scholarly journals Long-Time Asymptotics of a Three-Component Coupled mKdV System

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 573 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Hilbert Problem ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Time Asymptotics ◽  
Long Time ◽  
Long Time Asymptotics ◽  
Matrix Spectral Problem ◽  
Riemann Hilbert Problem ◽  
Nonlinear Steepest Descent

We present an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4 × 4 matrix spectral problem. An integrable coupled mKdV hierarchy with three potentials is first generated. Based on the corresponding oscillatory Riemann-Hilbert problem, the leading asympototics of the three-component mKdV system is then evaluated by using the nonlinear steepest descent method.

scholarly journals LONG-TIME ASYMPTOTICS OF THE TODA LATTICE FOR DECAYING INITIAL DATA REVISITED

2009 ◽  
Vol 21 (01) ◽  
pp. 61-109 ◽  
Author(s):  
HELGE KRÜGER ◽  
GERALD TESCHL
Keyword(s):  
Initial Data ◽  
Toda Lattice ◽  
Steepest Descent ◽  
Time Asymptotics ◽  
Long Time ◽  
Long Time Asymptotics ◽  
Nonlinear Steepest Descent

The purpose of this article is to give a streamlined and self-contained treatment of the long-time asymptotics of the Toda lattice for decaying initial data in the soliton and in the similarity region via the method of nonlinear steepest descent.

THE RIEMANN–HILBERT APPROACH TO GLOBAL ASYMPTOTICS OF DISCRETE ORTHOGONAL POLYNOMIALS WITH INFINITE NODES

2010 ◽  
Vol 08 (03) ◽  
pp. 247-286 ◽  
Author(s):  
CHUNHUA OU ◽  
R. WONG
Keyword(s):  
Orthogonal Polynomials ◽  
Hilbert Problem ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Asymptotic Formulas ◽  
Charlier Polynomials ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal ◽  
Global Asymptotics ◽  
Riemann Hilbert Problem

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials [Formula: see text]. We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou ([8]) can be applied. Finally, we produce three Airy-type asymptotic expansions for [Formula: see text] in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller ([3]), there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong ([4]). Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials.

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