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

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 perturbations are orthogonal to the odd threshold resonance of the linearized operator.