Bifurcation diagram of a one-parameter family of one-dimensional nonlinear dispersive waves

The KdV equation with small dispersion is a model for the formation and propagation of dispersive shock waves. Dispersive shock waves are characterized by the appearance of modulated oscillations nearby the breaking point. The modulation in time and space of the amplitude, the frequencies and the wave-numbers of these oscillations is described by the g-phase Whitham equations. We study the initial value problem of the g-phase Whitham equations for a one-parameter family of monotone decreasing initial data. We use a variational principle for the g-phase Whitham equations recently proposed by Dubrovin: the minimizer of a functional on a certain infinite-dimensional space formally solves the initial value problem for each point of the (x, t) plane. For each value of the parameter of the initial data, we study the number of phases involved in the solution of the initial value problem and we classify the topological type of bifurcation diagram of the genus g(x, t).