For Some Nonlinear Exact Solutions And Conservation Laws

In this paper, by using the compatibility method, the direct symmetry method and the classical Lie group approach, we study the symmetry and new explicit solutions of some nonlinear evolution equations, including traveling solutions, solitary wave and similarly solutions and so on. We also discuss the conservation laws of some nonlinear evolution equations.In Chapter 2, the (3+1)-dimensional generalized Zakharov-Kuznetsov equation is studied. Using the compatibility method, we get the symmetries and the corresponding vector field, obtained three kinds of similarity reduction and many new exact solutions of the equation, including bell solitary wave solutions, trigonometric solutions, rational function solutions, Weierstrass elliptic double periodic function solutions, and so on. At the same time we also obtain the infinite conservation laws by N. H. Ibragimov method and the given vector field.In Chapter 3, using the Lie groups method, we obtain some types of symmetry of the (2+1)-dimensional breaking soliton equation. By solving the corresponding characteristic equations associated with symmetry equations, symmetry reductions of (2+1)-dimensional breaking soliton equation are given first,then some special types of similarity solutions are constructed.In Chapter 4, by using the classical Lie group method, the infinite-dimensional symmetry group of the (2+1)-dimensional Broer-Kaup (BK) equations is found and the characterization of the group properties is given. We successfully obtain some exact solutions for (2+1)-dimensional BK equation by the reducing equations and derive the conservation laws.