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 Schr\"{o}dinger 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 $$q(x,t)=u_{sol}(x,t| \sigma_{d}(\mathcal{I})) + t^{-1/2} h + \mathcal{O} (t^{-3/4}). $$ 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 $$ u(x,t)= u_{sol}(x,t| \sigma_{d}(\mathcal{I})) + \mathcal{O}(t^{-1}).$$ 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 $$ u(x,t)= \frac{1}{t^{1/3}} u_{P} \left(\frac{x}{t^{1/3}} \right) + \mathcal{O} \left(t^{2/(3p)-1/2} \right), \qquad 4<p < \infty.$$ 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.