Dispersive Wave Equation Posedness Of The Problem

In the dissertation, the Cauchy problems of some dispersive equations are considered by the Fourier restriction norm method, which was first introduced by J.Bourgain. It is important to point out that the phase functions, their first-order derivatives and second-order derivatives have non-zero singular points, which makes the problem much more difficult. However, we can overcome the difficulties by the Fourier restriction operators to separate these singular points. Therefore, some known results can be improved. The dissertation consists of four chapters.In Chapter 2, the Cauchy Problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local well-posedness for data in Hs(R)(s > -1/8) and global well-posedness for data in L2(R) are obtained.In Chapter 3, the Cauchy problem of Hirota equation is studied. For the equation with derivative in nonlinear terms, the Cauchy problem is locally well-posed for data in Hs(R){s > 1/4) and globally well-posed for data in Hs(R)(s > 1). For the equation without derivative, the Cauchy problem is locally well-posed for data in Hs(R)(s > -1/4) and globally well-posed for data in Hs(R)(s > 0). The main idea for the global well-posedness, based on the generalized trilinear estimates, is that the existence time of the solution in Hs(s > 1)( Hs(R)(s > 0)) only depends on the norm of initial data in H1 (L2).In Chapter 4, the Cauchy problem for the Fourth-order nonlinear Schrodinger equation related to the vortex filament space is considered. Local result for data in Hs(R)(s > 1/2) is obtained under certain coefficient condition.