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Soliton resolution and asymptotic stability of $N$-soliton solutions for the defocusing mKdV equation with a non-vanishing background

Zechuan Zhang, Taiyang Xu, Engui Fan

TL;DR

The paper analyzes the defocusing mKdV equation with finite-density initial data in the nonzero-background setting and proves soliton-resolution-type large-time behavior within the solitonic cone $|x/t+4|<2$. It employs a $\bar{\partial}$-generalized nonlinear steepest descent method on a Riemann-Hilbert problem to establish both asymptotic stability of $N$-soliton solutions and a precise decomposition of the solution into soliton contributions plus radiative terms with $\mathcal{O}(t^{-1})$ error. A detailed RH deformation combines pole-to-jump conversions, lens opening, and a mixed $\bar{\partial}$-problem that is split into a small-norm part and a pure $\bar{\partial}$-part, with rigorous bounds on all error terms. The results advance the mathematical understanding of long-time dynamics for integrable systems with nonvanishing backgrounds, providing rigorous soliton resolution and stability statements in a setting with nontrivial discrete spectrum and background radiation.

Abstract

We analytically study the large-time asymptotics of the solution of the defocusing modified Korteweg-de Vries (mKdV) equation under a symmetric non-vanishing background, which supports the emergence of solitons. It is demonstrated that the asymptotic expansion of the solution at the large time could verify the renowned soliton resolution conjecture. Moreover, the asymptotic stability of $N$-soliton solution is also exhibited in the present work. We establish our results by performing a $\bar{\partial}$-nonlinear steepest descent analysis to the associated Riemann-Hilbert (RH) problem.

Soliton resolution and asymptotic stability of $N$-soliton solutions for the defocusing mKdV equation with a non-vanishing background

TL;DR

The paper analyzes the defocusing mKdV equation with finite-density initial data in the nonzero-background setting and proves soliton-resolution-type large-time behavior within the solitonic cone . It employs a -generalized nonlinear steepest descent method on a Riemann-Hilbert problem to establish both asymptotic stability of -soliton solutions and a precise decomposition of the solution into soliton contributions plus radiative terms with error. A detailed RH deformation combines pole-to-jump conversions, lens opening, and a mixed -problem that is split into a small-norm part and a pure -part, with rigorous bounds on all error terms. The results advance the mathematical understanding of long-time dynamics for integrable systems with nonvanishing backgrounds, providing rigorous soliton resolution and stability statements in a setting with nontrivial discrete spectrum and background radiation.

Abstract

We analytically study the large-time asymptotics of the solution of the defocusing modified Korteweg-de Vries (mKdV) equation under a symmetric non-vanishing background, which supports the emergence of solitons. It is demonstrated that the asymptotic expansion of the solution at the large time could verify the renowned soliton resolution conjecture. Moreover, the asymptotic stability of -soliton solution is also exhibited in the present work. We establish our results by performing a -nonlinear steepest descent analysis to the associated Riemann-Hilbert (RH) problem.

Paper Structure

This paper contains 18 sections, 21 theorems, 184 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. Main results.
  4. Preliminaries
  5. Lax pair and Jost solutions
  6. Existence and differentiability of Jost functions.
  7. Asymptotics and symmetries of the Jost functions.
  8. Scattering map from initial data to reflection coefficient
  9. An RH characterization of the mKdV equation
  10. Signature tables
  11. Deformation of the RH Problem
  12. Interpolation and conjugation
  13. Opening $\bar{\partial}$ lenses
  14. The large-time asymptotic analysis
  15. Asymptotics of $N$-soliton solution
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose the initial data $q_0\mp1\in H^{4,4}(\mathbb{R}^{{\pm}})$ with scattering data $\left\{r(z),\{z_j,c_j\}_{j=0}^{N-1}\right\}$. Order $z_j$ such that and define $\xi=\frac{x}{t}$. Let $q^{(sol),N}(x,t)$ be the $N$-soliton solution whose scattering data can be denoted by $\{\tilde{r}\equiv0,\{z_j,\tilde{c}_j\}_{j=0}^{N-1}\}$, where For fixed $\xi_0\in(0,2)$, there exist constants $t_0=t_0(q

Figures (7)

  • Figure 1: The $(x,t)$-plane is divided into three kinds of asymptotic regions: Solitonic region, $-6 <\xi\le-2$; Solitonless region, $\xi<-6$ and $\xi>-2$; Transition region, $\xi\approx -6$.Here, $\xi:=x/t$.
  • Figure 2: Distribution of the discrete spectrum on circle $|z|=1$: the jump contour $\mathbb{R}$, the red dots ($\centerdot$) and green dots ($\centerdot$) represent the zeros of $a(z)$, they are corresponding to breathers and solitons respectively. The blue dots ($\centerdot$) represent singularities.
  • Figure 3: Plots of the distributions for saddle points: $\textbf{(a)}$$\xi<-6$, $\textbf{(b)}$$-6<\xi<6$, $\textbf{(c)}$$\xi>6$. The red curve represents $\mathop{\rm Re}\nolimits \theta'(z)=0$, and the green curve represents $\mathop{\rm Im}\nolimits \theta'(z)=0$. The intersection points are the saddle points which represent the zeros of $\theta'(z)=0$.
  • Figure 4: Signature table of $\mathop{\rm Re}\nolimits (2\text{i} t \theta(z))$ with different $\xi$: $\textbf{(a)}$$\xi<-6$, $\textbf{(b)}$$-6<\xi<-2$, $\textbf{(c)}$$\xi>-2$. In the purple region, $\mathop{\rm Re}\nolimits(2\text{i} t\theta)<0$, while in the white region, $\mathop{\rm Re}\nolimits(2\text{i} t\theta)>0$. The purple dashed curves are the critical curves.
  • Figure 5: The dashed lines are where $\mathop{\rm Re}\nolimits z=\pm\xi_0$. We divide the discrete spectrum in the first quadrant $D_1$ into three sets: $\triangle\setminus\Lambda$, $\nabla\setminus\Lambda$ and $\Lambda$, with the poles reserved in $\Lambda$. By the symmetries, the discrete spectrum in the second quadrant is also divided into three sets.
  • ...and 2 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • ...and 32 more