Long time asymptotics for the defocusing mKdV equation under nonzero boundary conditions in solitonic region without phase points

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries equation for finite density type initial data. With the ∂̄ generalization of the nonlinear steepest descent method of Deift and Zhou, we extrapolate the leading order approximation to the solution of mKdV for large time in the solitonic region of space-time, |x/t| < 6, and we give bounds for the error which decay as t → ∞ for a general class of initial data whose difference from the non-vanishing background possesses a fixed number of finite moments. Our results provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for mKdV equation.

[1]  Zhenya Yan,et al.  Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions , 2020 .

[2]  S. Cuccagna,et al.  On the Asymptotic Stability of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}$$\end{document}-Soliton Solutions , 2016, Communications in Mathematical Physics.

[3]  Scipio Cuccagna,et al.  On weak interaction between a ground state and a trapping potential , 2013, 1309.4878.

[4]  D. Pelinovsky,et al.  The asymptotic stability of solitons in the cubic NLS equation on the line , 2013, 1302.1215.

[5]  Frank Merle,et al.  Asymptotic stability of solitons of the gKdV equations with general nonlinearity , 2007, 0706.1174.

[6]  Yvan Martel,et al.  Linear Problems Related to Asymptotic Stability of Solitons of the Generalized KdV Equations , 2006, SIAM J. Math. Anal..

[7]  K. Lonngren,et al.  On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line , 2001 .

[8]  L. Pratt,et al.  Predicting eddy detachment for an equivalent barotropic thin jet , 1994 .

[9]  P. Deift,et al.  A steepest descent method for oscillatory Riemann-Hilbert problems , 1992, math/9201261.

[10]  Kenji Ohkuma,et al.  Multiple-Pole Solutions of the Modified Korteweg-de Vries Equation , 1982 .

[11]  Hiroaki Ono.  On a Modified Korteweg-de Vries Equation , 1974 .

[12]  Hiroaki Ono,et al.  Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma , 1969 .

[13]  L. Debnath.  Solitons and the Inverse Scattering Transform , 2012 .

[14]  P. Deift,et al.  Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data , 2011 .

[15]  A. B. D. Monvel,et al.  Initial boundary value problem for the mKdV equation on a finite interval , 2004 .

[16]  P. Deift,et al.  The toda rarefaction problem , 1996 .

[17]  Wolfgang K. Schief,et al.  An infinite hierarchy of symmetries associated with hyperbolic surfaces , 1995 .