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.

Cet article explore les excursions dans le temps et leurs implications.

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