## 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.

### MSC:

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

 [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
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