Ansatzes for the Navier-Stokes field are described. These ansatzes reduce the Navier-Stokes equations to system of differential equations in three, two, and one independent variables. The large sets of exact solutions of the Navier-Stokes equations are constructed.

A new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries. Higher symmetries of the Schrödinger equation with an arbitrary potential are investigated. Nonlinear determining equations for potentials are solved using reductions to Weierstrass, Painlevé, and Riccati forms. Algebraic properties of higher order symmetry operators are analyzed. Combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear Schrödinger equations.

All systems of (n + 1)-dimensional quasilinear evolution second-order equations invariant under chain of algebras AG(1, n) ⊂ AG1(1, n) ⊂ AG2(1, n) are described. The results obtained are illustrated by the examples of the nonlinear Schrödinger equations, Hamilton-Jacobi-type systems and of reaction-diffusion equations.

We construct a number of ansatzes that reduce one-dimensional nonlinear heat equations to systems of ordinary differential equations. Integrating these, we obtain new exact solution of nonlinear heat equations with various nonlinearities.3/2 1 ϕ 3 + ϕ 4 , ϕ i = ϕ i (x 0 , x 2 ).(4)

6 Symmetry properties and exact solutions of system (3.12)As was mentioned in Sec. 3, ansatzes (3.4)-(3.7) reduce the NSEs (1.1) to the systems of PDEs of a similar structure that have the general form (see (3.12)):

We present a detailed account of symmetry properties of SU (2) Yang-Mills equations. Using a subgroup structure of the Poincaré group P (1, 3) we have constructed all P (1, 3)-inequivalent ansatzes for the Yang-Mills field which are invariant under the three-dimensional subgroups of the Poincaré group. With the aid of these ansatzes reduction of Yang-Mills equations to systems of ordinary differential equations is carried out and wide families of their exact solutions are constructed.

We have studied the maximal s m m e t r y group admitted by the non-linear polywave equation O'u = F(u). In particular, we establish that the equation in question admits theSymmetry reduction for the biwave equation U2u = Au -3 is carried out and some exact solutions are obtained.

We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The AiG beam describes in a more realistic way the propagation of the Airy wave packets because AiG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation.