On the large-time asymptotics of the defocusing mKdV equation with step-like initial data

TL;DR

This work analyzes the defocusing mKdV equation with step-like boundaries $q(x,t) o c_l$ and $c_r$ ($c_l>c_r>0$) using the inverse scattering transform and a rigorous nonlinear steepest-descent RH framework. By formulating a time-evolving RH problem and introducing region-adapted $g$-functions and analytic approximations, the authors derive precise large-time asymptotics in three principal space-time regimes: RI with a $t^{-1/2}$-oscillatory approach to $c_l$, RII with a slowly varying profile bridging the two constants, and RIII with a $t^{-2}$-scale approach to $c_r$ (including a subregion $ ext{RIII}_a$ with a Beurling-type local parametrix). The analysis employs global and local parametrices (parabolic-cylinder, Airy, and Bessel models) and small-norm RH theory to obtain leading terms, sub-leading corrections, and explicit error bounds, thereby establishing a rigorous description of rarefaction-wave dynamics for nonzero-step data. The results extend the inverse-scattering approach for step-like data and provide a robust framework for comparing rarefaction waves with dispersive shocks in defocusing integrable PDEs. They also illustrate how asymptotic data can be extracted from boundary-dependent scattering information via analytic approximations when exact closed-form scattering data are unavailable. The findings have implications for understanding long-time dispersive dynamics in media with nonzero far-field states and for developing sharp asymptotics in integrable models with non-vanishing boundaries.

Abstract

We study the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial data approaching nonzero constants $c_l$ and $c_r$ as $x \to -\infty$ and $x\to+\infty$, respectively. Assuming $c_l>c_r>0$, the solution exhibits a rarefaction wave structure. We first develop the inverse scattering transform for the solution satisfying these step-like boundary conditions. Using the associated scattering data, we prove that there exists a unique global solution of the Cauchy problem and characterize it in terms of a Riemann-Hilbert (RH) problem. By applying the nonlinear steepest descent method to this RH problem, we rigorously obtain large-time asymptotics of rarefaction wave solution in three distinct space-time regions, each characterized by a different leading order behavior. They are: (I) a left-field region where the solution approaches the left background constant, modulo a small oscillatory correction, (II) a central region where the solution exhibits a slowly varying profile that transitions between the two constants, and (III) a right-field region where the solution tends to the right background constant, up to an algebraically small correction. Rigorous derivations of the leading terms, sub-leading terms as well as the error bounds are presented.

Figures (17)

• Figure 1: The evolution of $q(x,t)$ for the step-like initial data \ref{['Initial data']} with $c_l=1.2 > c_r=0.4$. (left) Heatmap of $q(x,t)$. (right) The black dashed curve shows the initial profile $q(x,0)$, while the blue curve displays $q(x,t)$ at $t=5$. The solution develops a rarefaction wave, exhibiting a slowly varied profile between the two constant states.
• Figure 2: Evolution of $q(x,t)$ for the step-like initial data \ref{['Initial data']} with $c_l=0.4 < c_r=1.2$. (left) Heatmap of $q(x,t)$. (right) The black dashed curve shows the initial profile $q(x,0)$, while the blue curve displays $q(x,t)$ at $t=5$. The solution develops a dispersive shock wave, featuring a region of rapid oscillations between the two constant states.
• Figure 3: The different asymptotic regions of the $(x,t)$-half plane.
• Figure 4: Signature table of $\text{\upshape Im\,} g_{\textup{I}}(\xi;k)$ for $\xi\in\mathcal{R}_{\textup{I}}$.
• Figure 5: The jump contours $\Gamma^{(3)}$ and regions $U^{(3)}_j,U^{(3)*}_j,\;j=1,2,3$ of RH problem for $M^{(3)}$ when $\xi\in\mathcal{R}_\textup{I}$.

Theorems & Definitions (40)

• Definition 2.1: lenellsmkdvfourier
• Theorem 2.2
• proof
• Definition 2.3
• Theorem 2.4
• proof
• Proposition 3.1
• Remark 3.2
• Proposition 3.3
• proof