Existence and Stability of Standing Waves for Nonlinear Schrodinger Equation with or without Confinement

Abstract

In this Master's thesis, we mainly consider semi-linear Schrodinger equations of the form iu_t+∆u-V(x)u=f(u) where u=u(t,x),t∈R,x∈R^N,f:C→C and V:R^N→R is the Potential. For the convenience of the reader, we first recall the principle results and methods for the model case V(x)=0 and f(u)=±|u|^(p-1) u with p>1, but the methods of the proofs can be applied to more general problems. Some of possible generalizations and modifications will be mentioned as remarks, other can be found in the subsequent chapters. In particular, we will focus on the stability of standing waves which are solutions of the stationary problem -∆φ+V(x)φ+ωφ+f(φ)=0, where x∈R^N,ω∈R . The first chapter is devoted to study the linear case of the Schrodinger equation iu_t+∆u=F(t,x) for both homogeneous and nonhomogeneous cases and state the basic properties. In the next chapter, we study the local/global existence and uniqueness for the Cauchy problem when the initial data belongs to the Sobolev space 〖 H〗^s (R^N),H^1 (R^N), and f(u)=±|u|^(p-1) u,p>1. We give a main result that asserts the well-posedness under suitable condition on p,s. We also investigate the well-posedness for some general potential V which is well-posedness in the focusing case. The third chapter is concerned with the stationary problem -∆φ+V(x)φ+ωφ+f(φ)=0 and the solution is called a ground state. The main result in this chapter asserts that the problem has a nontrivial solution under suitable conditions on ω,f which is the existence and uniqueness of ground state with or without confinement. The last chapter talks about the stability of the standing waves solutions (SWS) e^itω φ(x). We will investigate essentially the case f(u)=-|u|^(p-1) u and the stability, unstable, strongly unstable, and the orbital stability of the (SWS).