Long-time asymptotics of the good Boussinesq equation and its modified version: Painlev\'{e} region

TL;DR

This paper analyzes the long-time asymptotic behavior of the good and modified Boussinesq equations in the Painlevé region using Riemann-Hilbert problem techniques, deriving formulas linked to Painlevé IV solutions.

Contribution

It develops a novel asymptotic analysis for these equations in the Painlevé region, connecting solutions to Painlevé IV and employing the Deift-Zhou method.

Findings

Asymptotic formulas for the modified Boussinesq equation in Painlevé regions.

Leading-order asymptotics for the good Boussinesq equation via Miura transformation.

Validation of theoretical results through numerical simulations.

Abstract

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