Stability of standing waves for a nonlinear Klein–Gordon equation with delta potentials

Elek Csobo, Francois Genoud, Masahito Ohta, Julien Royer

Research output: Contribution to journalArticleScientificpeer-review

Abstract

In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein–Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike the unperturbed case, a noteworthy difficulty here arises from the possible non-unitarity of the semigroup generating the corresponding linear evolution. We then show that the equation is Hamiltonian and we establish several stability/instability results for its standing waves. Our analysis relies on a detailed study of the spectral properties of the linearization of the equation, and on the well-known ‘slope condition’ for orbital stability.

Original languageEnglish
Pages (from-to)353-388
Number of pages36
JournalJournal of Differential Equations
Volume268
Issue number1
DOIs
Publication statusPublished - 2019

Keywords

  • Delta potential
  • Nonlinear Klein–Gordon equation
  • Orbital stability
  • Standing waves

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