A Steepest Descent Method for Oscillatory Riemann--Hilbert Problems. Asymptotics for the MKdV Equation

Percy Deift, Xin Zhou

Type: Article

Publication Date: 1993-03-01

Citations: 1323

DOI: https://doi.org/10.2307/2946540

Abstract

In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg-de Vries (MKdV) equation

Locations

arXiv (Cornell University) - PDF
Annals of Mathematics - View

Similar Works

Action	Title	Year	Authors
+	A steepest descent method for oscillatory Riemann-Hilbert problems	1992	Percy Deift Xin Zhou
+ PDF Chat	A steepest descent method for oscillatory Riemann-Hilbert problems	1992	Percy Deift Xin Zhou
+	A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems	2020	Fudong Wang Wen‐Xiu Ma
+	An extension of the steepest descent method for Riemann-Hilbert problems: The small dispersion limit of the Korteweg-de Vries (KdV) equation	1998	Percy Deift Stephanos Venakides Xin Zhou
+	dispersion limit of the Korteweg-de Vries (KdV) equation An extension of the steepest descent method for Riemann-Hilbert problems: The small	2007	Percy Deift Stephanos Venakides Xin Zhou
+	Large time asymptotics for the cylindrical Korteweg–de Vries equation. I.	2020	A. R. Its V. V. Sukhanov
+	The long-time asymptotic analysis of integrable systems via the Riemann-Hilbert method	2007	Alexander Its
+ PDF Chat	Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent	2009	Katrin Grunert Gerald Teschl
+	The nonlinear steepest descent method for Riemann-Hilbert problems of low regularity	2015	Jonatan Lenells
+ PDF Chat	The nonlinear steepest descent method for Riemann-Hilbert problems of low regularity	2017	Jonatan Lenells
+	The nonlinear steepest descent method for Riemann-Hilbert problems of low regularity	2015	Jonatan Lenells
+ PDF Chat	Long-time asymptotic behavior for an extended modified Korteweg-de Vries equation	2019	Nan Liu Boling Guo Deng‐Shan Wang Yu‐Feng Wang
+	Long-time asymptotic behavior for an extended modified Korteweg-de Vries equation	2019	Nan Liu Boling Guo Deng‐Shan Wang Yu‐Feng Wang
+ PDF Chat	On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking	2008	S. V. Manakov P. Santini
+ PDF Chat	Long-time asymptotic analysis of the Korteweg–de Vries equation via the dbar steepest descent method: the soliton region	2017	Pietro Giavedoni
+ PDF Chat	Long-time asymptotics for the Korteweg–de Vries equation with step-like initial data	2013	Iryna Egorova Zoya Gladka Vladimir Kotlyarov Gerald Teschl
+	Rarefaction waves of the Korteweg–de Vries equation via nonlinear steepest descent	2016	K.M. Andreiev Iryna Egorova Till Luc Lange Gerald Teschl
+	Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions	2023	Zhaoyu Wang Taiyang Xu Engui Fan
+	A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems	2009	Yen Do
+	The complex mKdV equation with step-like initial data: Large time asymptotic analysis	2022	Zhaoyu Wang Kai Xu Engui Fan

Works That Cite This (1184)

Action	Title	Year	Authors
+ PDF Chat	On the gap probability of the tacnode process	2024	Luming Yao Lun Zhang
+	Long‐time asymptotics of solution for the fifth‐order modified KdV equation in the presence of discrete spectrum	2024	Nan Liu Mingjuan Chen Boling Guo
+ PDF Chat	Large Deformations of the Tracy–Widom Distribution I: Non-oscillatory Asymptotics	2017	Thomas Bothner Robert Buckingham
+ PDF Chat	Soliton Resolution for the Wadati–Konno–Ichikawa Equation with Weighted Sobolev Initial Data	2022	Zhiqiang Li Shou‐Fu Tian Jin‐Jie Yang
+	Long-time asymptotics of a three-component coupled nonlinear Schrödinger system	2020	Wen‐Xiu Ma
+ PDF Chat	Long-time asymptotics for the Toda lattice in the soliton region	2008	Helge Krüger Gerald Teschl
+	Orthogonal polynomials in the normal matrix model with a cubic potential	2012	Pavel Bleher Arno B. J. Kuijlaars
+ PDF Chat	The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions	2016	Gino Biondini Daniel Kraus B. Prinari
+	Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data	2020	Yan Rybalko Dmitry Shepelsky
+	On the long-time asymptotics of the modified Camassa-Holm equation in space-time solitonic regions	2022	Yiling Yang Engui Fan

Works Cited by This (12)

Action	Title	Year	Authors
+	Asymptotic behavior of solutions of the Korteweg-de Vries equation	1986	V. S. Buslaev
+	Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III	1981	Michio Jimbo Tetsuji Miwa
+	Scattering and inverse scattering for first order systems	1984	Richard Beals Ronald R. Coifman
+ PDF Chat	Monodromy- and spectrum-preserving deformations I	1980	H. Flaschka Alan C. Newell
+	The decay of the continuous spectrum for solutions of the Korteweg-deVries equation	1973	Mark J. Ablowitz Alan C. Newell
+	The Riemann–Hilbert Problem and Inverse Scattering	1989	Xin Zhou
+	Handbook of Mathematical Functions	1966	Donald A. McQuarrie
+ PDF Chat	A steepest descent method for oscillatory Riemann-Hilbert problems	1992	Percy Deift Xin Zhou
+	Asymptotic expansions for ordinary differential equations	1965	Wolfgang Wasow
+	Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II	1981	Michio Jimbo Tetsuji Miwa