Bifurcation diagram of a one-parameter family of dispersive waves

Abstract

The Korteweg de Vries (KdV) equation with small dispersion is a model for the formation and propagation of dispersive shock waves in one dimension. Dispersive shock waves in KdV are characterized by the appearance of zones of rapid modulated oscillations in the solution of the Cauchy problem with smooth initial data. The modulation in time and space of the amplitudes, the frequencies and the wave-numbers of these oscillations and their interactions is approximately described by the $g$-phase Whitham equations. We study the initial value problem for the Whitham equations for a one parameter family of monotone decreasing initial data. We obtain the bifurcation diagram of the number $g$ of interacting oscillatory zones.