Derivative-dependent Functional Variable Separation Approach For A Class Of (1+1)-dimensional Nonlinear Evolution Equation

With the perfectness of the linear theory, the nonlinear science has greatly developed in many fields, which becomes the focus of study. As nonlinear equationsarising from many fields, undoubtedly, how to reduce and solve those equations becomes the key and difficult problem in the study of nonlinear science. Unlike linear equations, there is no a general way to deal with nonlinear systems in essence, since the principle of linear superposition doesn't work. A kind of special solutions can be obtained by one or other ways indeed, but all kinds of special solutions can not be obtained via one approach. As a result, there is no universal means to solve nonlinear systems.It is well known that the derivative-dependent functional variable separationapproach is an efficient way to study nonlinear partial differential equations(PDEs). There are a lot of approaches to study nonlinear PDEs, such as symmetry group method, variable separation approach, group foliationmethod, etc.In this paper, we have applied the derivative-dependent functional variableseparation approach to discuss the following (1+1)-dimensional nonlinear evolution equation with mixed partial derivatives which has certain physical contexts:E≡E(t,x,u,u₁,u₂,…,U_m,U_1t,U_2t,…,U_nt)=0．we have obtained some important results:(1) We have proposed the new theory of the derivative-dependent functionalvariable separation for the (1+1)-dimensional nonlinear evolution equationwith mixed partial derivatives;(2)We have constructed a relation between the derivative-dependent functionalvariable separation for the above class of equations and the functional variable separation for whose corresponding systems of PDEs;(3)As an application, we have obtained complete classification of the general nonlinear evolution equations u_xt=A(u,u_x)u_xxx+B(u,u_x) that admit DDFSSsand some DDFSSs.