Direct and inverse scattering problems of the modified Sawada–Kotera equation: Riemann–Hilbert approach

Abstract

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.