Extended Symmetry Reduction Method For Solving Several Types Of Shallow Water Wave Equation

With the development of the Nonlinear Science, lots of Nonlinear Evolution Equations (NLEEs) arefound, and they are playing important roles in many different physics fields. Seeking the solutions of NLEEs is now a very important subject in Nonlinear Science.The Lie group method is a Powerful tool to study the nonlinear differentical equations. By using the classical or nonclassical Lie symmetry method we can get amount of exact solutions of the nonlinear differential equations. There are also many method to abtain the the exact solutions of the nonlinear PDE, such as the direct method, generalized condition symmetry, variable separation approach and so on. In this artical, we utilize the extended direct method to sove shallow water wave equations based on physics, and obtain several new similarity solutions of them.In the chapter 1, We introduce the shallow water wave equations and the produce of the direct method.In the chapter 2, We arrange and analyze the existing work, to solve the shallow water wave equations with the direct method.In the chapter 3, We utilize the extended direct method to solve the shallow water wave equations and obtain several new similarity solutions.In the chapter 4, We apply the direct method to generalize the existing solutions of the (2+1) dimensional shallow water wave eqution.