Generalizations of the Korteweg-de Vries equation are considered, and some explicit solutions are presented. There are situations where solutions engender the interesting property of being chiral, that is, of having velocity determined in terms of the parameters that define the generalized equation, with a definite sign.

The bifurcation analysis of the K (m, n) equation, which serves as a generalized model for the Korteweg-de Vries equation describing the dynamics of shallow water waves on ocean beaches and lake shores, is carried out in this paper. The phase portraits are given and solitary wave solutions are obtained. Singular periodic wave solutions are also given in this work.

The dynamics of the poles of the two–soliton solutions of the modified Korteweg–de Vries equation ut + 6u ux + uxxx = 0 are determined. A consequence of this study is the existence of classes of smooth, complex–valued solutions of this equation, defined for−∞ < x < ∞, exponentially decreasing to zero as |x| → ∞, that blow up in finite time.

An alternate generalized Korteweg-de Vries system is studied here. A procedure for generating solutions is given. A theorem is presented, which is subsequently applied to this equation to obtain a type of Bäcklund transformation for several specific cases of the power of the derivative term appearing in the equation. In the process, several interesting, new, ordinary, differential equations are...

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy (L-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately sat...

Considered herein is the stability problem of solitary wave solutions of a generalized Ostrovsky equation, which is a modification of the Korteweg-de Vries equation widely used to describe the effect of rotation on surface and internal solitary waves or capillary waves.

This paper is concerned with interacting wave packet dynamics for long waves. The Kortweg-de Vries equation is the most well-known model for weakly nonlinear long waves. Although the dynamics of a single wave packet in this model is governed by the defocusing nonlinear Schrödinger equation, implying that a plane wave is modulationally stable, the dynamics of two interacting wave packets is gove...

Abstract: Certain explicit solutions to the Korteweg-de Vries equation in the first quadrant of the xt-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schrödinger equation. In the reflectionless case such solutions reduce to pure N -soliton solutions. An il...

We propose a new family of complex PT -symmetric extensions of the Korteweg-de Vries equation. The deformed equations can be associated to a sequence of non-Hermitian Hamiltonians. The first charges related to the conservation of mass, momentum and energy are constructed. We investigate solitary wave solutions of the equation of motion for various boundary conditions.