Conditional Symmetry Approach Research Articles

A Precise Definition of Reduction of Partial Differential Equations

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1+3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear wave equations.

Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations

In this paper, a class of nonlinear diffusion-convection equations, ut = (D(u)uxn)x+P(u)ux , which has quite a large number of physical applications, is analysed by using symmetry group methods which include the classical method, the potential symmetry method and the generalized conditional symmetry method. A complete classification of the functional forms of the diffusion and convection coefficients is presented when the equation admits Lie's point symmetry groups and potential symmetry groups. The separation of variables for the equation is investigated using the generalized conditional symmetry approach. For some interesting cases, exact solutions using the method of separation of variables are discussed in detail.

Non-Lie Symmetries and Supersymmetries

Appeared more than one century ago, the classical Lie approach serves as a powerful toolin investigations of symmetries of partial differential equations. In the last three decadesthere appear essential generalizations of this approach. They are the modern version ofthe Lie-B¨acklund symmetries [1], the non-Lie approach [2–4], the conditional symmetryapproach [5–7], etc.I will speak about non-Lie symmetries of equations of mathematical physics. The term”non-Lie symmetries” has appeared in papers of Fushchych [2], the main ideas and basicresults of the non-Lie approach are outlined in our books [3, 4].To formulate the general idea of non-Lie symmetries, we consider a linear differentialequationL