Related papers: The Coupled Modified Korteweg-de Vries Equations
The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invarint solution for it are obtained by means of this technique. Polynomial, trigonometric and elliptic function solutions can be…
We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for…
A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and…
A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and…
Periodic waves of the modified Korteweg-de Vries (mKdV) equation are identified in the context of a new variational problem with two constraints. The advantage of this variational problem is that its non-degenerate local minimizers are…
The Korteweg-de Vries equation u_t+uu_x+u_{xxx}=0 is PT symmetric (invariant under space-time reflection). Therefore, it can be generalized and extended into the complex domain in such a way as to preserve the PT symmetry. The result is the…
An extension of the super Korteweg-de Vries integrable system in terms of operator valued functions is obtained. In particular the extension contains the $N=1$ Super KdV and coupled systems with functions valued on a symplectic space. We…
This paper represents a new perspective in understanding the controllability of the Korteweg-de Vries (KdV) equation on unbounded domains. By studying the equation on both the right and left half-line with a single control input, we show…
This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg-de Vries equation, the Kuramoto-Sivashinsky equation, the generalized Korteweg-de Vries-Kuramoto-Sivashinski equation and the non…
In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in…
We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the…
We study the Korteweg-de Vries-type equation dt u=-dx(dx^2 u+f(u)-B(t,x)u), where B is a small and bounded, slowly varying function and f is a nonlinearity. Many variable coefficient KdV-type equations can be rescaled into this equation. We…
We consider the Cauchy problem for the Korteweg--de Vries equation with real initial data $q$ that is both $L^1$ and $L^2$ summable and supported on (0,\infty). Using the left reflection coefficient and Hankel operators on the Hardy space…
We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves…
The Korteweg-de Vries (KdV) equation is known as a universal equation describing various long waves in dispersive systems. In this article, we prove that in a certain scaling regime, a large class of rough solutions to the Boussinesq…
Results on well-posedness of three inverse problems with integral conditions on a bounded interval for the generalized Korteweg-de Vries equation without any restrictions on the growth rate of nonlinearity are established. Either the…
We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow…
In this paper we consider the long time behavior of solutions to the modified Korteweg-de Vries equation on R. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotics without relying on…
Using the group-theoretical approach to the inverse scattering method the supersymmetric Korteweg-de Vries equation is obtained by application of the Drinfeld-Sokolov reduction to osp(1|2) loop superalgebra. The direct and inverse…
This paper is dedicated to finding the solutions of the equation of the loaded modified Korteweg-de Vries. By the way, it is shown to find the solutions via $(G'/G)$-expansion method that is one of the most effective ways of finding…