Long time and Painlevé-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions
Long time and Painlevé-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions
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孤子时空域 Sasa-Satsuma 方程的长时和 Painlevé 型渐近
DOI：
10.1016/j.jde.2022.05.006
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发表时间：
2021-08
期刊：
影响因子：
--
通讯作者：
Engui Fan
中科院分区：
文献类型：
--
作者：
Weikang Xun;Engui Fan
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 $x0, |x/t|=mathcal{O}(1)$, 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 $overlinepartial$ 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 $overlinepartial$ equation.
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 $x0, |x/t|=mathcal{O}(1)$, 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 $overlinepartial$ 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 $overlinepartial$ equation.
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DOI：
10.1016/j.physd.2014.11.006
发表时间：
2015-02
期刊：
Physica D: Nonlinear Phenomena
影响因子：
--
作者：
N. Akhmediev;J. Soto-Crespo;N. Devine;N. Hoffmann
通讯作者：
N. Akhmediev;J. Soto-Crespo;N. Devine;N. Hoffmann
影响因子：
1.7
作者：
A. B. D. Monvel;D. Shepelsky
通讯作者：
A. B. D. Monvel;D. Shepelsky
影响因子：
1.3
作者：
Jianke Yang;D. Kaup
通讯作者：
Jianke Yang;D. Kaup
影响因子：
2.4
作者：
Jian Xu;E. Fan
通讯作者：
Jian Xu;E. Fan
影响因子：
2.4
作者：
Huan Liu;X. Geng;Bo Xue
通讯作者：
Huan Liu;X. Geng;Bo Xue