which can be considered as a coupling between the KdV (with respect to u) and the mKdV (with respect to v) equations. The coupled KdV-mKdV equations were proposed by Kersten and Krasil’shchik [1] and originate from a supersymmetric extension of the classical KdV [2]. It also can be considered as a coupling between the KdV and mKdV equations: By setting v = 0 we obtain the KdV equation ut + uxxx...

in this paper , the quintic b-spline collocation scheme is employed to approximate numerical solution of the kdv-like rosenau equation . this scheme is based on the crank-nicolson formulation for time integration and quintic b-spline functions for space integration . the unconditional stability of the present method is proved using von- neumann approach . since we do not know the exact solution...

The Rotation-Modified Korteweg-deVries (RMKdV) equation differs from the ordinary KdV equation only through an extra undifferentiated term due to Coriolis force. This article describes the steadily-travelling, spatially-periodic solutions which have peaks of identical size. These generalize the “cnoidal” waves of the KdV equation. There are five overlapping regimes in the parameter space. We de...

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, includin...

In this paper, we study a new negative-order KdV-CBS equation in (3+1) dimensions which is combination of the Korteweg-de Vries (KdV) and Calogero–Bogoyavlenskii–Schiff (CBS) equation. Firstly, determine Lie point symmetries conservation laws by using multiplier method. The will be used to obtain triple reduction second order ordinary differential (ODE), lead line travelling waves soliton solut...

Computationally efficient, structure-preserving reduced-order methods are developed for the Korteweg de Vries (KdV) equations in Hamiltonian form. The KdV equation is discretized space by finite differences. resulting skew-gradient system of ordinary differential (ODEs) integrated with linearly implicit Kahan's method, which preserves approximately. We have shown, using proper orthogonal decomp...

The Gardner equation is an extension of the Korteweg–de Vries (KdV) equation. It exhibits basically the same properties as the classical KdV, but extends its range of validity to a wider interval of the parameters of the internal wave motion for a given environment. In this paper, we derive exact solitary wave solutions for the generalized Gardner equation that includes nonlinear terms of any o...

Two classes of rational solutions to a KdV-like nonlinear differential equation are constructed. The basic object is a generalized bilinear differential equation based on a prime number p 1⁄4 3. A conjecture is made that the two presented classes of rational solutions contain all rations solutions to the considered KdV-like equation, which are generated from polynomial solutions to the correspo...

Algebro-geometric Poisson brackets for real, finite-zone solutions of the Korteweg–de Vries (KdV) equation were studied in [1]. The transfer of this theory to the Toda lattice and the sinh-Gordon equation is more or less obvious. The complex part of the finite-zone theory for the nonlinear Schrödinger equation (NS) and the sine-Gordon equation (SG) is analogous to KdV, but conditions that solut...

A typical and effective way to construct a higher dimensional integrable equation is to extend the Lax pair for a (1 + 1) dimensional equation known as integrable to higher dimensions. Here we construct an alternative modified KdV equation in (2+1) dimensions by the higherdimensional extension of a Lax pair. And it is shown that this higher dimensional modified KdV equation passes the Painlevé ...