Study On Painlevé Test With Symbolic Computation

Completely integrable differential equations model such physically interesting phe-nomena as reaction-diffusion systems, population and molecular dynamics, nonlinear net-works, chemical reactions, and material science (in particular solid mechanics and elastic materials). The two primary methods for solving a completely integrable nonlinear evolu-tion equation are by explicit transformations into a linear equation or by using the inverse scattering transform. The inverse scattering transform is a non- trivial exercise in analysis with no systematic way to determine a priori if it will be successful. However, passing the Painleve test is a strong indicator that the differential equation will be solvable using the inverse scattering transformation. Moreover, Painleve test may provide much useful information for solving nonlinear differential equations, such as bilinear transformation, Backlund transformation, Lax pairs and Darboux transformation. This dissertation con-sists of the following parts.Chapter 1 is the research background related to the dissertation. First the develop-ment of Painleve test is briefly outlined. Then various implementation of Painleve test in algebra systems are reviewed.Chapter 2 is devoted to basic ideas and concepts of Painleve Test. We focus on the WTC algorithm for the Painleve test of PDEs. It is crucial to make distinction between the Painleve property and Painleve test. Some counterexamples of Painleve test are given.In Chapter 3, we propose a modified WTC algorithm for the Painleve test of nonlin-ear variable coefficient partial differential equations. This new algorithm further simpli-fies the computation in third step of Painleve test compared to the Kruskal's simplification algorithm. Two examples illustrate its computing procedure and its efficiency.In Chapter 4, we construct a connection between the (G'/G)-expansion method and the truncated Painleve expansion method, which says that the (G'/G)-expansion method is a special form of the truncated Painleve expansion method. Hence, a small step is made towards the direction of unifying several kinds of algebra method to construct exact solutions of nonlinear evolution equations from the viewpoint of the Painleve analysis.