The Wentzel-Kramers-Brillouin approximation of semiclassical eigenvalues of the Zakharov–Shabat problem

Abstract

The WKB approximation of semiclassical eigenvalues of the non-self-adjoint Zakharov–Shabat problem is a standard element in the theory of the integrable focusing nonlinear Schrödinger equation; this approximation is based on a reformulation of the eigenvalue problem as the eigenvalue problem for a self-adjoint Schrödinger operator with a correction term that depends on the spectral parameter. The approximation results from neglecting the correction term. We perform a numerical experiment which gives new evidence to support the validity of this procedure in the context of the semiclassical limit problem for the focusing nonlinear Schrödinger equation. In particular, our results suggest that the rate of convergence of the approximate eigenvalues to the true ones is of the order of the square of the small parameter. This information is relevant to the task of rigorously incorporating this approximation into the asymptotic analysis of the singular limit for the focusing nonlinear Schrödinger equation.