Painlevé Analysis And Function Expansion Method For Solving Nonlinear Partial Diferential Equations

With the rapid development of nonlinear science, searching for the exact solutions ofnonlinear partial diferential equations plays an importent role in soliton theory. The trun-cated Painlev′e expansion approach and the function expansion method are two very simpleand powerful methods. In this thesis, we will use the two methods to fnd the exact so-lutions of the (2+1)-Dimensional BKK equation, the nonlinear Boussinesq equation, the(2+1)-Dimensional Boussinesq equation.In the study of the (2+1)-Dimensional BKK equation, we will use the generalized tanhfunction expansion method to obtain the Ba¨cklund transformation frstly. Then some exactsoliton solutions of the (2+1)-Dimensional BKK equation are obtained from the seed solution.In the study of the nonlinear Boussinesq equation, the residual symmetry of the nonlinearBoussinesq equation is given by the truncated Painlev′e expansion approach. The residualsymmetries can be localized to Lie point symmetries after introducing suitable prolongedsystems. The fnite transformations of the residual symmetries are equivalent to the secondtype of Darboux-Ba¨cklund transformations. Then with the help of the truncated Painlev′eexpansion approach and the generalized tanh function expansion method, many interactionsolutions among soliton and other types of nonlinear excitations of the nonlinear Boussinesqequation can be obtained, particularly, the soliton-cnoidal wave interaction solutions areobtained.In the study of the (2+1)-Dimensional Boussinesq equation, frstly we will introduce anew concept called consistent Riccati expansion and solvability. Then by using the consistentRiccati expansion approaches, the consistent equation can be given. Many exact solutionswith special structure of the the (2+1)-Dimensional Boussinesq equation can be obtained.