Solitons for the nonlinear Klein-Gordon equation. (English) Zbl 1200.35248

Summary: We study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the ratio energy/charge of a function, which turns out to be a useful approach for many field equations.


35Q53 KdV equations (Korteweg-de Vries equations)
47J35 Nonlinear evolution equations
35C08 Soliton solutions
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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[1] Shatah, Instability of nonlinear bound states, Math Phys pp 173– (1985) · Zbl 0603.35007
[2] Benci, Solitary waves in the nonlinear wave equation and in gauge theories, Fixed Point Theory Appl pp 1– (2007)
[3] Grillakis, Stability theory of solitary waves in the presence of symmetry, Funct Anal pp 160– (1987) · Zbl 0656.35122
[4] Shatah, Stable standing waves of non - linear Klein - Gordon equations, Commun Math Phys pp 313– (1983) · Zbl 0539.35067
[5] Coleman, Action minima among solutions to a class of euclidean scalar field equation, Math Phys pp 211– (1978)
[6] Rosen, Particle - like solutions to nonlinear complex scalar field theories with positive - definite energy densities, Math Phys pp 996– (1968)
[7] Cazenave, Orbital stability of standing waves for some nonlinear Schro dinger equations, Math Phys pp 549– (1982) · Zbl 0513.35007
[8] Berestycki, Nonlinear scalar field equations Existence of a ground state Arch Rational, Mech Anal pp 313– (1982)
[9] Bellazzini, On the existence of the fun - damental eigenvalue of an elliptic problem in RN, Nonlinear Stud pp 439– (2007)
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