Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces
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Authors
Min Zhang
- College of Science, China University of Petroleum, Qingdao, Shandong 266580, People’s Republic of China.
Jie Rui
- College of Science, China University of Petroleum, Qingdao, Shandong 266580, People’s Republic of China.
Abstract
In this paper, d-dimensional (dD) quasi-periodically forced nonlinear Schrödinger equation with a general nonlinearity
\[iu_t - \Delta u +M_\xi u + \varepsilon\phi (t)(u + h(|u| ^2)u) = 0, \quad x\in \mathbb{T}^d,\quad t\in \mathbb{R}\]
under periodic boundary conditions is studied, where \(M_\xi\) is a real Fourier multiplier and \(\varepsilon\) is a small positive parameter, \(\phi (t)\)
is a real analytic quasi-periodic function in t with frequency vector \(\omega=(\omega_1,\omega_2,...,\omega_m)\) , and \(h(|u| ^2)\) is a real analytic function
near \(u = 0\) with \(h(0) = 0\). It is shown that, under suitable hypothesis on \(\phi (t)\), there are many quasi-periodic solutions for the
above equation via KAM theory.
Share and Cite
ISRP Style
Min Zhang, Jie Rui, Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3670--3693
AMA Style
Zhang Min, Rui Jie, Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces. J. Nonlinear Sci. Appl. (2017); 10(7):3670--3693
Chicago/Turabian Style
Zhang, Min, Rui, Jie. "Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3670--3693
Keywords
- Quasi-periodically forced
- KAM theory
- Schrödinger equation
- quasi-periodic solutions.
MSC
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