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

TL;DR

This paper investigates the integrability of the KdV hierarchy with self-consistent sources using squared solutions, identifying conditions for integrability preservation and analyzing the impact on scattering data and soliton solutions.

Contribution

It introduces a method to analyze the integrability of KdV equations with self-consistent sources and formulates conditions under which integrability is maintained.

Findings

Certain perturbations preserve integrability and eigenvalues.

Some solutions cannot be obtained via inverse scattering transform.

Perturbed equations can have time-dependent eigenvalues and non-integrable solutions.

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.