Study on the solutions of Kawahara, and complex-valued Burgers and KdV-Burgers equations

Abstract

The KdV equation is a nonlinear partial differential equation. The real-valued as well as complex-valued KdV equations have wide physical applications and very rich mathematical theory. The work in this dissertation studies two important problems.

First, the initial- and boundary-value problem for the Kawahara equation, a fifth order KdV type equation, is studied in weighted Sobolev spaces. This functional framework is based on the dual-Petrov-Galerkin algorithm, a numerical method proposed by Shen to solve third and higher odd-order partial differential equations. The theory presented here includes the existence and uniqueness of a local mild solution and of a global strong solution in these weighted spaces. If the L 2 -norm of the initial data is sufficiently small, these solutions decay exponentially in time. Numerical computations are performed to complement the theory.

Second, spatially periodic complex-valued solutions of the Burgers and KdV-Burgers equations are studied in detail. It is shown that for any sufficiently large time T, there exists an explicit initial data such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of two types of series solutions of the KdV-Burgers equation are established for initial data satisfying mild conditions.