The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2-based Sobolev spaces H s where local well-posedness is presently known, apart from the H 1 4 (R) endpoint for mKdV. The result for KdV relies on a new method for co...

based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the korteweg-de vries (kdv) equation are first constructed by the known darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude de...

In the present article, a numerical method is proposed for the numerical solution of the KdV equation by using a new approach by combining cubic B-spline functions. In this paper we convert the KdV equation to system of two equations. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms L2, L∞ are computed. Three invariants of motion are...

In this paper we find a explicit moving frame along curves of Lagrangian planes invariant under the action of the symplectic group. We use the moving frame to find a family of independent and generating differential invariants. We then construct geometric Hamiltonian structures in the space of differential invariants and prove that, if we restrict them to a certain Poisson submanifold, they bec...

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for H initial data, s > −1/2, and for any s1 < min(3s+1, s+ 1), the difference of the nonlinear and linear evolutions is in H1 for all times, with at most polynomially growing H1 norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case s ≥ 0. Our resu...

in this paper, we have studied on the solutions of modied kdv equation andalso on the stability of them. we use the tanh method for this investigationand given solutions are good-behavior. the solution is shock wave and can beused in the physical investigations

We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV equation to other NPDEs. The method is based upon a linearization principle which can be applied on nonlinearities which have a polynomial form. We illustrate the ...

1. As was shown in the remarkable communication [4] the Cauchy problem for the Korteweg–de Vries (KdV) equation ut = 6uux−uxxx, familiar in theory of nonlinear waves, is closely linked with a study of the spectral properties of the Sturm–Liouville operator Lψ = Eψ, where L = −d/dx + u(x). For rapidly decreasing initial conditions u(x, 0), where ∫∞ −∞ u(x, 0)(1 + |x|)dx < ∞, the KdV equation can...

The higher order wave equation of KdV type, which describes many important physical phenomena, has been investigated widely in last several decades. In this work, multisymplectic formulations for the higher order wave equation of KdV type are presented, and the local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-symplectic discretization of e...