Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type

Abstract

We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion and with monotonically increasing initial data using the Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero dispersion limit, is obtained using the steepest descent method for oscillatory Riemann-Hilbert problems. The asymptotic solution is completely described by a scalar function $\g$ that satisfies a scalar RH problem and a set of algebraic equations constrained by algebraic inequalities. The scalar function $\g$ is equivalent to the solution of the Lax-Levermore maximization problem. The solution of the set of algebraic equations satisfies the Whitham equations. We show that the scalar function $\g$ and the Lax-Levermore maximizer can be expressed as the solution of a linear overdetermined system of equations of Euler-Poisson-Darboux type. We also show that the set of algebraic equations and algebraic inequalities can be expressed in terms of the solution of a different set of linear overdetermined systems of equations of Euler-Poisson-Darboux type. Furthermore we show that the set of algebraic equations is equivalent to the classical solution of the Whitham equations expressed by the hodograph transformation.