Determinant Form of the Complex Phase Function of the Steepest Descent Analysis of Riemann-Hilbert Problems and Its Application to the Focusing Nonlinear Schrodinger Equation

Abstract

We derive a determinant formula for the g-function that plays a key role in the steepest descent asymptotic analysis of the solution of matrix Riemann–Hilbert problems (RHPs) and is closely related to a hyperelliptic Riemann surface. We formulate a system of transcendental equations in determinant form (modulation equations), that govern the dependence of the branchpoints of the Riemann surface on a set of external parameters. We prove that, subject to the modulation equations, is identically zero for all the branchpoints. Modulation equations are also obtained in the form of ordinary differential equations with respect to external parameters; some applications of these equations to the semiclassical limit of the focusing nonlinear Schrödinger equation (NLS) are discussed.