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Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-...
We prove that the initial value problem (IVP) associated to the fifth order KdV equation {equation} \label{05KdV} \partial_tu-\alpha\partial^5_x u=c_1\partial_xu\partial_x^2u+c_2\partial_x(u\partial_x...
In this letter, based on the computerized symbolic computation, we extend the Clarkson and Kruskal (CK) method to search for new soliton-type solutions and auto-B¨acklund transformation of partial dif...
Abstract: The matrix KdV equation with a negative dispersion term is considered in the right upper quarter--plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linea...
Abstract: We establish the existence of real pole-free solutions to all even members of the Painlev\'e I hierarchy. We also obtain asymptotics for those solutions and describe their relevance in the d...
We construct generalized solutions to the ultradiscrete KdV equation,including the so-called negative solition solutions. The method is based on the ultradiscretization of soliton solutions to the dis...
In this letter, we will consider variational iteration method (VIM) for finding analytical solutions of the coupled Schrödinger–KdV equation. The available analytical solutions of the coupled Sch...
We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V0 which is decaying sufficiently fast at +¥ and arbitrarily enough (i.e., no decay or patte...
We consider the Cauchy problem of the fifth order KdV equation with low regularity initial data. We cannot apply the iteration argument to this problem when initial data is given in the Sobolev space...
We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V0 which is decaying sufficiently fast at +¥ and arbitrarily enough (i.e., no decay or patte...
We consider the generalized Korteweg-de Vries (gKdV) equation @tu + @3 xu + μ@x(uk+1) = 0, where k ≥ 5 is an integer number and μ = ±1.In the focusing case (μ = 1), we show that if the initial data u0...
The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small but finite amplitude electrostatic waves. The Lagrangian of the time fractional KdV equatio...
We prove that P.Mathieu's Open problem on constructing Gardner's deformation for the N=2 supersymmetric a=4-Korteweg-de Vries equation has no supersymmetry invariant solutions, whenever it is assumed ...
This paper is concerned with a generalized type of Darboux transformations defined in terms of a twisted derivation $D$ satisfying $D(AB)=D(A)+\sigma(A)B$ where $\sigma$ is a homomorphism. Such twiste...
We construct and analyze a family of semi-discretized difference schemes with two parameters for the Korteweg-de Vries (KdV) equation. The scheme possesses the first four near-conserved quantities f...

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