Existence Of Discrete Breathers For DNLS Equation And Quasi-Periodic Discrete Breathers In Coupled Map Lattices

In this paper, we first study the existence of discrete breathers in high-dimensional nonlinear Schrodinger (DNLS) equation subjected to damping and periodically driving :We give a continuation theorem of the zeros of a map defined in R l. We use the implicit function theorem to prove that the zeros are continuable, and that the continuations are exponentially decaying. We apply the continuation theorem to the DNLS equation and prove that when the damping force 6 and the driving force h satisfy h > , the discrete multibreathers with the rotation frequency exist, where > .Next we are concerned with the existence of quasi-periodic breathers for the coupled map lattices:where the local map f is the delayed Logistic map. When = 2 and = 0, the system has degenerate equilibria. We give a center manifold theorem in the neighbourhood of the equilibrium point, and prove that the center manifold is exponentially decaying. We use the Poincare-Andronov-Hopf Bifurcation theorem to prove that there is an invariant curve in the center manifold when the coupling coefficient e is small enough. The numerical simulations demonstrate that there is a quasi-periodic rotation on the invariant curve when the coupling coefficient e is small enough. That is to say, the bifurcations of the quasi-periodic breathers in the lattice system occur with weak spatial coupling.