ch asympto tic line so liton of a given N -soliton solu tion is u niquely iden tified with an index pair p n = [ e n , g n ] , 1 ≤ e n < g n ≤ 2 N , which also lab els a particular chor d in a chord diagram . This pair ing map (Eq. (2.7)) plays a crucial rule in establishing a corresponde nce betwe en an N -soliton equiv alen ce clas s and a particu- lar cho rd diagram of M N . The soliton p airing en coded in Prop osition 2.5 can be d eri ved via a systematic asym ptotic analysis (for fixed t ) of the N -soliton τ -functio n, which is sum of r eal exponen tials with positi ve co ef ficien ts, by identi- fying t hose phase co mbinations Θ ( m 1 , . . . , m N ) that are do minant in different regions of the xy - plane as | y | → ∞ . Since a d iscussion o f the asy mptotics o f th e τ -fun ction is beyon d the main f ocus of the present article, it has been om itted here. 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Y ag lom, Challenging Mathematical Pr oblems W ith Elementary Solutions , V ol. I: Combinatorial Analysis and Probability Theory , Holden-Day , San Francisco, CA, 1964 . D E PA RT M E N T O F M A T H E M ATI C S , U N I V E R S I T Y O F C O L O R A D O , C O L O R A D O S P R I N G S , C O 8 0 9 3 3 E-mail addre ss : chuck@math.uc cs.edu D E PA RT M E N T O F M A T H E M ATI C S , O H I O S TA T E U N I V E R S I T Y , C O L U M B U S , O H 4 3 2 1 0 E-mail addre ss : kodama@math.o hio-stat e.edu 16