On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation (Q6547226)

The focusing nonlinear Klein-Gordon equation\N\[\N\partial_t^2\phi-\partial_x^2 \phi+\phi=|\phi|^{p-1}\phi\N\]\Nwith \(p>1\) is a widely studied model in the theory of dispersive equations. The soliton solution \(Q\) is known explicitly:\N\[\NQ(x)=\frac {(\alpha+1)^{\frac 1{2\alpha}}}{\cosh^{\frac 1\alpha}(\alpha x)}, \quad \alpha=\frac{p-1}{2}.\N\]\NThe present paper deals with the codimension one asymptotic stability of the soliton under even perturbation which are initially small in a weighted Sobolev space in the important case \(p=3\). The main result is the \(L_x^\infty\) modified decay of \(u(t,x)=\phi(t,x)-Q(x)\) at the rate \(t^{-1/2}\log(2+ t)\) along the center stable manifold. This result is valid up to times of order \(t\leq e^{\, c\epsilon^{-1/3}}\) where \(\epsilon\) measures the size of the initial perturbation. The main difficulty, and the reason why the result is valid on a finite time interval, is the presence of the even threshold resonance for the linearized operator around the soliton. The method applied follows an earlier work of the both authors and relies on the Darboux factorization of the linearized operator.