Differential Equations Invariant Under Conditional Symmetries

Abstract

Nonlinear PDE's having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous Korteweg-de Vries like equations are given to show and clarify the methodology introduced.

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

“…We should mention that one could also investigate the inverse problem for conditional symmetries: that is, for a given number of dependent and independent variables and a given order of the equation, and given a certain vector field X, determine all the differential equations ∆ which admit X as a conditional symmetry. This was considered by Levi, Rodriguez and Thomova in [55], and more recently by Pucci and Saccomandi [56,57]. We are not able to provide stochastic extensions of this analysis at the moment.…”

The discretized Boussinesq equation and its conditional symmetry reduction

“…As C = 0 appears in (4) as a condition imposed on the determining equations one has called the resulting symmetries conditional symmetries [6,18,19,32,40]. It has been shown in [30,31] that a PDE E = 0 can be written in terms of the invariants I j of its conditional symmetries X as…”

“…In the following we present an overview of the results obtained earlier [30,31], namely the construction of the Boussinesq equation from the invariants of three chosen conditional symmetries and the reduction with respect to these conditional symmetries. These are the cases we shall discretize in section 3.…”

Using Symmetries à Rebours

“…Then we choose Φ (0,0) = + ( , ), Φ (1,0) = , Φ (3,0) in so doing we recover a particular case of the famous nonclassical reduction for the Boussinesq equation. 15,16…”