Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces


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

  • Share on Facebook
  • Share on Twitter
  • Share on LinkedIn
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


MSC


References