The Painlevé-type asymptotics of defocusing complex mKdV equation with finite density initial data

TL;DR

This work establishes Painlevé-II type long-time asymptotics for the defocusing complex mKdV equation with finite density initial data in a transition region where the scaled stationary phase structure interacts with the nonzero boundary. It develops a refined $\bar\partial$-steepest descent framework, including pole removal, a hybrid $\bar\partial$-RHP deformation, and a local Painlevé-II model near $z=\pm1$, to obtain a precise asymptotic formula for $q(x,t)$. The main result expresses $q(x,t)$ in terms of Painlevé-II transcendent $u(s)$, with explicit dependence on scattering data and a computable phase parameter, plus controlled error terms, thereby linking nonlinear dispersive dynamics to integrable Painlevé-type structures. The approach accommodates finite-density data and demonstrates the applicability of hybrid $\bar\partial$-RH methods to transition regions beyond soliton-free or purely radiation regimes, enhancing understanding of universal Painlevé-type behavior in dispersive waves.

Abstract

We consider the Cauchy problem for the defocusing complex mKdV equation with finite density initial data \begin{align*} &q_t+\frac{1}{2}q_{xxx}-3|q|^2q_{x}=0,\\ &q(x,0)=q_{0}(x) \sim \pm 1, \ x\to \pm\infty, \end{align*} which can be formulated into a Riemann-Hilbert (RH) problem. With $\bar\partial$-generation of the nonlinear steepest descent approach and a double scaling limit technique, in the transition region $$\mathcal{D}:=\left\{(x,t)\in\mathbb{R}\times\mathbb{R}^+\big|-C< \left(x/(2t)+3/2\right) t^{2/3}<0, C\in\mathbb{R}^+\right\},$$ we find that the long-time asymptotics of the solution $q(x,t)$ to the Cauchy problem is associated with the Painlevé-II transcendents.

Figures (10)

• Figure 1: The space-time regions of $x$ and $t$, the red solid line is $\xi=-3/2$. The two red dashed lines which belong to the regions $\mathcal{D}_1$ and $\mathcal{D}_2$, respectively, approach to the red solid line, these can be expressed as $\xi\approx-3/2$. The region passed through by the dashed lines is the transition region.
• Figure 2: The discrete spectrum is restricted to the unit circle $\mathcal{G}: \{\nu_n\in\mathbb{C}^+, \overline{\nu}_n\in\mathbb{C}^-\mid|\nu_n|=|\overline{\nu}_n|=1\}$. The real axis $\mathbb{R}$ is the jump contour of the RH problem \ref{['RHP0']}.
• Figure 3: The signature table of $\mathop{\mathrm{Re}}\limits [2i\theta(z)]$. The gray dashed lines denote the critical line $\mathcal{Y}:\mathop{\mathrm{Re}}\limits (2i\theta(z))=0$. In the gray regions, we have $\mathop{\mathrm{Re}}\limits [2i\theta(z)]>0$, which implies that $\left|e^{-2it\theta(z)}\right| \to 0$ as $t\to \infty$. In the white regions, $\mathop{\mathrm{Re}}\limits [2i\theta(z)]<0$, which implies that $\left|e^{2it\theta(z)}\right| \to 0$ as $t\to \infty$. The green dashed circle is the unit circle $\mathcal{G}:$$\{z\in\mathbb{C}\mid|z|=1\}$.
• Figure 4: The jump contour $\Sigma^{(1)}$ of $M^{(1)}(z)$.
• Figure 5: The jump contour $\Gamma$ be opened as $\Sigma$ .

Theorems & Definitions (35)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• Proposition 4.1