In 2003, Gilson et al studied the bilinearization and multisoliton soltions of the Sasa-Satsuma equation. It was shown that the Sasa-Satsuma equation is solvable by the means of inverse scattering transformation IST, and the soliton solution can propagate steadily. Early in 99, Sasa and Satsuma present the Sasa-Satsuma equation as a new-type high-order NLS equation. In the past few years, Sasa-Satsuma equation has attracted much attention and widely been studied. The Sasa-Satsuma equation is one of the integrable extensions of the nonlinear Schrödinger NLS equation, and it plays an important role in a number of physical science areas due to its rich mathematical structure and physical background such as deep water waves and dispersive nonlinear media. ux, 0 = u 0 x SR.2 where initial data u 0 x, 0 belongs to the Schwarz space SR. 3 For the region x/t /3 = O, the Painleve asymptotic is found by ux, t = x t /3 u P t /3 +O t 2/3p /2, 4 0, x/t = O A mixed -RH problem Solution of the mixed -RH problem Long time asymptotic behaviors in region II Painleve asymptotics in Region III: x/t /3 = O 38 2ģ Introduction In this paper, we focus on the long time asymptotics of the Cauchy problem for the Sasa- Satsuma equation u t + u xxx + 6 u 2 u x + 3u u 2 x = 0, x, t R R +. in which the leading term is NI solitons, the second term is a residual error from a equation. 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: For the region x 0, x/t = O, the long time asymptotic is given by ux, t = u sol x, t σ d I+Ot. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. 1 arxiv: v2 3 Sep 202 Long time and Painleve-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions Weikang Xun and Engui FAN * Abstract The Sasa-Satsuma equation with 3 3 Lax representation is one of the integrable extensions of the nonlinear Schrödinger equation.
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