Orbits homoclinic to centre manifolds of conservative PDEs

In this paper, perturbations to orbits homoclinic to saddle-centres for conservative systems are considered. We prove that if the Hessian of the conserved energy at the saddle-centre is positive definite in the centre directions, then either single bump orbits homoclinic to the saddle-centre persist or its centre-unstable and centre-stable manifolds intersect transversally, where the centre manifold is stable. The return map induced by the homoclinic orbits to the centre manifold and applications to sine-Gordon breathers, homoclinic orbits for nonlinear Schrödinger equations, and periodic travelling waves for Klein-Gordon equations are discussed.