Stability of Standing Waves for a Nonlinear Klein-Gordon Equation with Delta Potentials

Elek Csobo (1) , François Genoud (2) , Masahito Ohta (3) , Julien Royer (4)

1 DIAM - Delft Institute of Applied Mathematics
2 EPFL - Ecole Polytechnique Fédérale de Lausanne
3 Tokyo University of Science [Tokyo]
4 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: Orbital stability; Nonlinear Klein-Gordon Equation; Delta potential; Standing waves

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Elek Csobo, François Genoud, Masahito Ohta, Julien Royer. Stability of Standing Waves for a Nonlinear Klein-Gordon Equation with Delta Potentials. 2019. ⟨hal-01890232v3⟩