| In this paper, by using the Lie group method, the direct symmetry method and the modified CK's direct method to solve a (2+1)-dimensional nonlinear evolution equation, the SK-KP equation, the MKP-II equation and the BK equation, we obtain these equations'symmetries and explicit solutions. We also obtain conservation laws of the SK-KP equation and the BK equation. By using the direct construction method, we get soliton solutions and periodic solutions of the Burgers equation and the KP equation.In Chapter 1, by applying a direct symmetry method, we obtain the symmetry reduction and some new exact solutions of the (2+1)-dimensional nonlinear evolution equation, which include Jacobi elliptic function, hyperbolic function, trigonometric function and so on. These exact solutions will be helpful to a better understanding of some practical problems in physics.In Chapter 2, the SK-KP equation is discussed. Firstly, assuming the form of symmetry. Secondly, by using the form of symmetry and the SK-KP equation, we resent the Lie symmetry. Similarity reductions and many new exact solutions of the SK-KP equation are obtained. At last, we also give the conservation laws of SK-KP equation.In Chapter 3, we obtain the exact solutions of the MKP-II equation and the BK equation through the generalized CK direct method. A relationship is constructed between the new solutions and the old ones of the MKP-II equation and the BK equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the BK equation. Then we get the reductions using the symmetry and give some exact solutions of the BK equation.In Chapter 4, we study the Burgers equation and the KP equation. We construct a frame of soliton solutions and periodic solutions to nonlinear equations. And we successfully solve the (2+1)-dimensional Burgers and (3+1)-Dimensional KP equation. And a lot of exact traveling solutions to the Burgers and KP equations are obtained. |
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