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

Abstract

The Sasa-Satsuma equation with $3\times 3$ 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 Rieamnn-Hilbert problem characterization for the Cauchy problem and the $\overline{\partial }$-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 $x<0,|x/t|=\mathcal{O}(1)$, the long time asymptotic is given by $$q(x,t)=u_{sol}(x,t|\ddot{I}{ƒ}_d(\mathcal{I}))+t^{-1/2}h+\mathcal{O}(t^{-3/4}).$$ 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 $\overline{\partial }$ equation. (2)\ For the region $x>0,|x/t|=\mathcal{O}(1)$, the long time asymptotic is given by $$u(x,t)=u_{sol}(x,t|\ddot{I}{ƒ}_d(\mathcal{I}))+\mathcal{O}(t^{-1}).$$ in which the leading term is $N(I)$ solitons, the second term is a residual error from a $\overline{\partial }$ equation. (3) \ For the region $|x/t^{1/3}|=\mathcal{O}(1)$, the Painleve asymptotic is found by $$u(x,t)=\frac{1}{t^{1/3}}{u}_P\left(\frac{x}{t^{1/3}}\right)+\mathcal{O}\left(t^{2/(3p)-1/2}\right),4<p<\infty .$$ in which the leading term is a solution to a modified Painleve $\mathrm{II}$ equation, the second term is a residual error from a $\overline{\partial }$ equation.

46 pages