A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE

作     者：Spyridon Kamvissis 

作者机构：Department of Applied MathematicsUniversity of Crete 

出 版 物：《Acta Mathematica Scientia》 (数学物理学报（B辑英文版）)

年 卷 期：2011年第31卷第6期

页      面：2233-2246页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported in part by the ESF program MISGAM and the EU program ACMAC at the University of Crete 

主　　题：Riemann-Hilbert problem Toda lattice 

摘      要：One of the inspirations behind Peter Lax＇s interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann＇s conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs. Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints （an ＂equilibrium measure＂ problem）. Later, it was found that the asyraptotic method of Deift and Zhou （analysis of the associated Riemann-Hilbert factorization problem in the complex plane） could apply to previously intractable problems and also produce more detailed information. Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solu- tions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background. Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift- Zhou ＂nonlinear stationary phase method＂ to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!