TY - JOUR

T1 - On the integrability of KdV hierarchy with self-consistent sources

AU - Gerdjikov, Vladimir S.

AU - Grahovski, Georgi G.

AU - Ivanov, Rossen I.

PY - 2012/7

Y1 - 2012/7

N2 - 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.

AB - 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.

KW - Inverse scattering method

KW - KdV hierarchy

KW - KdV6 equation

KW - Self-consistent sources

KW - Soliton perturbations

UR - http://www.scopus.com/inward/record.url?scp=84870589005&partnerID=8YFLogxK

U2 - 10.3934/cpaa.2012.11.1439

DO - 10.3934/cpaa.2012.11.1439

M3 - Article

AN - SCOPUS:84870589005

SN - 1534-0392

VL - 11

SP - 1439

EP - 1452

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

IS - 4

ER -