On the integrability of KdV hierarchy  with  self-consistent sources

Vladimir S. Gerdjikov; Georgi Grahovski; Rossen Ivanov

doi:10.3934/cpaa.2012.11.1439

Article Contents

2012, Volume 11, Issue 4: 1439-1452. Doi: 10.3934/cpaa.2012.11.1439

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On the integrability of KdV hierarchy with self-consistent sources

1.
Institute for Nuclear Research and Nuclear Energy, Bulgarian academy of sciences, 72 Tsarigradsko chaussee, 1784 Sofia
2.
School of Electronic Engineering, Dublin City University, Glasnevin, Dublin 9
3.
School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8

Received: March 31, 2011

Published: June 2012

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Abstract

Non-holonomic deformations of integrable equations of the KdV 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.

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Mathematics Subject Classification: Primary: 37K15, 37K40, 37K55; Secondary: 35P10, 35P25, 35P30.

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