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A precise definition of reduction of partial differential equations. (English) Zbl 0936.35012

By reduction in the title here is meant Lie symmetry reduction of partial differential equations to ordinary differential equations, more specifically reduction under conditional symmetries. Such types of symmetry came into practical use in the late eighties through the work of P. J. Olver and P. Rosenau [Phys. Lett. A114, 107-112 (1986)] and V. I. Fushchich and I. M. Tsifra [J. Phys. A 20, L45–L48 (1987; Zbl 0663.35045)] based on an earlier idea of G. W. Bluman and J. D. Cole [J. Math. Mech. 18, 1025-1042 (1969; Zbl 0187.03502)].

A partial differential equation, $L=0$ is conditionally invariant under a set of involutive operators $\left\{{Q}_{a}|a=1...n\right\}$ if the application of their prologations to $L$ vanishes not on the submanifold of jet space defined by the equation but only on its intersection with those associated with the ${Q}_{a}$’s. The authors point out that in practice the idea is more easily used for equations with only two independent variables, and their aim is to provide a basis for a less ad hoc treatment of higher dimensional equations. They achieve this by constructing an ansatz adapted to the general solution of the symmetry defining system: ${Q}_{a}\equiv 0$. This is a clear, straightforward treatment in which the reduction ansatz implies the presence of a conditional symmetry and vice versa. They apply their results to systems if the form $\square u=F\left(u\right)$ where $\square$ is the d’Alembertian in four dimensions, which reduce to d’Alembert-Hamilton systems. They consider special cases and recover some new solutions.

##### MSC:
 35A30 Geometric theory for PDE, characteristics, transformations 58J70 Invariance and symmetry properties