A Precise Definition of Reduction of Partial Differential Equations

Abstract

We give a comprehensive analysis of interrelations between the basic concepts of Ž . the modern theory of symmetry classical and non-classical reductions of partial differential equations. Using the introduced definition of reduction of differential Ž . equations we establish equivalence of the non-classical conditional symmetry and Ž . 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 q 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.

Equivalence problem and integrability of the Riccati equations

“…The invariants of equivalence transformations and infinitesimal technique of finding these invariants has been introduced by Ovsiannikov [3,4]. It is known that invariants and differential invariants of Lie group of symmetry of differential equation and Lie group of equivalence transformations of a class of differential equations are very 372 T. Czyżycki and J. Hrivnák NoDEA important in investigations of their properties such as reduction and invariant form [5][6][7][8][9], as well as isomorphisms of the studied Lie algebras [10]. In this paper we find the Lie group of equivalence transformations of the class of the general real Riccati equations and study its properties.…”

Classical and nonclassical symmetries of a nonlinear differential equation for describing waves in a liquid with gas bubbles

“…Можно показать, что условие инвариантно-сти для уравнения (10) выполняется при любых значениях компонент касательного векторного поля ξ, τ и η (см. работы [11,12]). Отметим, что если X является оператором неклассической симметрии, то λX также является оператором неклассической симметрии для любой неравной тожде-ственно нулю функции λ(x, t, u) [11,12].…”

“…работы [11,12]). Отметим, что если X является оператором неклассической симметрии, то λX также является оператором неклассической симметрии для любой неравной тожде-ственно нулю функции λ(x, t, u) [11,12]. Следовательно, в дальнейшем необходимо рассмотреть два случая неклассических операторов X. Первым из них является слу-чай τ = 0, где без ограничения общности можно считать, что τ = 1.…”

On the Notion of Conditional Symmetry of Differential Equations

“…involving only the variables z and functions depending only on z (see [24] for a detailed discussion on the reduction procedure). In particular, in the case of a single PDE for a single unknown function depending on two variables, the PDE is reduced to an ODE, as well known, and as in Example 2 above.…”