Well-posedness For Higher-order Nonlinear Schr(?)dinger Equations With Potentials In H~s

Quantum mechanics is one of the most important scientific discovery in the last century, whose basic equation is the Schrodinger equation.Because of Schrodinger equation has such a strong physical background, the study of which and the corresponding Schrodinger opera-tor has been the focus of attention in the past century. These studies focused on the Lp-Lq estimates, local smoothing estimates, Strichartz estimates of the solution of Schrodinger equation and the self-adjoint, spectral theory, scattering theory of the Schrodinger operators. The posedness of Schrodinger equation studied in this thesis, describes the stability of the state of the particle in a certain period of time, which has important physical meaning. The well-posedness for nonlinear Schrodinger equation inⅡ3(Rn), which was studied by Kato at first. The best condition of s was obtained by T. Cazenave,when the solution of nonlinear Schrodinger equation is local well-posedness in Hs(Rm). However,the research results of the well-posedness for high-order Schrodinger equation are not so much. Especially for the case with potential, there are still many problems waiting to be solved. The main work is to study when the potentials V(x, t) satisfy suitable conditions,the local well-posedness for higher-order nonlinear Schrodinger equations with potentials in Hs(Ra).The whole thesis will be into three chapters. The first chapter is an introduction to ex-plain the physical background of Schrodinger equation and related results on well-posedness in Hs(Rn) which are presented. The second chapter is prior knowledge,describing some estimates of higher-order Schrodinger equations,which prepare the proof of the third chapter. The third chapter is the description and proof of the main conclusions of this thesis,which considers the local well-posedness for higher-order nonlinear Schrodinger equations in with potentials Hs(Rn), when the potentials satisfy suitable conditions. In a demonstration, by constructing the role of space, combined with Strichartz estimates, using Banach fixed point theorem to prove the main results to be obtained.