Long-time asymptotics of the good Boussinesq equation and its modified version: Painlev\'e 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 Painlev\'e region. The Deift-Zhou steepest descent method is used to deform the associated 3 × 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.
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