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Nonholonomic 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 a perturbed model with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV6 equation are analysed. This allows for a formulation of conditions on the perturbation terms that preserve its integrability. The perturbation corrections to the scattering data and to the corresponding action-angle (canonical) variables are studied. The analysis shows that although many nontrivial solutions of KdV6 can be obtained by the Inverse Scattering Transform (IST), there are solutions that in principle can not be obtained via IST. Thus the equation in general is not completely integrable.
Gerdjikov, V., Grahovski, G., Ivanov, R.: On the integrability of KdV hierarchy with self-consistent sources, Commun. Pure Appl. Analysis, vol. 11 (2012), pp. 1439-1452. doi: 10.3934/cpaa.2012.11.1439