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

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.