Exact Solutions And Conservation Laws To Some Nonlinear Evolution Equations

In this paper, by using Lie group method and the modified CK's direct method, we study the general symmetry groups and new explicit solutions of some nonlinear evolution equations, including traveling wave solutions, solitary wave solutions and similarity solutions and so on. We also discuss conservation laws of some nonlinear evolution equations.In Chapter one, the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP) equation is considered. By using the direct symmetry method, we get the symmetry of the equation, five types of similarity reductions and many new exact solutions of the equation, including rational solutions, hyperbolic function solutions, Jacobi elliptic function solutions and triangular periodic solutions. We also get the conservation laws of the equation. And we give some cases in which certain Lie algebra is associated with the conservation laws.In Chapter two, using the compatibility method, we obtain the symmetries and the equivalent vector of the (3+1)-dimensional Kadomtsev-Petviashvilli (KP), four types of similarity reductions and many new exact solutions of the equation, including rational solutions, bell shaped solitary wave solutions, triangular periodic solutions, Weierstrass elliptic doubly periodic solutions and so on. We also obtain the conservation laws by N. H. Ibragimov's theorem and the given vector. One of theadvantages of the compatibility method is that it is capable of greatly reducing the complexity in the computational process, in comparison with the existing techniques such as the nonclassical method . The method can also be applied to other nonlinear evolution equation in mathematical physics.In Chapter three, using the modified CK's direct method, we obtain the general symmetry groups and Lie symmetry of the (2+1)-dimensional dispersive long water equation. The Lie group obtained by the classical Lie group approach is only special case of our results with respect to symmetry groups. And the Lie symmetry here can also be obtained by the classical Lie group approach by which the computation is much more complicated. Based on the obtained theorem for the general symmetry groups, we also found the relationship of the old solutions and the new solutions and get a lot of new solutions from the known ones. We also illuminate the asymptotical property of some solution as time passes. Using the obtained Lie symmetry, the conservation laws is obtained.