@MISC{Olver94directreduction, author = {Peter Olver}, title = {Direct Reduction and Differential Constraints}, year = {1994} }

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Abstract

. Direct reductions of partial differential equations to systems of ordinary differential equations are in one-to-one correspondence with compatible differential constraints. The differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions, but admits a noncharacteristic second order reduction if and only if it is linearizable. 1. Introduction. One of the most useful methods for determining particular explicit solutions to partial differential equations is to reduce them to ordinary differential equations (which are presumably easier to solve) through a suitably inspired ansatz. The classical Lie method for finding group-invariant solutions, first described in full generality in [7], generalizes and includes well-known methods for finding similarity solutions, travelling wave solutions, and other basic reduction methods. For example, the solutions which are invariant under a one-parameter symmetry...