The nonlinear steepest descent approach to the singular asymptotics of the second Painlevé transcendent

2012 ◽  
Vol 241 (23-24) ◽  
pp. 2204-2225 ◽  
Author(s):  
Thomas Bothner ◽  
Alexander Its
Keyword(s):  
Steepest Descent ◽  
Painlevé Transcendent ◽  
Nonlinear Steepest Descent

scholarly journals Rarefaction waves of the Korteweg–de Vries equation via nonlinear steepest descent

2016 ◽  
Vol 261 (10) ◽  
pp. 5371-5410 ◽  
Author(s):  
Kyrylo Andreiev ◽  
Iryna Egorova ◽  
Till Luc Lange ◽  
Gerald Teschl
Keyword(s):  
Vries Equation ◽  
Steepest Descent ◽  
Rarefaction Waves ◽  
De Vries ◽  
Nonlinear Steepest Descent

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.

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