Modulational Instability Of Nonlinear Schrdinger Equations

With the rapid development of science and technology, a large number of non-linear problem, which appeared in many fields, can be finally transformed to solving a class of nonlinear evolution equations and analyzing the properties of the solutions. As a result, nonlinear evolution equations play more and more important role in me-chanics, physics, applied mathematics, geoscience, and so on. It is considered as a hot and frontier subjects, the study of the exact solution of nonlinear equation and stability of solutions caused lots of attention. The nonlinear Schrdinger equation is a kind of important nonlinear evolution equation in mathematical physics and it is widely used in many domains such as solid state physics, quantum mechanics, theory of plasma and bose Einstein condensation. Consequently, solving and analyzing stability of nonlinear evolution equation becomes a significant and valuable work.This paper mainly discusses the solution and analyze the stability of some nonlin-ear partial differential equations and it is divided into five chapters.The first chapter is the introduction part, which briefly describes the basic situation and background of the partial differential equation, and the main results of this work are stated in this chapter.The second chapter of this paper introduces the homogeneous balance method and Jacobi elliptic function method of the concrete solving steps.The third chapter studies a class of modulational instability of plane-wave solu-tions of two-dimensional cubic-quintic Schrodinger equation, using the Jacobi elliptic function to obtain some traveling wave solutions and analysis the stability of the solu-tions, in addition, some of them are numerically simulated by mathematica.The fourth chapter studies a class of modulational instability of plane-wave solu-tions of the perturbed nonlinear coupled Schrodinger equation with a dissipative, using the homogeneous balance method to get some traveling wave solutions and analysis the stability of them.The fifth chapter of this paper, I summarize the full thesis and point out the further research work.