Solving Nonlinear Partial Differential Equation And Symmetry Reduction

In this dissertation, under the guidance of mathematics mechanization and by means ofsymbolic computation software, some problems of solving some significant nonlinear evolutionequations are discussed and some methods for constructing the exact solutions of nonlinearevolution equations are presented and improved. These presented methods are realized onsymbolic computation system Maple. At the same time, the symmetry reductions of high-dimensional equations are studied. Group explanation of the direct method for the Lie symmetrygroup of the Lax integrabel system is given and the method is improved.Chapter 1 is to introduce the history and development of the soliton theory, mathematicsmechanization and symbolic computation, the exact solutions of nonlinear evolution equation(s)and the symmetry theory of differetial equation(s). As well as the works and achievements thathave been obtained at home and abroad are listed. Our main works are listed at last.Chapter 2 concerns the AC=BD model and the theory of C-D pair for solving nonlineardifferential equations. Based on the theory of AC=BD and pseudo-differential division withremainder, a direct construct method for operator C is presented. The operator C is notonly depend on the independent variables of the original equations, but also depend on thedependent ones. The method is not only applied to continuous systems but also discrete ones.Some examples indecate that the method covers some existing methods, such as B(?)cklundtransformation, Cole-Hopf transformation, function expand method, Lou direct method, Burgersequation method, and so on. Then, the theory of AC=BD is applied to the symmetry analysisof nonlinear differential equations. These greatly enlarge the theory of AC=BD.Based on the idea of mathematics mechanization, in Chapter 3, we firstly present the threeRiccati equation expansion method, then improve it to uniformly construct the exact solutionsof nonlinear evolution equations. Rich exact solutions of higher-order Schr(?)dinger equationand classical Boussinesq equations are obtained by these methods, respectively. And tanh-sech method is presented to construct more genearl solutions. The (2+1)-dimensional Burgersequation is chosen to prove effective. At last, the hyperbolic function expansion method isimproved to study differential-difference equations and many explicit exact solutions of (1+1)-dimensional Toda lattice and Volterra lattice equations are obtained by the improved method.In chapter 4, based on the symbolic computation, we combine the new constuctive methodwith the classical Lie symmetry method to seek new type of solutions of the nonlinear evolution equations. With the combined method, some new types of solutions of the (2+1)-dimensionalcubic nonlinear Schr(?)dinger equation and Davey-Stewartson equations are obtained. And wecombine the general projective Riccati equation method with the method presented by Lou tostudy (2+1)-dimensional dispersive long wave equation and obtain some new results.As for the symmetry reduction of high dimensional systems, the similarity reductions aris-ing from the classical Lie point symmetries not only of some (2+1)-dimensional differentialequations, but also of their Lax pairs, is presented in Chapter 5. Comparing the reduced Laxpair's compatibility and the reduced differential equations, we find that not all the reduced Laxpair is just the reduced equations' Lax pair. In general, the reduced equations are subsets ofthe compatibility conditions of the reduced Lax pair. The spectral problems of the reduceddifferential equations are obtained and the spectral parameter is just one of the admitted in-finitesimal generators by the Lax pairs. In the last, based on the Lie symmetry group theory,group explanation of the direct method for the Lie symmetry group of the Lax integrabel systemwhich was presented by Professor Lou Senyue, is given and Lou's direct method is improved.