Periodic Solutions of Nonlinear Schrödinger Equations and the Nash-Moser Method

  • Walter Craig
  • C. Eugene Wayne
Part of the NATO ASI Series book series (NSSB, volume 331)

Abstract

In this paper we construct time-periodic solutions of the nonlinear Schrödinger equation
$$ - i{\partial _t}u = \left( { - {\partial _{xx}}u + v\left( x \right)} \right)u + g\left( {u,\bar u,x} \right),0 < x < \pi ,t \in \mathbb{R}. $$
(1.1)
satisfying either periodic or Dirichlet boundary conditions. We assume that the non-linearity g is of the form \( g\left( {u,{\mkern 1mu} \bar u,{\mkern 1mu} x} \right) = {\partial _{\bar u}}\mathcal{G}\left( {u,{\mkern 1mu} \bar u,{\mkern 1mu} x} \right) \), for some real valued function G. This equation can then be viewed as a Hamiltonian system with infinitely many degrees of freedom, and the problem of time periodic solutions is related to perturbation theory in the neighborhood of elliptic stationary points of such systems. In contrast to the case of finite dimensional Hamiltonian systems, even the problem of periodic solutions for (1.1) exhibits the phenomenon of small denominators. Our results imply that the effect of these small denominators may be overcome, resulting in the construction of Cantor-like families of periodic solutions in a neighborhood of the equilibrium u = 0. We will present the details here only for the case of periodic boundary conditions, since this case is technically more difficult, allows for interesting resonances between the linear modes, and because the case of Dirichlet boundary conditions has already been treated by Kuksin (1988, 1993) using KAM methods. We note that in the special case in which g depends on u and ū only through the combination |u|2, one does not encounter small denominators, and one can construct periodic solutions with an ordinary implicit function theorem.

Keywords

Periodic Solution Nonlinear Wave Equation Frequency Sequence Neumann Series Time Periodic Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Walter Craig
    • 1
  • C. Eugene Wayne
    • 2
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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