Determinant form of the complex phase function of the steepest descent analysis of Riemann-Hilbert problems and its application to the focusing nonlinear schrödinger equation

We derive a determinant formula for the g-function that plays a key role in the steepest descent asymptotic analysis of the solution of 2 × 2 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 αj of the Riemann surface on a set of external parameters. We prove that, subject to the modulation equations, ∂g/∂αj 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. © The Author 2009.

Duke Scholars

Author Stephanos Venakides Mathematics