Symbolic Computation And Softwares For Some Nonlinear Systems

As a new research area of artificial intelligence, symbolic computation has been one of the most effective tools on the study of nonlinear systems. Nonlinear science studies those math-ematical systems and natural phenomena such as soliton, chaos, fractal and complex network systems, which can be boiled down into solving nonlinear functions (including nonlinear Ordi-nary Differential Equations, nonlinear Partial Differential Equations and nonlinear Differential Difference-Equations). Computerized symbolic computation can implement the operations on the nonlinear systems algorithmically and effectively. Symbolic computation algorithms and their software implementation has become one of the most important research topics for inter-disciplinary subjects between computer and nonlinear sciences.With the aid of symbolic computational software Maple, this dissertation studies sym-bolic computation of exact solutions and conservation laws for nonlinear partial differential and differential-difference equations from the perspectives of construction, algorithm.automation and visualization. The major results are as follows:1. A new modified Novikov equation is introduced. Multiple exact travelling wave solutions are obtained by using the extended-tanh function method. Furthermore, based on the ho-mogeneous balance method, an auto-Backlund transformation is derived and some solitary wave solutions to this new equation are established.2. A new algorithm and its Maple implementation AEFM are presented to find exact solitary wave solutions of nonlinear partial differential equations (PDEs) in terms of Exp-function method. AEFM can perform Exp-method automatically and outputs directly solitary solu-tions to (n+1) dimensional PDEs. Furthermore, AEFM allows users to control the desired solutions by given different input parameters.3. A new recursive combination algorithm (RCA) is given to construct candidate densities for computation of conservation laws of (1+1),(2+1) and (3+1) dimensional Partial Differential Equations. Its implementation ICLAWMD written in Maple is presented. ICLAWMD has a special user-friendly interface and can automatically implement the computation under different conditions. The output of the program can be shown in a Windows Form or described in a detailed analysis report. The effectiveness of the package is illustrated by applying it to a variety of equations. 4. A leading order integration condition (LOIC) analysis based algorithm is presented for con-structing densities of (1+1) dimensional Differential-Difference Equations. A maple pack-age ICLAWDDE for automatically implementing the algorithm is provided. ICLAWDDE can generate results in a Windows Form or in a detailed analysis report. The effectiveness of the package is illustrated by applying it to a variety of equations.Our methods and softwares are useful for mathematicians and physicists to solve integra-bility problems and gain insight into nonlinear models.