Long-Time Asymptotics of the Sawada–Kotera Equation on the Line

Source abstract

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 Painlevé region, and the rapid decay region. Notably, the Painlevé region is governed by the F-XVIII equation in the Painlevé classification of fourth-order ordinary differential equations, a fourth-order analogue of the Painlevé 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 Painlevé 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.