Ground State Solutions In The Coupled Discrete Nonlinear Schrodinger Equations

The existence of ground state solutions of the coupled discrete nonlinear Schrodinger equations is studied by using critical point theory in this dissertation. It provides the the-ory basis to observe discrete solitons and their properties in experiments.This dissertation is composed of five chapters. The content of the dissertation is as follows.Chapter1gives a brief introduction to the historical background, status and the up-to-date progress for all investigated problems together with main results and preliminary tools in this dissertation.In chapter2, we demonstrate the existence of ground state solutions in coupled dis-crete nonlinear Schrodinger equations (CDNLS) with periodic potentials. We consider two types of solutions to CDNLS periodic and vanishing at infinity. Calculus of varia-tions and the Nehari manifolds are employed to establish the existence of the periodic solutions. Using periodic approximations, we present sufficient conditions on the exis-tence of ground state solutions which are vanishing at infinity, and then we show that both of the components of this ground state solutions are not zero.The existence of ground state solutions in coupled discrete nonlinear Schrodinger equations with unbounded potentials is investigated in Chapter3.By using the Nehari manifold approach and the compact embedding theorem, we find simple conditions that are sufficient to ensure existence of ground state solutions with both components being non-trivial. Our results extend some known results in the literature.In chapter4, the existence of ground state solutions in coupled discrete nonlinear Schrodinger equations with saturable nonlinearity is studied. By establishing variational structure and applying the Nehari manifold approach, we establish sufficient conditions on the existence of the periodic solutions. Using periodic approximations, the non-trivial ground state solutions which are vanishing at infinity to the coupled discrete nonlinear Schrodinger equations with saturable nonlinearity are obtained.The summary of this dissertation and the outlook for the future research work are stated in Chapter5.