Purely nonlinear instability of standing waves with minimal energy

  1. Andrew Comech1,
  2. Dmitry Pelinovsky2

Article first published online: 5 SEP 2003

DOI: 10.1002/cpa.10104

Issue

Communications on Pure and Applied Mathematics

Communications on Pure and Applied Mathematics

Volume 56, Issue 11, pages 1565–1607, November 2003

Additional Information

How to Cite

Comech, A. and Pelinovsky, D. (2003), Purely nonlinear instability of standing waves with minimal energy. Comm. Pure Appl. Math., 56: 1565–1607. doi: 10.1002/cpa.10104

Author Information

  1. 1

    University of North Carolina, Mathematics Department, Chapel Hill, NC 27599

  2. 2

    McMaster University, Department of Mathematics, Hamilton, Ontario L8S 4K1, Canada

Publication History

  1. Issue published online: 5 SEP 2003
  2. Article first published online: 5 SEP 2003
  3. Manuscript Received: SEP 2002

Funded by

  • NSF. Grant Numbers: 0296036, 0200880
  • NSERC Grant. Grant Number: 5-36694

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Abstract

We consider Hamiltonian systems with U(1) symmetry. We prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable, in spite of the absence of linear instability. Essentially, the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system. We apply our theory to the nonlinear Schrödinger equation. © 2003 Wiley Periodicals, Inc.

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