Inverse scattering transform for N-wave interaction problem with a dispersive term in two spatial dimensions
aa r X i v : . [ n li n . S I] A ug INVERSE SCATTERING TRANSFORM FOR N-WAVEINTERACTION PROBLEM WITH A DISPERSIVE TERM INTWO SPATIAL DIMENSIONS
MANSUR I ISMAILOV ∗ ) , ∗∗ ) Abstract.
In this work, we introduce a dispersive N -wave interaction prob-lem ( N = 2 n, n ∈ N ) involving n velocities in two spatial dimensions andone temporal dimension. Exact solutions of the problem are exhibited. Thisis a generalization of the N -wave interaction problem and matrix Davey-Stewartson equation with 2+1 dimensions that examines the Benney-typemodel of interactions between short and long waves. Accordingly, associ-ated with the solutions of two dimensional analog of the Manakov system, aGelfand-Levitan-Marchenko (GLM)-type, or so-called inversion-like, equationis constructed. It is shown that the presence of the degenerate kernel readsexact soliton-like solutions of the dispersive N -wave interaction problem.Wealso mention the unique solution of the Cauchy problem on an arbitrary timeinterval for small initial data. Introduction
The inverse scattering transform (IST) for nonlinear evolution equations with2+1, i.e., two spatial and one temporal dimensions has been started with the pa-pers by Zakharov and Shabat [18, 19]. For the general case of evolutional partialdifferential equations in 2+1 dimensions the IST requires a novel approach, namelyeither a nonlocal Riemann–Hilbert (RH) [3] or a ∂ -bar formalism [1, 4], howeverfor certain nonlinear two-dimensional equations, the classical approach of the ISTvia the GLM equation is still applicable [2, 11, 12]. The IST can be employedto the initial value problem for a variety of physically significant equations whichare related to the inverse scattering problem for first order systems of partial dif-ferential equations. Concrete results with a wide class of the exact solutions forvarious forms of Davey–Stewartson, Kadomtsev–Petviashvili equations and the N -wave interaction in 2 + 1 dimension, has been obtained in [7, 13] on the basis ofthe analysis of GLM type integral equations, in [5, 17] on the basis of the nonlocalRiemann–Hilbert problem and in [6] via the ∂ -bar method.This paper considers the two-dimensional spatial dispersive 2 n -wave interactionproblem with n velocities which is generalized the N − wave interaction problem of[7] and two component Davey–Stewartson equation of [13]. This nonlinear equationadmits a Lax-type representation. Therefore we use the IST via the GLM equationfor its integration. As the inverse problem we set the two dimensional inverse-scattering problem for the following Manakov system, studied in detail in [8]: Date : 20, August, 2020.2000
Mathematics Subject Classification.
Primary 37K15; Secondary 35L50, 35R30, 35Q58.
Key words and phrases.
Inverse scattering method; Manakov-type system; Dispersive N-waveinteraction problem. ∗ ) , ∗∗ ) (1.1) ∂∂y ψ − σ ∂∂x ψ + Q ( x, y ) ψ = , where σ = (cid:18) I n n × × n − (cid:19) is constant n × n diagonal matrix with the identitymatrix I n of the order n and n × n × and 1 × n row × n zero vectors; Q = (cid:18) n q q (cid:19) is an off-diagonal matrix with the zero matrix entry n of theorder n and the n × q and 1 × n row q vector functions.The case n = 1 makes this system two component nonstationary Dirac system[14] (and also two dimesional analogue of Zakharov-Shabat (ZS) or AKNS system[15]). In the case n = 2 this system is two dimesional analogue of Manakov system[10] and for arbitrary positive integer n is the two dimesional analogue of Dirac-type sysytem [16]. Inverse scattering theory for the system (1.1) is satisfactorylyinvestigated in [14] for the case n = 1 and in [8] for arbitrary n >
1. This paperuse this inverse problem to solve dispersive 2 n -wave interaction problem with n velocities satisfied by additionally time dependent Q ( x, y ; t ), which is generalized2 n -wave interaction with n velocities and n × n matrix Davey-Stewartson equationin two spatial and one temporal dimensions.This article is organized as follows: In Section 2, the dispersive 2 n -wave inter-action problem with n velocities and its Lax representation is introduced. It wasclear that its spectral problem is the problem two dimensional analogue of Manakovsystem (1.1) that the inverse scattering problem is studied in detail in [8]. Section3, deals with the inverse problem associated with the linear equation (1.1) and thecorresponding multidimensional GLM equation with explicitly solvable degeneratekernel case. The aim of the Section 4 is to apply the results of [8] to the integrationof the dispersive 2n-wave interaction problem with n velocities by using the ISTmethod: The Cauchy problem is investigated the exact soliton-like solutions arederived.2. Dispersive 2n-wave interaction problem with n velocities and its Laxrepresentation
Consider the system of N − wave equations with N = 2 n in the following form: ∂∂t q k + α k ∂∂x q k + β k ∂∂y q k − iγ ∂ ∂x∂y q k = n X m =1 p km q m − pq k , (2.1) ∂∂t q n + k + α k ∂∂x q n + k + β k ∂∂y q n + k + iγ ∂ ∂x∂y q n + k = − n X m =1 p mk q n + m + pq n + k ,k = 1 , ..., n where α k , β k and γ are real numbers with β k − α k = β j − α j and β k + α k = β j + α j when k = j . The functions p km and p are solutions of the equations ∂∂ξ p = − iγ n X m =1 ∂∂x ( q m q n + m ) , ∂∂η p kk = − iγ ∂∂x ( q k q n + k ) , k = 1 , ..., n,∂∂η p km = ( β m − β k ) q k q n + m − iγ ∂∂x ( q k q n + m ) , m, k = 1 , ..., n ; m = k. (2.2) NVERSE SCATTERING TRANSFORM FOR N-WAVE INTERACTION PROBLEM 3 where ∂∂ξ = ∂∂y + ∂∂x , ∂∂η = ∂∂y − ∂∂x . This system (2.1) is the 2+1 dimensional N -wave interaction problem with thedispersive term iγ ∂ ∂x∂y q k and also with the quasi-potential (2.2). The aim of thispaper is the integrability of this system by using the suitable method of inversescattering transform. The case when the terms α k ∂∂x q k + β k ∂∂y q k absence theequation (2.1) becomes to 2+1 dimensional nonlinear Schrodinger equation andthe IST method for its integration is realized in [13] for n = 1 by using the inversescattering problem (ISP) for two component nonstationary Dirac equation, [14].The undispresive system (2.1) when γ = 0, can be integrate by using IST methodin [7, 9] which the integrate ISP is matrix nonstationary Dirac system of n + 1components with n >
1, [8].Generally, in physical systems, waves with different length scales appear. Theyare examine the interactions between waves for certain model partial differentialequations. In Benney model [20], the equations (2.1), (2.2) examines the interac-tions between short and long waves, where p is the long wave profile and q is, toleading order, the short wave envelope. The constants α k and β k are the groupvelocities of the short waves, γ is due to the linear dispersion.Let us denote q = col { q , ..., q n } , q = col { q n +1 , ..., q n } T , P = ( p km ) nk,m =1 .Then, the equation (2.1) and (2.2) are reduced to the matrix system ∂ t q − B ∂ ξ q + b∂ η q − iγ ( q ) xy = Pq − p q , (2.3) ∂ t q − ∂ ξ q B + b∂ η q + iγ ( q ) xy = p q − q P , and ∂ η P = [ q q , B ] − iγ ( q q ) x , (2.4) ∂ ξ p = − iγ ( q q ) x , respectively, where ∂ t = ∂∂t , ∂ ξ = ∂∂y + ∂∂x , ∂ η = ∂∂y − ∂∂x , B = diag ( b , . . . , b n ) with b k = β k − α k and b = − β k + α k .Let M and A be first order matrix operators: M = σ ∂∂x + Q , A = δ ∂∂x + iγ I n ∂ ∂x + Γ . Then the equation (2.3), (2.4) admits the following Lax representation:(2.5) (cid:20) ∂∂y − M , ∂∂t − A (cid:21) = 0 . Here δ and Γ are n + 1-th ( n ≥
2) order square matrices. The matrix δ be a realand diagonal: δ = (cid:18) B 0 n × × n b (cid:19) , where B = diag ( b , . . . , b n ) with b k = b j = b when k = j and Γ is the following form Γ = (cid:18) P ( B − b I n ) q q ( B − b I n ) p (cid:19) that obey the relation [ σ , Γ ] = [ δ , Q ] + 2 iγ Q x .Let us denote L = ∂∂y − M and D = ∂∂t − A . Lemma 1.
Let ψ be a solution of the system (1.1), whose the coefficients q and q satisfy system (2.3). Then the function ϕ = D ψ also satisfy the system (1.1). MANSUR I ISMAILOV ∗ ) , ∗∗ ) Proof.
From (2.5) we obtain:( LD − DL ) ψ = L ( D ψ ) − D ( L ψ ) = . Since L ψ = 0, then L ( D ψ ) = 0 . It means that D ψ is solution of system (1.1). (cid:3) Manakov-type systems on the plane
Let us consider the system (1.1) on the plane −∞ < x, y < + ∞ with thematrix function q and q has measurable complex-valued rapidly decreasing(Schwartz) entries. Notice that if the potential is independent on y , then by taking ψ ( x, y ) = ψ ( x ) exp ( iλy ), we can convert equation (1.1) into the Manakov systemgiven by − σ ddx ψ ( x ) + Q ( x ) ψ ( x ) = iλ ψ ( x )which is considered in [10].Throughout this chapter the following notations will be used : ◮ We partition ( n + 1) × ( n + 1) matrix A as follows: A = (cid:18) A A A A (cid:19) where A is n × n matrix A is and n × A is 1 × n row vectorand A is scalar. ◮ ̥ x denotes the ( n + 1) × ( n + 1) diagonal matrix shift operator, such that fora n + 1 dimensional vector function h ( t ) ̥ x h ( y ) = (cid:18) h ( y + x ) h ( y − x ) (cid:19) where h ( y ) is a vector function that consists of the first n component of vector h ( y ) , h ( y ) is a scalar function. ◮ We denote A ± ( x ) h ( x, y ) = ∓ Z ∓∞ y A ± ( x, y, τ ) h ( x, τ ) dτ by the upper and lower Volterra integral operators. For any a ± ( y ) ∈ L (cid:0) R , C n +1 (cid:1) there exist unique solutions in L (cid:0) R , C n +1 (cid:1) ofthe systems (1.1) with the conditions ψ ( x, y ) = ̥ x a ± ( y ) + o (1) , y −→ ±∞ andthese solutions admit the representations (3.1) ψ ( x, y ) = ( I + A ± ( x )) ̥ x a ∓ ( y ) , where I is identity operator and the kernels A ± ( x, y, τ ) = (cid:18) A ± ( x, y, τ ) A ± ( x, y, τ ) A ± ( x, y, τ ) A ± ( x, y, τ ) (cid:19) of the integral operators A ± ( x ) are uniquely determined by the coefficients of thesystem (1.1), and for the fixed x , these kernels are the Hilbert-Schmidt kernels. Inaddition, these kernels are connected with the potential by the following equalities NVERSE SCATTERING TRANSFORM FOR N-WAVE INTERACTION PROBLEM 5 (3.2) A ± ( x, y, y ) = ± q ( x, y ) , A ± ( x, y, y ) = ∓ q ( x, y ) . An operator S transforming the given incident waves a − ( y ) ∈ L (cid:0) R , C n +1 (cid:1) intothe scattered waves a + ( y ) ∈ L (cid:0) R , C n +1 (cid:1) is called the scattering operator for thesystem (1) on the plane: a + ( y ) = Sa − ( y )where a + ( y ) = a − ( y )+ R + ∞−∞ ̥ y − x − s ( Q ψ ) ( x, s ) ds . Operator S is ( n + 1) × ( n + 1)matrix linear operator on the space L (cid:0) R , C n +1 (cid:1) .From the representations (3.1) it follows that the next factorization results for S . For every x, the operator ̥ x S ̥ − x admits factorizations(3.3) ̥ x S ̥ − x = ( I + A − ( x )) − ( I + A + ( x )) , We can analogously introduce the next representations corresponding to asymp-totics ψ ( x, y ) = ̥ x b − ( y )+ o (1) , x −→ −∞ and ψ ( x, y ) = ̥ x b + ( y )+ o (1) , x −→ + ∞ . For any b ± ( y ) ∈ L ( R , C n ) there exist unique solutions in L (cid:0) R , C n (cid:1) of thesystems () and these solutions admit the representations (3.4) ψ ( x, y ) = ( I + B ± ( y )) ̥ x b ∓ ( y ) , where the kernels B ± ( x, y, τ ) = (cid:18) B ± ( x, y, τ ) B ± ( x, y, τ ) B ± ( x, y, τ ) B ± ( x, y, τ ) (cid:19) of the integraloperators B ± ( x ) are uniquely determined by the coefficients of the system (1.1)and these kernels are the Hilbert-Schmidt kernels for fixed x . In addition, thesekernels are connected with the potential by the following equalities (3.5) B ± ( x, y, x ) = ± q ( x, y ) , B ± ( x, y, x ) = ∓ q ( x, y ) . From the representation (3.4) follows the next factorization results for S . Forevery x, the operator ̥ x S ̥ − x admits factorizations(3.6) ̥ x S ̥ − x = ( I + K + ( x )) − ( I + K − ( x ))where matrix elements of the kernel of matrix integral operator K ± ( x ) are deter-mined by B ± ( y ) as follows K ± i ( x, y, τ ) = B ± i ( x, y, x − y + τ ) , (3.7) K ± i ( x, y, τ ) = B ± i ( x, y, x + y − τ ) , i = 1 , . It is possible the unique restoration of the potential by scattering operator.Let S be the scattering operator for the system (1.1) on the plane with thepotential Q ( x, y ) belonging to the Schwartz class. Then the potential Q ( x, y ) isuniquely determined by the known scattering operator S . The ISP is solved withthe following steps:1) Construct the operator ̥ x S ̥ − x ;2) Find the factorization factors A − ( x ) and A + ( x ) from the (3.3), since ̥ x S ̥ − x admits left factorization;3) Find the matrix coefficients of the system (1.1) with respect to the kernels A ± ( x, y, τ ) of the operators A ± ( x ), by formula (3.2). MANSUR I ISMAILOV ∗ ) , ∗∗ ) Let us denote the kernels of the matrix integral operators S − I and S − − I as F ( y, τ ) and G ( y, τ ), respectively. Let F ( y, τ ) = (cid:18) F ( y, τ ) F ( y, τ ) F ( y, τ ) F ( y, τ ) (cid:19) and G ( t, τ ) = (cid:18) G ( y, τ ) G ( y, τ ) G ( y, τ ) G ( y, τ ) (cid:19) . We will call the collection of functions { F ( y, τ ) , G ( y, τ ) } as the scattering data for the system (1.1) . Let us denote the kernels of the matrices ̥ x S ̥ − x − I and ̥ x S − ̥ − x − I as F ( x, y, τ ) and G ( x, y, τ ), respectively. It is clear that F (0 , y, τ ) = F ( y, τ ) and G (0 , y, τ ) = G ( y, τ ).Therefore, we obtain the following Gelfand- Levitan-Marchenko type matrix in-tegral equations from (3.6). K − ( x, y, τ ) − Z y −∞ (cid:20)Z + ∞ y K − ( x, y, z ) G ( z − x, s + x ) dz (cid:21) F ( s + x, τ − x ) ds = F ( y + x, τ − x ) , τ ≥ y, (3.8) K +21 ( x, y, τ ) − Z + ∞ y (cid:20)Z y −∞ K +21 ( x, y, z ) F ( z + x, s − x ) dz (cid:21) G ( s − x, τ + x ) ds = G ( y − x, τ + x ) , τ ≤ y, By the right factorization (3.6) of ̥ x S ̥ − x there exist unique solutions of theseequations.Considering the relationships (3.5) between the potential Q ( x, t ) and the oper-ators B ± ( y ), and also (3.7) between the operators B ± ( y ) and K ± ( x ) we obtainthe following results for the ISP in the plane: Let S = I + F be the scattering operator for the system (1.1) on the whole plane.Then there exists S − = I + G , where F and G are the Hilbert-Schmidt matrixintegral operators. Let us partition F = ( F ij ) i,j =1 , G = ( G ij ) i,j =1 and let thekernels of the operators F and G be given. Then there exists a unique solutionof the system of integral equations (3.8) and the solution of this system determinesthe potential by formulae (3.9) q ( x, y ) = − K − ( x, y, y ) , q ( x, y ) = − K +21 ( x, y, y ) . Thus, for the system (1.1) with the coefficient Q ( x, y ) there is scattering operator S with the scattering data F and G which are the Hilbert-Schmidt integraloperators with the kernels F ( x, y ) and G ( x, y ) that decrease quite fast withrespect to variables at infinity. This defines the mapping of the scattering data (cid:8) q ( x, y ) , q T ( x, y ) (cid:9) Π → (cid:8) F ( x, y ) , G T ( x, y ) (cid:9) . This operator mapping coefficients of the system (1.1) into the scattering data iscontinuous in L and its inverse Π − exists and is continuous and its action can beconstructively described by means of the uniquely solvable of systems (3.8). NVERSE SCATTERING TRANSFORM FOR N-WAVE INTERACTION PROBLEM 7 Inverse scattering method
To integrate the Cauchy problem for the system (2.1) with the initial condition(4.1) q k ( x, y, t ) | t =0 = q k ( x, y ) , k = 1 , ..., n by the inverse scattering method. We use the ISP for the system (1.1) with thepotential Q ( x, y ) = (cid:18) n q ( x, y ) q ( x, y ) 0 (cid:19) , where q = col (cid:8) q , ..., q n (cid:9) , q =col (cid:8) q n +1 , ..., q n (cid:9) T in the whole plane, that is given in Chapter 3. Let F ( x, y ) and G ( x, y ) are the scattering data for the system (1.1) with the coefficient Q ( x, y )which decrease quite fast with respect to variables at infinity. This defines themapping of the scattering data q = (cid:20) q (cid:0) q (cid:1) T (cid:21) Π → (cid:20) F (cid:0) G (cid:1) T (cid:21) . Let us investigate the evolution of this scattering data, when the coefficients ofthe operator L satisfy the equations (2.3).The pair { F , G } is denoted the scattering data correspond to operator L withthe coefficients q ( x, y ; t ) and q ( x, y ; t ) which are satisfy the system of equation(2.3). Theorem 1.
Let the coefficients q and q of the system (1.1) depend on t as aparameter and satisfy the system of equation (2.3). Besides that P ( x, + ∞ ) = , p ( x, −∞ ) = 0 − . Then the kernels F ( y, τ ; t ) , G ( y, τ ; t ) of the integral operators F , G corre-sponding to the scattering operator S for the system (1.1) on the plane satisfy thesystem of equations (4.5) and (4.6).Proof. By virtue of definition of the scattering operator S we get(4.2) ϕ + = S ϕ − , where ϕ ± = P ± a ± , P ± = ∂∂t − δσ ∂∂y − iγ I n ∂ ∂y , P ( x, ±∞ ) = , p ( x, ±∞ ) = 0.Since a + = Sa − , from (4.2) we obtain:(4.3) P + S = SP − . Analogously,(4.4) P − S − = S − P + . Since S = I + F and S − = I + G , where F f ( y ) = + ∞ Z −∞ F ( y, τ ; t ) f ( τ ) dτ , F ( y, τ ; t ) =( F ij ( y, τ ; t )) i,j =1 and G f ( y ) = + ∞ Z −∞ G ( y, τ ; t ) f ( τ ) dτ , G ( y, τ ; t ) = ( G ij ( y, τ ; t )) i,j =1 , from the matrix operator equation (4.3) it follows that the kernels of the integraloperator F satisfy the equation(4.5) ∂∂t F − (cid:18) B ∂∂y F − b ∂∂τ F (cid:19) − iγ (cid:18) ∂ ∂y F − ∂ ∂τ F (cid:19) = , MANSUR I ISMAILOV ∗ ) , ∗∗ ) The similarly equation for the kernels of the integral operator G follows from thematrix operator equation (4.4):(4.6) ∂∂t G + 2 (cid:18) b ∂∂y G − ∂∂τ G B (cid:19) + iγ (cid:18) ∂ ∂y G − ∂ ∂τ G (cid:19) = 0 . (cid:3) Now, let us give a procedure for the solution of the system (2.3) by inversescattering method.
Theorem 2.
Let functions F ( y, τ ; t ) and G ( y, τ ; t ) satisfy the equations (4.5)and (4.6) and these functions together with their derivatives with respect to t andtheir first and second derivatives with respect to y and τ belong to L ( R ) . Thenthe equations (4.7) are uniquely solvable and the functions q ( x, y ; t ) = − K − ( x, y, y ; t ) , q ( x, y ; t ) = − K + ( x, y, y ; t ) , is the solution of the non linear equation (2.3).Proof. If the coefficients of the system (1.1) depend on t as a parameter and satisfythe system of equation (2.3), then the kernels F ( y, τ ; t ) , G ( y, τ ; t ) of the integraloperators F , G satisfy the system of equations (4.5), (4.6). In addition, thecoefficients q , q of the system (1.1) is uniquely determined by (3.9) and theanalogues of the equations (3.8) K − ( x, y, τ ; t ) − R y −∞ K − ( x, y, z ; t ) (cid:16)R + ∞ y G ( z − x, s + x ; t ) F ( s + x, τ − x ; t ) ds (cid:17) dz = F ( y + x, τ − x ; t ) , τ ≥ y, (4.7) K + ( x, y, τ ; t ) − R + ∞ y K + ( x, y, z ; t ) (cid:16)R y −∞ F ( z + x, s − x ; t ) G ( s − x, τ + x ; t ) ds (cid:17) dz = G ( y − x, τ + x ; t ) , τ ≤ y, which are constructed by kernels F ( y, τ ; t ) , G ( y, τ ; t ) are uniquely solved by K − ( x, y, τ ) , K + ( x, y, τ ) (cid:3) The statement of this theorem is equivalent to the assertion that the function(4.8) q = (cid:20) q ( q ) T (cid:21) = Π − e − i A t Πq is the solution of equation (2.3) with the initial condition (4.1) if it is determined,where A = (cid:20) A n n A (cid:21) , A = 2 i (cid:16) B ∂∂y − b I n ∂∂τ (cid:17) − γ I n (cid:16) ∂ ∂y − ∂ ∂τ (cid:17) A = 2 i (cid:16) B ∂∂τ − b I n ∂∂y (cid:17) + γ I n (cid:16) ∂ ∂y − ∂ ∂τ (cid:17) . It is well known that Πq = (cid:20) F (cid:0) G (cid:1) T (cid:21) is direct problem of determining scatteringdata, e − i A t Πq is the evolution of scattering data and Π − e − i A t Πq is the inversescattering problem of finding q .The function q ( x, y ; t ) these functions together with their derivatives with respectto t and their first and second derivatives with respect to x and y belong to L ( R )is called the solution of Cauchy problem (1.1), (4.1) if at t = 0 it coincides with theinitial data q ( x, y,
0) = q ( x, y ). NVERSE SCATTERING TRANSFORM FOR N-WAVE INTERACTION PROBLEM 9
Theorem 3.
The solution of Cauchy problem (1.1), (1.4) is unique. The solutionof this Cauchy problem exists on an arbitrary interval of time for small initial data.Proof.
The uniqueness of the solution follows from the possibility of representingit in the form (4.8) that the assumption of the existence of solution requires therepresentation (4.8) which is expressed by initial data. If the initial data q aresufficiently small in (4.8) then this formula has a sense by virtue of continuity of Π and unitarity of e − i A t that (cid:13)(cid:13) e − i A t Πq (cid:13)(cid:13) is less than 1. (cid:3) Exact soliton-like solutions of the dispersive -wave interactionproblem Let n = 2 in (1.1) and(5.1) Q = (cid:18) q q (cid:19) , q = (cid:18) q q (cid:19) , q = (cid:0) q q (cid:1) . It is easy to see that the scattering data corresponding to the potential (5.1) is inthe form of F ( y, τ ; t ) = (cid:18) f ( y, τ ; t ) f ( y, τ ; t ) (cid:19) , G ( y, τ ; t ) = (cid:0) g ( y, τ ; t ) g ( y, τ ; t ) (cid:1) . The nonlinear system of equations (2.1) becomes to the form ∂ t q + α ∂ x q + β ∂ y q − iγ∂ xy q = ( p − p ) q + p q ,∂ t q + α ∂ x q + β ∂ y q − iγ∂ xy q = p q + ( p − p ) q , (5.2) ∂ t q + α ∂ x q + β ∂ y q + iγ∂ xy q = ( p − p ) q − p q ,∂ t q + α ∂ x q + β ∂ y q + iγ∂ xy q = ( p − p ) q − p q , where ∂∂ξ p = − iγ ∂∂x ( q q ) − iγ ∂∂x ( q q ) ,∂∂η p = − iγ ∂∂x ( q q ) , ∂∂η p = − iγ ∂∂x ( q q ) , (5.3) ∂∂η p km = ( β m − β k ) q k q m − iγ ∂∂x ( q k q m ) , m, k = 1 , m = k. In the case P + ( x ) = p − ( x ) = 0, these derivatives comes to form p = i γ q q ) + i γ q q ) − i γ ξ Z −∞ (cid:20) ∂∂η ( q q ) + ∂∂η ( q q ) (cid:21) ds,p = i γ q q ) − i γ + ∞ Z η ∂∂ζ ( q q ) dτ , p = i γ q q ) − i γ + ∞ Z η ∂∂ζ ( q q ) dτ , (5.4) p km = β m − β k q k q m + i γ q k q m ) − i γ + ∞ Z η ∂∂ζ ( q k q m ) dτ , m, k = 1 , m = k. ∗ ) , ∗∗ ) and after the elimination of p and p km , the system (5.2) represents a system ofintegro-differential equations.The evolution of the scattering data are in the following form according to (4.5)and (4.6): ∂ t f − b ∂ y f + 2 b∂ τ f − iγ (cid:0) ∂ y f − ∂ τ f (cid:1) = 0 ,∂ t f + 2 b ∂ y f − b∂ τ f − iγ (cid:0) ∂ y f − ∂ τ f (cid:1) = 0 , (5.5) ∂ t g + 2 b∂ y g − b ∂ τ g + iγ (cid:0) ∂ y f − ∂ τ f (cid:1) = 0 ,∂ t g + 2 b∂ y g − b ∂ τ g + iγ (cid:0) ∂ y f − ∂ τ f (cid:1) = 0 . We deduce explicit solutions of the system (5.2) by using the formulas for theexactly solvable case of the inverse-scattering problem for the system (1.1). Weget an elementary example for F ( y, τ ) = (cid:18) f ( y ; t ) f ( τ ; t ) f ( y ; t ) f ( τ ; t ) (cid:19) , G ( y, τ ) = (cid:0) g ( y ; t ) g ( τ ; t ) g ( y ; t ) g ( τ ; t ) (cid:1) , where the functions f k and g k satisfy theequations ∂ t f − b ∂ y f − iγ∂ y f = 0 , ∂ t f + 2 b∂ τ f + iγ∂ τ f = 0 ,∂ t f − b ∂ y f − iγ∂ y f = 0 , ∂ t f + 2 b∂ τ f + iγ∂ τ f = 0 , (5.6) ∂ t g + 2 b ∂ y g + iγ∂ y g = 0 , ∂ t g + 2 b ∂ τ g − iγ∂ τ g = 0 ,∂ t g + 2 b∂ y g + iγ∂ y g = 0 , ∂ t g − b ∂ τ g − iγ∂ τ g = 0 . Let K − = (cid:20) K − K − (cid:21) , K +21 = (cid:2) K +1 K +2 (cid:3) in (4.7) . Then K − ( x, y, τ ; t ) = a ( x, y ; t ) f ( τ − x ; t ) f ( y + x ; t ) + a ( x, y ; t ) f ( τ − x ; t ) f ( y + x ; t ) , (5.7) K − ( x, y, τ ; t ) = a ( x, y ; t ) f ( y + x ; t ) f ( τ − x ; t ) + a ( x, y ; t ) f ( y + x ) f ( τ − x ) , where a ( x, y ; t ) = 1 − α α − α α − α α + α α α α − α α α α ,a ( x, y ; t ) = α α − α α − α α + α α α α − α α α α ,a ( x, y ; t ) = α α − α α − α α + α α α α − α α α α ,a ( x, y ; t ) = 1 − α α − α α − α α + α α α α − α α α α with α = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds, α = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds,α = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds, α = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds, NVERSE SCATTERING TRANSFORM FOR N-WAVE INTERACTION PROBLEM 11 α = Z y −∞ g ( s + x ) f ( s + x ) ds, α = Z y −∞ g ( s + x ; t ) f ( s + x ; t ) ds,α = Z y −∞ g ( s + x ; t ) f ( s + x ; t ) ds, α = Z y −∞ g ( s + x ; t ) f ( s + x ; t ) ds and K +1 ( x, y, τ ; t ) = b ( x, y ; t ) g ( τ + x ; t ) g ( y − x ; t ) + b ( x, y ; t ) g ( τ + x ; t ) g ( y − x ; t ) , (5.8) K +2 ( x, y, τ ; t ) = b ( x, y ; t ) g ( τ + x ; t ) g ( y − x ; t ) + b ( x, y ; t ) g ( τ + x ; t ) g ( y − x ; t ) , where b ( x, y ; t ) = 1 − β β + β β β β − β β − β β + β β β β ,b ( x, y ; t ) = β β − β β − β β + β β β β ,b ( x, y ; t ) = β β − β β − β β + β β β β ,b ( x, y ; t ) = 1 − β β + β β β β − β β − β β + β β β β with β = Z y −∞ g ( s + x ; t ) f ( s + x ; t ) ds, β = Z y −∞ g ( s + x ; t ) f ( s + x ; t ) ds,β = Z y −∞ g ( s + x ; t ) f ( s + x ; t ) ds, β = Z y −∞ g ( s + x ; t ) f ( s + x ; t ) ds,β = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds, β = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds,β = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds, β = Z + ∞ y f ( s − x ; t ) g ( s − x ; t ) ds. Thus, the explicit solution of the equation (5.2) with the potential (5.1) havingthe scattering data in the degenerate form of (5.6), exists and it is in the followingform: q ( x, y, y ; t ) = − K − ( x, y, y ; t ) , q ( x, y, y ; t ) = − K − ( x, y, y ; t ) ,q ( x, y, y ; t ) = − K +1 ( x, y, y ; t ) , q ( x, y, y ; t ) = − K +2 ( x, y, y ; t ) , where K − ( x, y, τ ; t ) and K − ( x, y, τ ; t ) are determined by (5.7), K +1 ( x, y, τ ; t ) and K +2 ( x, y, τ ; t ) are determined by (5.8) . Conclusion
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