The Homoclinic Orbits For Some Nonlinear Schr?dingier Equations

In this thesis we consider the homoclinic orbits for the perturbed quintic and derivative nonlinear Schrodinger equations with periodic boundary conditions. More specifically,we combine geometric singular perturbation theory with Melnikov analysis and integrable theory to prove the persistence of homoclinic orbits.The quintic and derivative nonlinear Schrodinger equations has a extensive applications in various physical areas and can be considered the perturbation of the NLS equation. Further they are the near integrable systems. In chapter 1,we briefly introduce the near integrable systems,and give the background in physics and the developments in mathematics for the quintic and derivative nonlinear Schrodinger equations. In which the main work of the dissertation is also described. In chapter 2,we discuss the existence of homoclinic orbits for a perturbed cubic-quintic nonlinear Schrodinger (CQS) equation with even periodic boundary conditions. First,we take the analysis of phase-plane on the invariant plane for the perturbed and unperturbed systems. Next,we applied the singular perturbation theory to establish the persistence of invariant manifolds,as well as the " fiber represent at iones " of these manifolds. Finally,by using the global integrable theory of the unperturbed system and Melnikov measurement we obtain the existence of homoclinic orbits for the CQS equation under the generalized parameters conditions. In chapter 3,we study the derivative nonlinear Schrodinger (DNLS)equation with a three-order dispersion. We adopt a three mode Fourier truncation and get a six dimensional model. This model is considered and the persistence of the homoclinic orbits is obtained by Melnikov's analysis together with the geometrical singular perturbation theory.When we study the homoclinic orbits of the CQS equation,the quintic term is a perturbation. If the quintic term lies in the first terms,the unperturbed system is a quintic nonlinear Schrodinger (QNLS) equation and the CQS equation is not a near integrable systems,it only can be considered a perturbation of Hamiltonian system. In this case,we establish the persistence of invariant manifolds for certain perturbation of the CQS equation under even periodic boundaryconditions. For some quintic derivative nonlinear Schrodinger (QDS) equation,Kundu had shown their integrability through gauge transformations. By using the inverse scattering method and certain technique we get the Lax pair of these equations and identify their integrability.