Whitham equations, Bergman kernel and Lax-Levermore minimizer
The paper contains a detailed study of the phase transition in the solution of the Whitham equations describing the zero dispersion limit of the Cauchy problem, with step-like monotonically increasing smooth initial data, for the Korteweg-de Vries (KdV) equation (it is also shown that the semiclassical limit of other nonlinear equations, like the nonlinear Schrödinger equation, can be treated in the same spirit). The emphasis is made on the behavior of the solution near the transition points, which are points of gradient catastrophe for the Whitham equations. The author analyses the relation between two approaches to the problem: the analysis of the associated Tsarev system, which can be reduced to a linear over-determined system of Euler-Poisson-Darboux type, and a recent approach (introduced by Deift, Zhou, and Venakides) based on the reformulation of the Cauchy problem for the KdV equation as a Riemann-Hilbert problem. Combining these approaches, the author shows that if the function \(\Phi_g(\lambda;u)\), which, on one hand, governs the behavior of the solution near the transition point and, on the other hand, solves a system of Euler-Poisson-Darboux type, possesses certain properties as a function of \(\lambda\) near the points of gradient catastrophe, then the genus of the solution of the Whitham equations (the number of intervals in the support of the associated Lax-Levermore minimizer) changes by one near each point of gradient catastrophe. Furthermore, the author characterizes a class of initial data for which all the singular cases occur only a finite number of times (thus giving an upper bound to the genus of the solution of the Whitham equations). The main ingredients of the investigation are the introduction of a natural kernel for solving the Whitham equations, the observation that the associated hodograph transformation can be written in many equivalent forms, and the nontrivial computation of the determinant of the Jacobian of the hodograph transformation near the points of gradient catastrophe.
scientific article; zbMATH DE number 2093671