Well-posedness And Stability Of Breather Solution For The Wave Equations

In this dissertation,we study some characteristics of wave equations,including global well-posedness,orbital stability of standing waves and nonlinear stability of the breather solutions,etc.This dissertation is divided into three chapters.In Chapter 1,we consider the nonlinear fractional Schrodinger equation where(-?)? denotes the Laplacian operator of order a,0<?<1,s?3/2-2?.By using a suitably iterative scheme,we prove the global well-posedness for the nonlinear fractional Schrodinger equation with N=1,which extends the result of Guo and Huo in[1].Moreover,we obtain the orbital stability of standing waves for the above equation via establishing the profile decomposition of bounded sequences in Hs(RN)(0<s<1)with N?2.In Chapter 2,we are concerned with a negative order modified Korteweg-de Vries(nmKdV)equation which is in the negative flow of the mKdV hierarchy.We construct the breather solutions by Hirota's bilinear method and derive the infinite conservation laws through the Lax pair of the nmKdV equation.By constructing a new Lyapunov functional with the conservation laws,we obtain the nonlinear stability of breather solutions.In Chapter 3,we derive the infinite conservation laws through the Lax pair of the coupled modified Korteweg-de Vries(cmKdV)equations.Then,we construct a new Lyapunov functional with the conservation laws.Finally,we present the nonlinear stability of breather solutions to the cmKdV equations.