B(?)cklund Transformation And The Exact Solutions For The Nonlinear Partial Diferential Equations

By using tanh function expansion, dimensional analysis and residual symmetry, theBa¨cklund transformation and the exact solutions to the nonlinear partial diferential equationsare studied in this thesis.In the study of the dispersive long wave equation, with the help of the truncated Painlev′eexpansion and the generalized tanh function expansion approach, the interaction solutions ofthe dispersive long-wave equation are obtained. And we can also observe the fusion and thefssion phenomenon.Then, we apply the dimensional analysis to the generalized ffth-order KdV equation,and diferent results are given. In the study of the ffth-order KdV equation, by using thedimensional analysis, the same Schwarzian form of the ffth-order KdV equation correspondsto the two diferent exact solutions; In the Sawada-Kotera equation, the dimensional anal-ysis is introduced to get the two diferent Schwarzian forms of the Sawada-Kotera equationcorresponding to the same set of exact solutions; While in the Kaup-Kupershmidt equation,by using the method of dimensional analysis, two diferent Schwarzian forms of the Kaup-Kupershmidt equations correspond to two diferent sets of exact solutions. In short, the usageof dimensional analysis can readily fnd more signifcant exact solutions.Finally, in the study of the ffth-order KdV equation, the fnite symmetry transforma-tion group of the equation is studied by the residual symmetry method, and the residualsymmetries can be localized to Lie point symmetries after introducing suitable prolongedsystems. Then the Ba¨cklund transformation related to the residual symmetries and somespecial types of localized structure for the ffth-order KdV equation are obtained.