Group Foliation Methods And Functional Separable Solutions Of Nonlinear Equations

The nonlinear phenomena widely appear in almost all the scientific fields such as physics, chemistry, biology, society, economy and so on. With the development of sciencs, researchers on the nonlinear systems are making more and more progress. As a result, to deal with the nonlinear equations that describe the nonlinear systems becomes one of the main and hot topics for the researchers. Fourier analysis and the variable separation approach are effective means in studying the linear partial differential equation(PDE). Since the linear superposition principle is not valid, So there is not a general method to solve the nonlinear PDE.It is well-known that the method of separation of variables is a efficient way to sovle the linear PDE. For nonlinear PDE, a natural question is whether they admit the functional separation solutions(FSS) or generalized functional separation solutions(GFSS). In past decades, a lot of methods have been established and developed to attain GFSS to the nonlinear PDE,such as the Direct method, the geometric method , the Ansatz-based method, the Formal variable separation approach and the Multi-linear variable separation approach.In the chapter 2 and chapter3,we find the separation of variables solutions for nonlinear diffusion and wave equations(2. 4) (3. 1) by utilizing the group foliation method. We introduce a antomorphic systemLet G(x,u) = g(x)h(u) ,then(2. 4) (3. 1) implies a polynomial which each term in the expression is a product of u -dependent function and x -dependent function. Our aim is to find all set of functions A(x), B(x), g(x), D(u), Q (u), and h(u) that the original equation has functional separable solutions...