Research On The Existence Of Solutions Of The Several Weakly Coupled NLS-KdV Systems

In this thesis,we mainly study the existence and properties of solutions of the several weakly coupled NLS-KdV system.On the one hand,the existence and properties of solutions of the NLS-KdV system,NLS-KdV-KdV system and NLS-NLS-KdV system are proved by using variational methods.On the other hand,the existence of multiple solutions is studied by using local bifurcation theory.This thesis includes the following five Chapters:Chapter 1 describes the research background and main results.Chapter 2 introduces the notations and basic knowledge.Chapter 3 studies the NLS-KdV system{-?u+?1u=?1|?|2 u+?uv,? ? RN,-?v+ ?2v-?2|v|v+2/1?u2,? ? RN,and NLS-KdV-KdVsystem{-?u+?1u=?1u2+?12uv2+?13uw,? ? RN,-?v+?2v=?2v3+2/1?12u2v+ ?23vw,? ? RN,-?w+?3w=?3w2+2/1?13u2+2/1?23v2,? ? RN,First,the existence of nontrivial solutions of the NLS-KdV and NLS-KdV-KdV system are proved by comparing the energy functional with the variational methods when the parameters satisfying the appropriate conditions.Second,the existence of nontrivial solutions are proved by using the bifurcation theory.Finally,the asymptotic behavior of positive solutions of the NLS-KdV system are shown.Chapter 4 considers the NLS-NLS-KdV system{-?u+?1u=?1u2+?12uv2+?13uw,? ? RN,-?v+?2v=?2v3+2/1?12u2v+ ?23vw,? ? RN,-?w+?3w=?3w2+2/1?13u2+2/1?23v2,? ? RN,The existence of positive solutions of this system are obtained by combining the mountain path theorem on convex sets and Nehari manifold restriction methods.Chapter 5 summarizes the thesis.