On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources

Details On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources

2011-09-21

arxiv.org/abs/1109.4543v1

Comm. Pure and Applied Analysis 11 (2012), No. 4, 1439 - 1452

Nonlinear Sciences

Exactly Solvable and Integrable Systems

16 pages, no figures, LaTeX

Scientific paper

10.3934/cpaa.2012.11.1439

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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