  Both Sawada-Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are
integrable systems with third-order Lax operator. Moreover, they are related
with the same modified nonlinear equation (called modified SK-KK equation) by
Miura transformations. This work first constructs the Riemann-Hilbert problem
associated with the SK equation, KK equation and modified SK-KK equation by
direct and inverse scattering transforms. Then the long-time asymptotics of
these equations are studied based on Deift-Zhou steepest-descent method for
Riemann-Hilbert problem. Finally, it is shown that the asymptotic solutions
match very well with the results of direct numerical simulations.
