Frontiers of Mathematics in China >

2009 , Vol. 4 >Issue 3: 463 - 482

DOI: https://doi.org/10.1007/s11464-009-0029-1

RESEARCH ARTICLE

Almost periodic solutions for a class of higher dimensional Schrödinger equations

  • Jiansheng GENG
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  • Department of Mathematics and Institute of Mathematical Science, Nanjing University, Nanjing 210093, China

Received date: 06 Jun 2008

Accepted date: 11 Sep 2009

Published date: 05 Sep 2009

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

In this paper, we show that there are almost periodic solutions corresponding to full dimensional invariant tori for higher dimensional Schröodinger equations with Fourier multiplier iut-Δu+Mξu+f(|u|2)u = 0, subject to periodic boundary conditions, where the nonlinearity f is a realanalytic function near u = 0 with f(0) = 0.The proof is based on an improved infinite dimensional KAM theorem.

Key words： Schrodinger equation; almost-periodic solution; KAM method

Cite this article

Jiansheng GENG . Almost periodic solutions for a class of higher dimensional Schrödinger equations[J]. Frontiers of Mathematics in China, 2009 , 4(3) : 463 -482 . DOI: 10.1007/s11464-009-0029-1

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Bourgain J. Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. International Mathematics Research Notices, 1994, 11: 475-497

DOI

2
Bourgain J. Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom Funct Anal, 1995, 5: 629-639

DOI

3
Bourgain J. Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. Geom Funct Anal, 1996, 6: 201-230

DOI

4
Bourgain J. Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Annals of Mathematics, 1998, 148: 363-439

DOI

5
Bourgain J. Nonlinear Schrödinger Equations. Park City Series 5. Providence: American Mathematical Society, 1999

6
Bourgain J. On invariant tori of full dimension for 1D periodic NLS. J Funct Anal, 2005, 229: 62-94

DOI

7
Bourgain J. Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Annals of Mathematics Studies, Vol 158. Princeton: Princeton University Press, NJ, 2005

8
Craig W, Wayne C E. Newton’s method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math, 1993, 46: 1409-1498

DOI

9
Eliasson H L, Kuksin S B. KAM for the non-linear Schrödinger equation. Annals of Mathematics (to appear)

10
Geng J, You J. KAM tori of Hamiltonian perturbations of 1D linear beam equations. J Math Anal Appl, 2003, 277: 104-121

DOI

11
Geng J, You J. A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun Math Phys, 2006, 262: 343-372

DOI

12
Geng J, You J. KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity, 2006, 19: 2405-2423

DOI

13
Kuksin S B. Nearly Integrable Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, Vol 1556. Berlin: Springer, 1993

14
Kuksin S B, Pöschel J. Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation. Ann Math, 1996, 143: 149-179

DOI

15
Niu H, Geng J. Almost periodic solutions for a class of higher-dimensional beam equations. Nonlinearity, 2007, 20: 2499-2517

DOI

16
Pöschel J. Quasi-periodic solutions for a nonlinear wave equation. Comment Math Helvetici, 1993, 71: 269-296

DOI

17
Pöschel J. A KAM theorem for some nonlinear partial differential equations. Ann Sc Norm sup Pisa CI sci, 1996, 23: 119-148

18
Pöschel J. On the construction of almost periodic solutions for a nonlinear Schrödinger equations. Ergod Th and Dynam Syst, 2002, 22: 1537-1549

19
Yuan X. A KAM theorem with applications to partial differential equations of higher dimensions. Commun Math Phys, 2007, 275: 97-137

DOI

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