Dynamics and exact solutions of non-evolutionary partial differential equations

The article presents a new method for constructing exact solutions of non-evolutionary partial differential equations with two independent variables. The method is applied to the linear classical equations of mathematical physics: the Helmholtz equation and the variable type equation. The constructed method goes back to the theory of finite-dimensional dynamics proposed for evolutionary differential equations by B. Kruglikov, O. Lychagina and V. Lychagin. This theory is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of PDEs. The proposed method is used to construct exact particular solutions of linear differential equations (Helmholtz equations and equations of variable type).