It is known that both the Sawada–Kotera equation and the Kaup–Kupershmidt equation are related with the same modified equation by different Miura transformations. There is singularity at the origin in the spectral problems of the Sawada–Kotera equation and the Kaup–Kupershmidt equation. Instead, this work investigates the forward and inverse scattering problems of the modified Sawada–Kotera equation by Riemann–Hilbert approach to avoid the singularity at the origin. The Riemann–Hilbert problem along with the reconstructing formula of the modified Sawada–Kotera equation are proposed. Moreover, the properties of the reflection coefficients are analysed rigorously. The results in this paper make an important step toward the long-time asymptotics of the modified Sawada–Kotera equation.
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 q xx-term and its modified version.
The modified Sawada-Kotera equation is investigated by prolongation technique and Painleve singularity analysis. As a result, the Lax pair and conservation laws of the modified Sawada-Kotera equation are formulated. It is proved that this equation pasts the Painleve test in sense of having enough arbitrary functions at its resonant points. The auto-Backlund transformation and exact solutions of the modified Sawada-Kotera equation are obtained explicitly.
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