Title

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

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 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.

Publication Date

2-1-2009

Publication Title

International Mathematics Research Notices

Volume

2009

Issue

11

Number of Pages

2056-2080

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1093/imrn/rnp011

Socpus ID

67650799877 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/67650799877

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