Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition
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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 $\Omega =\{(x,t)\in \mathbb {R}^{2}|x>0, t>0\}$, 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.References
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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
- MR Author ID: 894745
- 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
- MR Author ID: 907456
- Email: ttzhang@cumt.edu.cn
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3754355