The inverse problem for a general N × N spectral equation

Caudrey, P. J.

doi:10.1016/0167-2789(82)90004-5

Cited by 89 publications

(136 citation statements)

References 4 publications

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“…Thus the IST problem is directly related to a spectral equation of third order, namely (4.2). The inverse problem for certain third-order spectral equations has been considered by Kaup [11] and Caudrey [12,13]. As expected (4.2) and (4.3) are similar to, but cannot be transformed into, the corresponding equations for the HSE (see Eqs.…”

Section: Formulation Of the Inverse Scattering Eigenvalue Problemmentioning

confidence: 86%

“…In this section we will show that the IST problem for the transformed VE in the form (2.6) has a third-order eigenvalue problem that is similar to the one associated with a higher-order KdV equation [11,16], a Boussinesq equation [11,12,17,18], and a model equation for shallow water waves [8,15].…”

Section: Formulation Of the Inverse Scattering Eigenvalue Problemmentioning

confidence: 99%

“…In Section 4 we find that the IST problem for the transformed VE involves a third-order eigenvalue problem. The inverse problem for certain third-order spectral equations has been considered by Kaup [11] and Caudrey [12,13]. In Section 5 we adapt the results obtained by these authors to the present problem and describe a procedure for using the IST to find the N -soliton solution to the transformed VE, and hence to the VE itself.…”

Section: Introductionmentioning

confidence: 99%

“…Following the procedure given in [12], the spectral equation (4.2) The matrix A has eigenvalues k j ðfÞ and left-and right-eigenvectorsṽ v j ðfÞ and v j ðfÞ, respectively, where Here T is regarded as a parameter; the T -evolution of the scattering data will be taken into account later. The solution of the direct problem is given by the equation system (4.5) in [12]. We shall restrict our attention to the Nsoliton solution.…”

Section: The N-soliton Solutionmentioning

confidence: 99%

“…The general theory of the inverse scattering problem for N spectral equations has been developed in [12]. Following the procedure given in [12], the spectral equation (4.2) The matrix A has eigenvalues k j ðfÞ and left-and right-eigenvectorsṽ v j ðfÞ and v j ðfÞ, respectively, where Here T is regarded as a parameter; the T -evolution of the scattering data will be taken into account later.…”

Section: The N-soliton Solutionmentioning

confidence: 99%

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The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method

Vakhnenko¹,

Parkes

2002

Chaos, Solitons & Fractals

119

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A B€ a acklund transformation both in bilinear form and in ordinary form for the transformed Vakhnenko equation is derived. An inverse scattering problem is formulated. The inverse scattering method has a third-order eigenvalue problem. A procedure for finding the exact N -soliton solution of the Vakhnenko equation via the inverse scattering method is described. The procedure is illustrated by considering the cases N ¼ 1 and N ¼ 2. Ó

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Section: Formulation Of the Inverse Scattering Eigenvalue Problemmentioning

confidence: 86%

Section: Formulation Of the Inverse Scattering Eigenvalue Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: The N-soliton Solutionmentioning

confidence: 99%

Section: The N-soliton Solutionmentioning

confidence: 99%

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The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method

Vakhnenko¹,

Parkes

2002

Chaos, Solitons & Fractals

119

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Direct and inverse scattering transforms with arbitrary spectral singularities

Zhou

1989

Comm Pure Appl Math

144

145

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IntroductionThe direct and inverse scattering problems for the n X n system (generalized the AKNS system) d -m + z [ J , m ] = q ( x ) m dx were solved by Beals and Coifman [3] for J diagonal with distinct eigenvalues and for potentials q, off-diagonal and generic. Partial solutions to the problems may be found in [13] and [7]. Certain bounded solutions m are piecewise meromorphic on the Riemann sphere relative to a contour 2, consisting of finitely many straight lines passing through the origin determined by J. Generic potentials are defined such that m extends continuously to the boundary from each component of C\Z (this implies that m has only finitely many poles) and only has simple poles of a special type. The Beals-Coifman scattering data are then defined by the multiplicative jump of m across I: and certain normalization constants at the poles. Following [8], we call z' E Z a spectral singularity of the scattering problem of q if the limit of m as z -+ z' from one of the components of C\X does not exist. Even for the potentials of Schwartz class, the poles may accumulate to spectral singularities and the spectral singularities themselves may accumulate (see Example 3.3.16). These problems (after the significant progress made by [3]) constitute the major obstacle to the complete solution of the direct and inverse scattering problems for first-order systems (including the 2 X 2 systems).When the inverse scattering method is applied in solving the Cauchy problem for the nonlinear evolution equations, the time evolution can be obtained by inverse scattering transform only if the initial q happens to be generic. Indeed, an a priori determination of all the generic potentials has not been obtained. This unpleasant situation apparently should not be ascribed to the evolution equations but to a deficiency of the existing theory of direct and inverse scattering transforms.In this paper, we find a way to overcome the deficiency by dropping the boundedness requirement for m. In our method, the bounded solutions of (1.1) are only required in a neighborhood of z = 00 (which is the only pole of zJ).According to [3], for q E L', the bounded solutions always exist there. Away

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The Lax Representation and the AKNS Approach

Gerdjikov¹,

Vilasi

Yanovski

Integrable Hamiltonian Hierarchies

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The inverse problem for a general N × N spectral equation

Cited by 89 publications

References 4 publications

The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method

The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method

Direct and inverse scattering transforms with arbitrary spectral singularities

The Lax Representation and the AKNS Approach

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