which can be considered as a coupling between the KdV (with respect to u) and the mKdV (with respect to v) equations. The coupled KdV-mKdV equations were proposed by Kersten and Krasil’shchik [1] and originate from a supersymmetric extension of the classical KdV [2]. It also can be considered as a coupling between the KdV and mKdV equations: By setting v = 0 we obtain the KdV equation ut + uxxx...

We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation ∂tv + v∂xv + ∂ 3 x v + δ ( ∂ x v + ∂ x v ) = 0, δ > 0, in the Korteweg-de Vries limit δ → 0, a canonical limit describing small-amplitude weakly unstable thin film flow. More precisely, we carry out a rigorous singular perturbation analysis reducing the problem of spectral ...

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure i...

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.

This article is concerned with initial-boundary value problems for the Korteweg-de Vries (KdV) equation on bounded intervals. For general linear boundary conditions and small initial data, we prove the existence and uniqueness of global regular solutions and its exponential decay, as t→∞.

It is demonstrated that the dynamics of small-amplitude dark solitons in optical fibers may be described by the wellknown Korteweg-de Vries equation. This approach allows us to explain analytically the temporal self-shift of dark solitons due to the Raman contribution to the nonlinear refractive index, which has been observed experimentally by Weiner et al. [Opt. Lett. 14, 868 (1989)].

The method of obtaining new integrable coupled equations through enlarging spectral problems of known integrable equations, which was recently proposed by W.-X. Ma, can produce nonintegrable systems as well. This phenomenon is demonstrated and explained by the example of the enlarged spectral problem of the Korteweg–de Vries equation.

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...

We prove the global existence and uniqueness of solutions both in the energy space and in the space of square integrable functions for a Korteweg-de Vries equation with noise. The noise is multiplicative, white in time, and is the muliplication by the solution of a homogeneous noise in the space variable.

Solutions of a boundary value problem for the Korteweg–de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at the boundaries. The Chebyshev method is found to be more efficient. © 2009 IMACS. Published by Elsevier...