Long time and Painlevé-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions

The Sasa-Satsuma equation with 3 × 3 Lax representation is one of the integrable extensions of the nonlinear Schrödinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. Based on the RieamnnHilbert problem characterization for the Cauchy problem and the ∂-nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1) For the region x < 0, |x/t| = O(1), the long time asymptotic is given by q(x, t) = usol(x, t|σd(I)) + t−1/2h +O(t−3/4). in which the leading term is N(I) solitons, the second term the second t−1/2 order term is soliton-radiation interactions and the third term is a residual error from a ∂ equation. (2) For the region x > 0, |x/t| = O(1), the long time asymptotic is given by u(x, t) = usol(x, t|σd(I)) +O(t−1). in which the leading term is N(I) solitons, the second term is a residual error from a ∂ equation. (3) For the region |x/t1/3| = O(1), the Painleve asymptotic is found by u(x, t) = 1 t1/3 uP ( x t1/3 ) +O ( t2/(3p)−1/2 ) , 4 < p < ∞. in which the leading term is a solution to a modified Painleve II equation, the second term is a residual error from a ∂ equation. 1 School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, P.R. China. * Corresponding author and email address: faneg@fudan.edu.cn