2021/08/31 by Xun, Weikang, Fan, Engui · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2108.13604
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 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)=usol(x,t| σd(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)= 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)= \frac1t1/3 uP (\fracxt1/3 ) + O (t2/(3p)-1/2 ), 4