Asymptotics of semiclassical soliton ensembles: Rigorous justification of the WKB approximation (Q2782258)

The author presents a new technique in the theory of ``steepest descent'' asymptotic analysis for matrix Riemann-Hilbert problems that solves in a general framework the following three problems:NEWLINENEWLINENEWLINE(1) The semiclassical limit of the focusing nonlinear Schrödinger hierarchy with decaying initial data;NEWLINENEWLINENEWLINE(2) The zero-dispersion limit of the Korteweg-de Vries equation with potential well initial data;NEWLINENEWLINENEWLINE(3) The large degree limit of certain systems of discrete orthogonal polynomials.NEWLINENEWLINENEWLINEThe method is illustrated in detail for the first case and it is shown the relation between the rigorous asymptotic analysis and the initial-value problem for the focusing nonlinear Schrödinger equation. A key point is to replace the true spectral data (which is not known) by its formal WKB approximation in the inverse scattering problem.