Abstract
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.
Original language | English |
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Pages (from-to) | 353-388 |
Number of pages | 36 |
Journal | Journal of Differential Equations |
Volume | 268 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2019 |
Keywords
- Delta potential
- Nonlinear Klein–Gordon equation
- Orbital stability
- Standing waves