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On the integrability of KdV hierarchy with self-consistent sources

Gerdjikov, VS and Grahovski, GG and Ivanov, RI (2012) 'On the integrability of KdV hierarchy with self-consistent sources.' Communications on Pure and Applied Analysis, 11 (4). 1439 - 1452. ISSN 1534-0392

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Abstract

Non-holonomic deformations of integrable equations of the Kd-V hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Jim Jamieson
Date Deposited: 02 Feb 2015 10:29
Last Modified: 23 Jan 2019 00:15
URI: http://repository.essex.ac.uk/id/eprint/11676

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