The Euler-Korteweg model is made of the standard Euler equations for compressible fluids supplemented with the Korteweg tensor, which is intended to take into account capillary effects. For nonmonotone ‘pressure’ laws, the Euler-Korteweg model is known to admit solitary waves, even though their physical significance remains unclear. In fact, several kinds of solitary waves, with various endstat...

Robbert Dijkgraaf, Sergei Gukov, Vladimir A. Kazakov, and Cumrun Vafa Institute for Theoretical Physics & Korteweg–de Vries Institute for Mathematics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts 02138, USA Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris CEDEX,...

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...