Some Methods For Exact Solutions Of Nonlinear Differential Equations And Symbolic Computation

In the dissertation, some constructive methods in the view of AC=BD model for solving nonlinear differential equations and its symbolic computation are studied under the guidance of mathematics mechanization. Closely around algorithmization. mechanization and visualiza-tion the dissertation focus on obtaining variable separation solutions for time fractional order nonlinear models. semi-discrete solutions of nonlinear differential-difference equations. multi-ple wave solutions to variable-coefficient nonlinear systems and a nonisospectral KdV equation hierarchy's inverse scattering solutions. simulates the behaviors of the obtained solutions and solves the related problems.Chapter1sketches out the origins. research status and future trends of mathematics mechanization and computer algebra and introduces the discovery of soliton. theoretical argu-ments, physical properties. state of development and its practical applications. Also the types of soliton solutions and its existence condition of nonlinear evolution equations and the close links between the soliton solutions and the integrability of equations are briefly introduced. In addition, the chapter summarizes some constructive methods of nonlinear evolution equations. research background and development situations in domestic and overseas. and further gives the content of the selected topic in this dissertation and its main research work.In Chapter2. the AC=BD model proposed and developed by Professor Zhang Hongqing for solving nonlinear differential equations is introduced and then is generalized to a fractional order subsidiary ordinary differential equation expansion method and inverse scattering trans-formation method. As a result, some exact variable separation solutions of a time fractional order biological population model are obtained. At the same time. the C-D pairs are found by which a nonisospectral KdV equation hierarchy is transformed into the time evolution ordi-nary differential equations of scattering data of it's corresponding linear Schrodinger spectral problem used in the inverse scattering transformation.Chapter3summarizes a general principle of constructing ansatz solutions to nonlinear differential-difference equations and then based on the principle improves the extended Tanh function method and Jacobi elliptic function expansion method for exactly solving nonlin-ear differential-difference equations. Applying the improved methods respectively to a (2+1)- dimensional Toda lattice equation and a discrete nonlinear Schrodinger equation yields semi-discretized formal solutions including hyperbolic function solutions, trigonometric function solutions. rational solutions and Jacobi elliptic function solutions. The evolution behaviors of some obtained solutions are described and their asymptotic properties are analyzed with sev-eral figures. The results show that the improved algorithm has its advantage over the original methods, which can be utilized to obtain more types of exact solutions including new ones.In Chapter4. two algorithms are proposed for using Exp-function method to construct multiple wave solutions of variable-coefficient nonlinear evolution equations and differential-difference equations by devising multiple rational expressions of exponential functions and are respectively applied to variable-coefficient KdV equations.(2+1)-dimensional Broer-Kaup system with variable coefficient and (1+1)-dimensional Toda lattice equation. As a result, new multiple wave solutions are obtained and the formulae of N-wave solutions are summarized which can translate into N-soliton solutions by Hirota's direct method. Algorithm design and concrete examples shows the proposed two algorithms are more direct than Hirota's method and are easy to realize mechanization which don't need the process of using Hirota's bilinear operator to turn the equations into bilinear form and the obtained solutions possess generality.Chapter5first derives a KdV equation hierarchy with coefficients depending on time t by a differential operator containing arbitrary functions and the compatibility condition of eigen-function from the Schrodinger spectral problem and therefore its Lax pairs are obtained. The variable-coefficient KdV equation hierarchy includes the isospectral and constant-coefficient KdV equation hierarchy as special case which contains some well known equations. equation hierarchies and new equations and equation hierarchies. Its more general case is a nonisospec-tral and variable-coefficient KdV equation hierarchy with self-consistent sources. Then an expression of the exact solutions and a formula of N-wave solutions with reflectionless poten-tial are obtained by the inverse scattering transformation, and the propagation characteristics and asymptotic properties of some obtained solutions are analyzed by describing their evolution behaviors. At the same time. the unsolved issue in research field to determine part scattering data of the Schrodinger spectral problem related to the nonisospectral KdV equation hierarchy is solved which further perfects the scattering theory for such issues.