| In Chapter 1, we derive estimates in high-order Sobolev spaces for the one-dimensional Zakharov system that are uniform as wave speed approaches infinity, by the technique of introducing a pseudo differential operator change of variable. This method has been previously applied to derivative nonlinear Schrodinger equations, the Davey-Stewartson system, and the Ishimori system by other authors, and the primary new obstacle here is commuting the pseudodifferential operator past the inverse wave operator. Applications to the convergence of solutions of the Zakharov system to the solution of the cubic nonlinear Schrodinger equation as wave speed approaches infinity are explored.; In Chapter 2, we prove local well-posedness of the initial-boundary value problem for the Korteweg-de Vries equation on the right half-line, the left half-line, and a line segment, in the low regularity setting. For the right half-line, one boundary condition is needed, and for the left half-line, two boundary conditions are needed.; In Chapter 3, we prove local well-posedness of the initial-boundary value problem for the one-dimensional nonlinear Schrodinger equation on the half-line.; The technique employed in Chapters 2--3 is an extension of that introduced by Colliander-Kenig (2002). |
