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

The Sasa-Satsuma equation with 3 × 3 matrix spectral problem 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 Rieamnn-Hilbert 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) = u s o l (x , t | σ d (I)) + t − 1 / 2 h + 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) = u s o l (x , t | σ d (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 / t 1 / 3 | = O (1) , the Painlevé asymptotic is found by u (x , t) = 1 t 1 / 3 u P (x t 1 / 3 ) + O (t 2 / (3 p) − 1 / 2) , 4 < p < ∞ , in which the leading term is a solution to a modified Painlevé II equation, the second term is a residual error from a ∂ ‾ -equation. [ABSTRACT FROM AUTHOR]