Long time asymptotics of a perturbed modified KdV equation

TL;DR

The paper analyzes the long-time behavior of a perturbed defocusing mKdV equation $u_t+u_{xxx}-6u^2u_x=\varepsilon u^{\ell}$ by embedding it in the inverse scattering framework and employing the $\overline{\partial}$-steepest descent method. It demonstrates that, for small $\varepsilon$ and initial data $u_0\in H^{2,1}$ with $\ell>9$, the reflection data converge to a limit $r_\infty$ and the solution $u(x,t)$ shares the same leading-order long-time asymptotics as the unperturbed mKdV, described regionwise across five space-time regions with Painlevé II behavior in Regions II–IV and a universal decay $\|u(\cdot,t)\|_{L^\infty}\lesssim (1+t)^{-1/3}$. The analysis relies on a refined Beals-Coifman RH problem for mKdV, a perturbative treatment of the scattering data under small forcing, and a $\overline{\partial}$-steepest descent framework to obtain sharp $L^\infty$ bounds and contraction estimates. This establishes the robustness of the long-time scattering behavior of mKdV under higher-order perturbations and provides explicit asymptotic formulas with region-specific leading terms. The results have implications for understanding stability of dispersive systems under small external forcing and for extending integrable-system perturbation theory to higher-order perturbations.

Abstract

We derive full asymptotics of the modified KdV equation (mKdV) with a higher-order perturbative term. We make use of the perturbative theory of infinite-dimensional integrable systems developed by P. Deift and X. Zhou \cite{DZ-2}, and some new and simpler proofs of certain $L^\infty$ bounds and $L^p$ a priori estimates developed recently in \cite{CLT}. We show that the perturbed equation exhibits the same long-time behavior as the completely integrable mKdV.

Figures (3)

• Figure 1: Five Regions
• Figure 2: $\Sigma^{\textrm{LC}}$
• Figure 3: $\Sigma$-Painlevé

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.3
• Corollary 1.4
• Proposition 1.6
• Theorem 1.7
• proof
• Theorem 1.8
• proof
• Corollary 1.9
• Remark 1.10