On the solution of the Whitham equations: An estimate of the genus

Abstract

The Korteweg de Vries (KdV) equation with small dispersion $$\left\{ {\begin{array}{*{20}{c}} {{{u}_{t}} + 6u{{u}_{x}} + {{\epsilon }^{2}}{{u}_{{xxx}}} = 0,{\text{ }}t,x \in \mathbb{R},{\text{ 0}} < \epsilon \ll 1} \hfill \\ {u\left( {x,t = 0,\epsilon } \right) = {{u}_{0}}\left( x \right)} \hfill \\ \end{array} } \right.$$ (1.1) and with initial data decreasing in the neighborhood of an inflection point, is a model for the formation and propagation of dispersive shock waves in one dimension [1]. The solutionu(x,t,e)of (1.1) develops, near the inflection point, an expanding region filled with rapid modulated oscillations. The evolution of the small e-asymptotic form of these modulated oscillations is described by a collection of systems of hyperbolic [2] partial differential equations in Riemann invariant form [[3,[4] $$\frac{{\partial {{u}_{i}}}}{{\partial t}} + {{\lambda }_{i}}\left( {{{u}_{1}},{{u}_{2}}, \ldots ,{{u}_{{2g + 1}}}} \right)\frac{{\partial {{u}_{i}}}}{{\partial x}} = 0,{\text{ }}i = 1,2, \ldots .2g + 1,{\text{ g}} \geqslant {\text{0}}$$ (1) which is called Whitham equations. For g =0 (1.2) coincides with the dispersion-less KdV equation $$ {u_t} + 6u{u_x} = 0, $$ (1.3) which is called Burgers equation.