Stability of Large Periodic Solutions of Klein–Gordon Near a Homoclinic Orbit

Abstract

We consider the Klein–Gordon equation (KG) on a Riemannian surface $$M$$ $$\begin{aligned} \partial ^{2}_t u-\Delta u-m^{2}u+u^{2p+1} =0,\quad p\in {\mathbb {N}} ^{*},\quad (t,x)\in {\mathbb {R}} \times M, \end{aligned}$$ which is globally well posed in the energy space. Viewed as a first order Hamiltonian system in the variables $$(u, v\equiv \partial _t u)$$ , the associated flow lets invariant the two-dimensional space of $$(u,v)$$ independent of $$x$$ . It turns out that in this invariant space, there is a homoclinic orbit to the origin and a family of periodic solutions inside the loops of the homoclinic orbit. In this paper, we study the stability of these periodic orbits under the (KG) flow, i.e., when turning on the nonlinear interaction with the nonstationary modes. By a shadowing method, we prove that around the periodic orbits, solutions stay close to them during a time of order $$(\ln {\eta })^2$$ , where $$\eta $$ is the distance between the periodic orbit considered and the homoclinic orbit.