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

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