Long-time asymptotics of the good Boussinesq equation with <i>q</i>_<i>xx</i>-term and its modified version

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