Exact solutions for the KdV6 and mKdV6 Equations via tanh-coth and sech Methods

The tanh-coth method is used to seek solutions to obtain solutions to the new integrable sixthorder Korteweg-de Vries equation (KdV6). Following the analogy between the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (MKdV) we construct a new system equivalent to KdV6 from which exact solutions to original equation and derived, during the sech method

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

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.Comment: 16 pages, no figures, LaTe

The exact traveling wave solutions to one integrable KDV6 equation

The traveling wave system of one integrable KdV6 equation is studied by using Cosgrove’s method. Some exact explicit traveling wave solutions are obtained. The local dynamical behavior of some known equilibria are discussed.Publisher's Versio

Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability

Recent concept of integrable nonholonomic deformation found for the KdV equation is extended to the mKdV equation and generalized to the AKNS system. For the deformed mKdV equation we find a matrix Lax pair, a novel two-fold integrable hierarchy and exact N-soliton solutions exhibiting unusual accelerating motion. We show that both the deformed KdV and mKdV systems possess infinitely many generalized symmetries, conserved quantities and a recursion operator.Comment: Latex, 2 figures, 16 pages. Revised with more explanations after Referees' feedback.To be published in J. Phys.

Computing Exact Solutions to a Generalized Lax-Sawada-Kotera-Ito Seventh-Order KdV Equation

The Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7)

Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation groups are given. In addition, a physical description of the conserved quantities is included. Finally, few travelling wave solutions have been obtained.Comment: 14 page

Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

Mathematics Subject Classi cation. Primary: 35B06, 35L65, 35C07; Secondary: 35Q53.We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation groups are given. In addition, a physical description of the conserved quantities is included. Finally, few travelling wave solutions have been obtained.11 página