Orbital stability and spectral properties of solitary waves of Klein-Gordon equation with concentrated nonlinearity. (English) Zbl 1478.35080

Summary: We obtain explicit characterization of orbital and spectral stability of solitary wave solutions to the \(\mathbf{U}(1)\)-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.


35C08 Soliton solutions
35B35 Stability in context of PDEs
35L15 Initial value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
Full Text: DOI


[1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, \(2^{nd}\) edition, American Mathematical Society, Providence, RI, 2005. · Zbl 1078.81003
[2] N. Boussaïd and A. Comech, Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schrödinger operators, preprint, arXiv: 2101.11979. · Zbl 1375.35427
[3] V. Buslaev; A. Komech; E. Kopylova; D. Stuart, On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Commun. Partial Differ. Equ., 33, 669-705 (2008) · Zbl 1185.35247
[4] E. Csobo; F. Genoud; M. Ohta; and J. Royer, Stability of standing waves for a nonlinear Klein-Gordon equation with delta potentials, J. Differ. Equ., 268, 353-388 (2019) · Zbl 1429.35026
[5] A. Comech; D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., 56, 1565-1607 (2003) · Zbl 1072.35165
[6] M. Grillakis; J. Shata; W. Strauss, Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., 74, 160-197 (1987) · Zbl 0656.35122
[7] A. Jensen; T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46, 583-611 (1979) · Zbl 0448.35080
[8] A. Komech; A. Komech, Global attractor for a nonlinear oscillator coupled to the Klein-Gordon field, Arch. Ration. Mech. Anal., 185, 105-142 (2007) · Zbl 1131.35003
[9] A. Komech; A. Komech, Global attraction to solitary waves for Klein-Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 855-868 (2009) · Zbl 1177.35201
[10] A. Komech; E. Kopylova; D. Stuart, On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11, 1063-1079 (2012) · Zbl 1282.35352
[11] A. Kolokolov, Stability of the dominant mode of the nonlinear wave equation in a cubic medium, J. Appl. Mech. Tech. Phys., 14, 426-428 (1973) · Zbl 0312.49004
[12] E. Kopylova, On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator, Nonlinear Analysis: Theory, Methods & Applications, 71, 3031-3046 (2009) · Zbl 1167.35515
[13] E. Kopylova, On asymptotic stability of solitary waves in discrete Klein-Gordon equation coupled to a nonlinear oscillator, Appl. Anal., 89, 1467-1492 (2010) · Zbl 1207.39021
[14] M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49, 10-56 (1982) · Zbl 0499.35019
[15] M. Ohta; G. Todorova, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal., 38, 1912-1931 (2007) · Zbl 1128.35074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.