Long-time asymptotics of the Sawada-Kotera equation on the line

Abstract

The Sawada-Kotera (SK) equation is an integrable system characterized by a third-order Lax operator and is related to the modified Sawada-Kotera (mSK) equation through a Miura transformation. This work formulates the Riemann-Hilbert problem associated with the SK and mSK equations by using direct and inverse scattering transforms. The long-time asymptotic behaviors of the solutions to these equations are then analyzed via the Deift-Zhou steepest descent method for Riemann-Hilbert problems. It is shown that the asymptotic solutions of the SK and mSK equations are categorized into four distinct regions: the decay region, the dispersive wave region, the Painlev\'{e} region, and the rapid decay region. Notably, the Painlev\'{e} region is governed by the F-XVIII equation in the Painlev\'{e} classification of fourth-order ordinary differential equations, a fourth-order analogue of the Painlev\'{e} transcendents. This connection is established through the Riemann-Hilbert formulation in this work. Similar to the KdV equation, the SK equation exhibits a transition region between the dispersive wave and Painlev\'{e} regions, arising from the special values of the reflection coefficients at the origin. Finally, numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.