In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are e...

The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota–Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.

In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are e...

In this paper, we apply the Miura transformation to construct the connection between a variablecoefficient KdV (vcKdV) equation and a variable-coefficient modified KdV (vcmKdV) equation under certain constraint. Solving the vcmKdV equation by use of the auxiliary equation method and using the Miura transformation, we find a rich variety of new exact solutions for the vcKdV equation, which inclu...

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 gener...

The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions. Perturbation theory is the ...

In this letter, a Painlevé integrable coupled KdV equation is proved to be also Lax integrable by a prolongation technique. The Miura transformation and the corresponding coupled modified KdV equation associated with this equation are derived. © 2010 Published by Elsevier Ltd

We study the relation between the centro-affine geometry of starshaped planar curves and the projective geometry of parametrized maps into RP. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwa...

The KdV equation arises in the framework of the Boussinesq scaling as a model equation for waves at the surface of an inviscid fluid. Encoded in the KdV model are relations that may be used to reconstruct the velocity field in the fluid below a given surface wave. In this paper, velocity fields associated to exact solutions of the KdV equation are found, and particle trajectories are computed n...

the thomas-fermi (tf) equation has proved to beuseful for the treatment of many physical phenomena. in this pa-per, the traveling wave solutions of the kdv equation is investi-gated by the simplest equation method. also, the effect of differentparameters on these solitary waves is considered. the numericalresults is conformed the good accuracy of presented method.