We study the Cauchy initial-value problem for the Benjamin-Ono equation in the zero-dispersion limit, and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by Lax and Levermore to analyze the corresponding limit for the Korteweg–de Vries equation. © 2010 Wiley Periodicals, Inc.

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 conventional Lie group approach is extended successfully to give out the group explanation to the new conditional similarity reductions obtained by modifying the Clarkson and Kruskal's (CK's) direct method for the (2+1)-dimensional Korteweg–de Vries (KdV) equation.

In this paper, we study a compound Korteweg-de Vries-Burgers equation with a higher-order nonlinearity. A class of solitary wave solutions is obtained by means of a series expansion.

We solve the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite.

In this letter, applying a series of coordinate transformations, we obtain a new class of solutions of the Korteweg–de Vries–Burgers equation, which arises in the theory of ferroelectricity. © 2005 Elsevier Ltd. All rights reserved.

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H(R) for −3/10 < s.

We solve the Cauchy problem for the Korteweg–de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finitegap potentials under the assumption that the respective spectral bands either coincide or are disjoint.

This paper surveys various aspects of the hydrodynamic formulation of the nonlinear Schrödinger equation obtained via the Madelung transform in connexion to models of quantum hydrodynamics and to compressible fluids of the Korteweg type.