Dispersion management for solitons in a Korteweg-de Vries system

The variable-coefficient Korteweg-de Vries is used to present a basic model, which has the form of a Korteweg-de Vries equation with a pe- riodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models describe, for instance, a periodically modulated waveguide for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly non- linear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial con- dition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion.