Potential Symmetries And Linearization Of Nonlinear Evolution Equations

In this thesis, we mainly discuss the potential symmetries and linearization of the nonlinear evolution equations.Firstly, We compute potential symmetries to systems of nonlinear diffusion equations u_t = (f(u,v)u_x+p(u,v)v_x)_x, v_t = (g(u,v)v_x+q(u,v)u_x)_x, which have physical applications in soil science, mathematical biology and invari-ant curve flows. That is. to determine functions f(u,v), g(u,v), p(u,v) and q(u,v) for which the systems admit potential symmetries. As a re-sult, a wide classes of partial differential equations (PDEs) of this form which admit potential symmetries are obtained by systematic classifica-tion methods. It is shown that certain PDEs of this form admit potential symmetries. Furthermore, some of them admit infinite-dimensional poten-tial symmetries. We then construct invertible mappings to linearize the auxiliary systems of the original systems which admit potential symmetries and the corresponding auxiliary systems satisfy the Bluman's linearization theory. It turns out that these invertible mappings lead to the nonlocal mappings which can linearize the original systems of diffusion equations.We then consider systems of n-component nonlinear diffusion equations u_it=[(?) f_ij(u₁,u₂…,u_n)u_jx]_x,i=1,…,n. The equivalence transformations of their auxiliary systems are derived, and the auxiliary systems which can be linearized by these equivalence transformations are com-pletely classified. It is further shown these equivalence transformations lead to the nonlocal transformations which connect the original nonlinear systems of the diffusion equations and the linear systems of diffusion equa-tions.Finally, we investigate nonlinear second-order and third-order evolution equations of the form u_t = F₁(x,t,u,u_x)u_xx + F₂(x,t,u,u_x) and u_t = F₁(x, t, u, u_x, u_xx)u_xxx + F₂(x, t, u, u_x) u_xx). For each of these equations we construct point transformations which connect them with second-order and third-order linear equations u_t = G₁(x)u_xx+G₂(x)u_x and u_t = G₁{x)u_xxx+ G₂(x)u_x, respectively. These point transformations are hodograph-type transformations which have the property that the new independent vari-ables depend on the old dependent variables and the new dependent vari-ables depend on the old independent variables. The hodograph-type trans-formations which relate the nonlinear PDEs and their corresponding linear PDEs are completely classified.