# Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry

@inproceedings{Li2022SolitonDF, title={Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry}, author={Yongming Li and Jonas Luhrmann}, year={2022} }

We establish the conditional asymptotic stability in a local energy norm of the unstable soliton for the one-dimensional quadratic Klein-Gordon equation under even perturbations. A key feature of the problem is the positive gap eigenvalue exhibited by the linearized operator around the soliton. Our proof is based on several virial-type estimates, combining techniques from the series of works [23–26,28], and an explicitly verified Fermi Golden Rule. The approach hinges on the fact that even…

## 3 Citations

### Asymptotic stability of kink with internal modes under odd perturbation

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We give a suﬃcient condition, in the spirit of Kowalczyk-Martel-Munoz-Van Den Bosch [26], for the local asymptotic stability of kinks under odd perturbations. In particular, we allow the existence of…

### Small energy stabilization for 1D Nonlinear Klein Gordon Equations

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We give a partial extension to dimension 1 of the result proved by Bambusi and Cuccagna [1] on the absence of small energy real valued periodic solutions for the NLKG in dimension 3. We combine the…

### Asymptotic stability near the soliton for quartic Klein-Gordon in 1D

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- 2022

. We consider the nonlinear focusing Klein-Gordon equation in 1 + 1 dimensions and the global space-time dynamics of solutions near the unstable soliton. Our main result is a proof of optimal decay,…

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