Long-time asymptotics of the good Boussinesq equation with q xx -term and its modified version

Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with qxx-term and its modified version.