Asymptotics of semiclassical soliton ensembles: rigorous justification of the WKB approximation

Rigorous pointwise asymptotics are established for semiclassical soliton ensembles (SSEs) of the focusing nonlinear Schrödinger equation using techniques of asymptotic analysis of matrix Riemann-Hilbert problems. The accumulation of poles in the eigenfunction is handled using a new method in which the residues are simultaneously interpolated at the poles by two distinct interpolants. The results justify the WKB approximation for the nonselfadjoint Zakharov-Shabat operator with real-analytic, bell-shaped, even potentials. The new technique introduced in this paper is applicable to other problems as well: (i) it can be used to provide a unified treatment by Riemann-Hilbert methods of the zero-dispersion limit of the Korteweg-de Vries equation with negative (soliton generating) initial data as studied by Lax, Levermore, and Venakides, and (ii) it allows one to compute rigorous strong asymptotics for systems of discrete orthogonal polynomials.