Asymptotics for the Sasa--Satsuma equation in terms of a modified Painlev\'e II transcendent

Abstract

We consider the initial-value problem for the Sasa-Satsuma equation on the line with decaying initial data. Using a Riemann-Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the sector $|x| \leq M t^{1/3}$, $M$ constant. It turns out that the asymptotics can be expressed in terms of the solution of a modified Painlev\'e II equation. Whereas the standard Painlev\'e II equation is related to a $2 \times 2$ matrix Riemann-Hilbert problem, this modified Painlev\'e II equation is related to a $3 \times 3$ matrix Riemann--Hilbert problem.