Long-time asymptotics of the good Boussinesq equation and its modified version: Painlevé region

TL;DR

This work resolves the long-standing question of the middle-field, Painlevé-region asymptotics for the good and modified Boussinesq equations by reducing a $3\times3$ Riemann-Hilbert problem to a Painlevé IV model via the Deift-Zhou steepest descent method. The authors derive explicit leading-order formulas for the modified Boussinesq equation in the Painlevé region and through the Miura transformation obtain corresponding Painlevé IV asymptotics for the good Boussinesq equation, with precise error bounds in the Painlevé region and two Painlevé transition subregions. The Clarkson-McLeod Painlevé IV solution enters prominently, with $y=-\frac{\sqrt{3}\,x}{2\sqrt{t}}$ and parameters $\alpha=-\tfrac{1}{6}$, $\beta=-\tfrac{2}{3}$, producing the universal Painlevé profile that governs the long-time dynamics. The results are validated against direct numerical simulations, confirming the accuracy of the Painlevé-based asymptotics and their consistency with the known dispersive-wave regime. The study also clarifies the role of the Miura transformation in connecting the good and modified Boussinesq equations in this asymptotic regime.

Abstract

This work investigates the long-time asymptotic behaviors of initial value problem for the good Boussinesq equation and the modified Boussinesq equation in Painlevé region. The Deift-Zhou steepest descent method is used to deform the associated $3 \times 3$ Riemann-Hilbert problem to the Painlevé IV model. Then asymptotic formulas for the modified Boussinesq equation in both the Painlevé region and the Painlevé transition region are derived, characterized by the Clarkson-McLeod solution of the Painlevé IV equation. Additionally, the leading-order term of the good Boussinesq equation in Painlevé region is obtained via the Miura transformation. The theoretical asymptotic solutions are validated against direct numerical simulations, confirming the accuracy of the asymptotic analysis.

Figures (8)

• Figure 1: The jump contour $\Sigma=\cup_{i=1}^{6}\Sigma_i$ and the six open sets $D_{j}$ for $j=1,2,\cdots,6$.
• Figure 2: The asymptotics regions in the upper half $(x,t)$-plane for the modified Boussinesq equation (\ref{['mbequation']}).
• Figure 3: Comparisons of the asymptotic solutions based on the Painlevé $\rm IV$ equation in (\ref{['mbresult']})-(\ref{['mbresult-3']}) with the full numerical simulations of the modified Boussinesq equation (\ref{['mbequation']}) under the initial condition (\ref{['initialmb']}) at $t = 100$ and $t = 300$. The solid green and blue lines represent the results of direct numerical simulations on $q(x,t)$ and $p(x,t)$, respectively, while the dashed red lines represent the asymptotic solutions.
• Figure 4: Comparisons of the asymptotic solutions based on the Painlevé $\rm IV$ equation in (\ref{['gbresult']}) with the full numerical simulations of the good Boussinesq equation (\ref{['good-boussinesq-1']}) at $t = 100$ and $t = 300$, respectively. The left and right panels correspond to two respective time points.
• Figure 5: From left to right: The signature tables and saddle points of the new phase functions $\Phi_{21}$, $\Phi_{31}$, and $\Phi_{32}$ for $\zeta=1$. The grey regions correspond to $\{k \mid \Re \Phi_{ij} > 0\}$, while the white regions correspond to $\{k \mid \Re \Phi_{ij} < 0\}$.

Theorems & Definitions (31)

• Proposition 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Remark 2.6
• Remark 2.7
• Lemma 3.1
• proof
• Proposition 3.2