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Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition


Authors: Shou-Fu Tian and Tian-Tian Zhang
Journal: Proc. Amer. Math. Soc. 146 (2018), 1713-1729
MSC (2010): Primary 35Q55, 35Q51; Secondary 35P30, 81Q05
DOI: https://doi.org/10.1090/proc/13917
Published electronically: December 4, 2017
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Abstract: The Gerdjikov-Ivanov (GI) type of derivative nonlinear Schrödinger equation is considered on the quarter plane whose initial data vanish at infinity while boundary data are time-periodic, of the form $ ae^{i\delta }e^{2i\omega t}$. The main purpose of this paper is to provide the long-time asymptotics of the solution to the initial-boundary value problems for the equation. For $ \omega <a^{2}(\frac {1}{4}a^{2}+3b-1)$ with $ 0<b<\frac {a^{2}}{4}$, our results show that different regions are distinguished in the quarter plane $ \Omega =\{(x,t)\in \mathbb{R}^{2}\vert x>0, t>0\}$, on which the asymptotics admit qualitatively different forms. In the region $ x>4tb$, the solution is asymptotic to a slowly decaying self-similar wave of Zakharov-Manakov type. In the region $ 0< x <4t\left (b-\sqrt {2a^{2}\left (\frac {a^{2}}{4}-b\right )}\right )$, the solution takes the form of a plane wave. In the region $ 4t\left (b-\sqrt {2a^{2}\left (\frac {a^{2}}{4}-b\right )}\right )<x<4tb$, the solution takes the form of a modulated elliptic wave.


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Additional Information

Shou-Fu Tian
Affiliation: School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China –and– Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
Address at time of publication: School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China
Email: sftian@cumt.edu.cn, shoufu2006@126.com

Tian-Tian Zhang
Affiliation: School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China
Email: ttzhang@cumt.edu.cn

DOI: https://doi.org/10.1090/proc/13917
Keywords: Integrable system, Riemann-Hilbert problem, initial-boundary value problem, long-time asymptotic, nonlinear steepest-descent method.
Received by editor(s): June 20, 2017
Published electronically: December 4, 2017
Additional Notes: Project supported by the Fundamental Research Fund for the Central Universities under Grant No. 2017XKQY101.
Shou-fu Tian and Tian-Tian Zhang serve as corresponding authors.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society

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