Ablowitz-Kaup-Newel-Segur Formalism and N-Soliton Solutions of Generalized Shallow Water Wave Equation

We apply Ablowitz-Kaup-Newel-Segur hierarchy to derive the generalized shallow water wave equation and we also investigate N−soliton solutions of the derived equation using Inverse Scattering Transform method and Hirota’s bilinear method. Introduction Studying integrable systems by the methods introduced by Zakharov and Shabat (ZS) [1] and by Ablowitz-Kaup-Newel-Segur (AKNS) [2] has been the focus of considerable attention through the past many years. These approaches basically deal with inverse scattering transform (IST) [3, 7, 8] and Lax-type conditions. By a simple extension of the Lax integrability condition the desired nonlinear evolution equation is generated. In recent times it is an accepted fact that existence of a Lax-pair is indeed a decisive trademark of integrable systems. ZS introduced this method to study Nonlinear Schrödinger equation and soon there after AKNS showed that one relatively minor modification of the ZS approach recovers the theory of the famous KdV equation [2], while another leads to an IST analysis for a well-known evolution equation, the Sine-Gordon equation [4]. The purpose of this work is to extend the well-known generalized shallow water wave (GSWW) equation in the variable coefficient form using the AKNS formalism and to investigate the novel N−soliton solutions of the derived equations with constant coefficients using IST andHirota’s bilinear method. The usual GSWW equation (that is with constant coefficient) reads [5, 6] uxxxt + αuxuxt + βutuxx − μuxt − ηuxx = 0, (1) where α, β ∈ R − {0}. Clarkson and Mansfield [6] showed some time ago, by investigating the Painlevé property that equation (1) is completely integrable if and only if (α−2β)(α−β) = 0. These correspond to the specific types α = β : uxxt + αuxut − ut − ux + f(t) = 0 , (2) α = 2β : uxxxt + αuxuxt + α 2 utuxx − μuxt − ηuxx = 0 , (3) which we call GSWWI equation and GSWWII equation respectively. In (2), f(t) is an arbitrary function of time t. Our concern will be to introduce variable coefficients in (2) and (3) in the manner uxxt + αuxut − η(t)ux − γut + f(t) = 0 (4) and uxxxt + α(t)uxuxt + α(t) 2 utuxx − μ(t)uxt − η(t)uxx = 0, (5) where α(t), β(t), μ(t) and η(t) are variable functions of time. The multi soliton solution of (4) and (5) will be derived using IST for (3) and Hirota’s bilinear method for (2). Bulletin of Mathematical Sciences and Applications Submitted: 2016-03-04 ISSN: 2278-9634, Vol. 20, pp 25-35 Revised: 2016-09-14 doi:10.18052/www.scipress.com/BMSA.20.25 Accepted: 2018-10-04 2018 SciPress Ltd, Switzerland Online: 2018-11-21 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ GSWW Equation with Variable Coefficients using AKNS Hierarchy and Explicit Soliton Solutions Case-I (α = 2β) : The linear eigenvalue problem is given by L[u]ψ(x, t) = λψ(x, t), (6) where λ = eigenvalue and ψ(x, t) evolve with time in a prescribed manner determined by ∂tψ(x, t) = A[u]ψ(x, t). (7) Consider the eigenvalue problem analogous to (6) ψx =Mψ; ψ = [ ψ1 ψ2 ] ; M = [ −iξ q(x, t) r(x, t) iξ ] . The time dependence of ψ analogous to (7) is given by ψt = Nψ; N = [ A B C D ] . The compatibility condition ψxxt = ψtxx gives Nxx −Mxt + 2NxM −MMt −MtM + [N,Mx] + [N,MM ] = Θ, (8) where [N,M ] = NM −MN and Θ is the null matrix of order 2× 2. Equation (8) can be split into the following equations Axx = qxC − rxB − 2rBx + (qr)t + 2iξAx , (9) Bxx = qxD − qxA− 2qAx − 2iξBx + qxt , (10) Cxx = rxA− rxD − 2rDx + 2iξCx + rxt , (11) Dxx = −qxC + rxB − 2qCx + (qr)t − 2iξDx . (12) Expanding A, B, C and D as


Introduction
Studying integrable systems by the methods introduced by Zakharov and Shabat (ZS) [1] and by Ablowitz-Kaup-Newel-Segur (AKNS) [2] has been the focus of considerable attention through the past many years. These approaches basically deal with inverse scattering transform (IST) [3,7,8] and Lax-type conditions. By a simple extension of the Lax integrability condition the desired nonlinear evolution equation is generated. In recent times it is an accepted fact that existence of a Lax-pair is indeed a decisive trademark of integrable systems. ZS introduced this method to study Nonlinear Schrödinger equation and soon there after AKNS showed that one relatively minor modification of the ZS approach recovers the theory of the famous KdV equation [2], while another leads to an IST analysis for a well-known evolution equation, the Sine-Gordon equation [4].
The purpose of this work is to extend the well-known generalized shallow water wave (GSWW) equation in the variable coefficient form using the AKNS formalism and to investigate the novel N −soliton solutions of the derived equations with constant coefficients using IST and Hirota's bilinear method.
The multi soliton solution of (4) and (5) will be derived using IST for (3) and Hirota's bilinear method for (2).

GSWW Equation with Variable Coefficients using AKNS Hierarchy and Explicit Soliton Solutions
Case-I (α = 2β) : The linear eigenvalue problem is given by where λ = eigenvalue and ψ(x, t) evolve with time in a prescribed manner determined by Consider the eigenvalue problem analogous to (6) The time dependence of ψ analogous to (7) is given by The compatibility condition ψ xxt = ψ txx gives Expanding A, B, C and D as we have from equations (9) to (12) For j = 0 equations (10) and (11) lead to the following equations
As |x| → ∞, the time evolution equation (17) reduces to We impose on ψ(x, t) the boundary conditions Equations (19), (20) and (21) implies For the standard type of scattering solutions we take where the reflection coefficient R(k, t) is given by The eigenfunctions ψ n (x, t), n = 1, 2, 3, ..., N , corresponding to the eigenvalues (22) satisfy the time evolution equations We have with the normalization condition The normalization constants C n (t) can be evaluated from (23), (24) and (25) as The scattering data S(t) corresponding to the potential v(x, t) evolves from S n (0) of the initial data v(x, 0) is given as From the scattering data S(t) in (26) at time t, one can invert the data and obtain uniquely v(x, t) of the Schrödinger type spectral problem (16), by solving the Gelfand-Levitan-Marchenko (GLM) integral equation [9,10] K(x, y, t)

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where the time variable t enters only as a parameter and Solving the GLM equation (26), one can recover the potential v(

One-soliton solution (N=1)
To evaluate one-soliton solution of GSWWII equation (3) we consider the special case of reflectionless potential (R(k, t) = 0) with only one bound state, N = 1, specified by

The GLM integral equation (26) for this case becomes
The GLM integral equation (29) can be put as Substituting (30) into the GLM integral equation (29), we have Equation (28) gives the one-soliton solution of (3) ).

Two-soliton solution (N=2)
To evaluate two-soliton solution of the equation (3), we need to consider again the special case of reflectionless potential (R(k, t) = 0) with two bound states, N = 2, specified by the discrete eigenvalues k 1 and k 2 and the corresponding normalization constants C j (t) = C j0 e k j 1−4k 2 j t , C j0 = C j (0), j = 1, 2. In this case the GLM integral equation (26) can be put as

Bulletin of Mathematical Sciences and Applications Vol. 20 29
To solve GLM integral equation (31), we take K(x, y, t) as Substituting this into the GLM integral equation (31) and equating the coefficients of e −k 1 y and e −k 2 y from both sides separately, we obtain Equation (28) gives the two-soliton solution of (3) ), j = 1, 2.

N −soliton solution
Now we investigate N −soliton solution of the equation (3) and for which we need reflectionless potentials (R(k, t) = 0) with N −bound states as earlier i.e. from (27) we express F in separation of variables form as where We also take K in separation of variables form Substituting (32) and (33) into the GLM integral equation (26), we obtain

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Defining the matrices equation (34) can be recast as the matrix equation From equation (33), one may get Equation (35) can be put as Equations (36) and (37) together imply where Q ml is the cofactor matrix and |Q| is the determinant of Q.

Case-II (α = β) :
The linear eigenvalue problem is given by where λ = eigenvalue and ψ(x, t) evolve with time in a prescribed manner determined by Consider the eigenvalue problem analogous to (39) [13] where q(x, t) and r(x, t) are potentials and ξ is the eigenvalue. The time dependence of ψ analogous to (40) is given by The compatibility condition ψ xt = ψ tx gives where [N, M ] = N M − M N and Θ is the null matrix of order 3 × 3.

Equation (41) can be split into the following equations
To find GSWWI equation (4) with variable coefficient let us assume using which, the equations (42)-(50) can be put as equations (51)-(53) reduce to the GSWWI equation (4) with variable coefficients. The scattering problem for the GSWWI equation (2) is where λ = ξ.
Clearly the scattering problem (54) is not of Schrödinger type. To evaluate explicit soliton solutions of the GSWWI equation (2) we use the Hirota's bilinear [11] form of equation (2).
We introduce the transformation to a new dependent variable W by Substituting (55) in (2) and integrating once gives equation (2) can be transformed to the bilinear form

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where the D−operator is defined as (n, m ≥ 0) To derive explicit soliton solutions of equation (2), we need the following bilinear identities where F is an arbitrary continuous function of independent variables x, t and η j = k j x+w j t, j = 1, 2 is the dispersion relation. The dispersion relation can be obtained by substituting into the linear terms of equation (2) as

One-soltion solution
To obtain one-soliton solution of equation (2), the seed solutions of the corresponding bilinear equation (57) are taken as Using (58), it is clear that F 0 and F 1 satisfies Taking F = F 0 + F 1 in (57) and using (60), we have With the help of the bilinear transformation (56), the one-soliton solution of equation (2) can be put as

Bulletin of Mathematical Sciences and Applications Vol. 20 33 N −soliton solution
In general the similar process can be done continuously and will yield a series of explicit soliton solutions. To obtain N −soliton solution, we choose where The N −soliton solution of equation (2) can be obtained from where F is given in (63).

Summary
In this work we have employed the AKNS scheme to derive the generalized shallow water wave equation for the complete integrable cases. We also obtained N −soliton solutions using IST and Hirota's bilinear method.