Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity

Andrew Comech; Elena Kopylova

doi:10.3934/cpaa.2021063

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2021, Volume 20, Issue 6: 2187-2209. Doi: 10.3934/cpaa.2021063

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Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity

Andrew Comech^1, and
Elena Kopylova^2, ,

1.
Texas A & M University, College Station, TX, Institute for Information Transmission Problems, Moscow, Russia
2.
Vienna University, Vienna, Austria, Institute for Information Transmission Problems, Moscow, Russia

^* Corresponding author
^* Corresponding author

Received: July 31, 2020

Revised: January 31, 2021

Early access: April 2021

Published: June 2021

The second author is supported by the Austrian Science Fund (FWF), Grant P 34177-N

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Abstract

We obtain explicit characterization of orbital and spectral stability of solitary wave solutions to the $ {\bf{U}}(1) $-invariant 1D Klein–Gordon equation coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.

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Mathematics Subject Classification: Primary: 35L70; Secondary: 47F05.

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Figure 1. The relation between $ \lambda\in\sigma(H_\kappa(\omega)) $ and $ \Lambda\in\sigma(L_\kappa(\omega)) $ for $ \omega\in(-m,m)\setminus\{0\} $, $ \kappa>0 $

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Figure 2. Location of simple nonzero eigenvalues $ \pm\lambda $

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