"Involution and Lie Symmetry Analysis of Differential Equations"
Vladimir Gerdt
Laboratory of Computing Techniques and Automation
Joint Institute for Nuclear Research
141980 Dubna, Russia
Email: gerdt@jinr.ru
ABSTRACT: We present a general algorithmic approach to completion to
involution of linear systems of partial differential equations. We
consider an algorithm for computation of the minimal involutive
system. An important application of the new algorithm is Lie symmetry
analysis of nonlinear differential equations. For construction of Lie
symmetry generators one needs to integrate their determining equations
which form a system of linear partial differential equations. This
step of Lie symmetry analysis is generally the most difficult one, and
completion of the determining equations to involution is the most
universal algorithmic tool of their integration. Another important
application of the involutive method is posing of an initial value
problem providing the unique solution of a system of partial
differential equations. For linear involutive systems we formulate
such a well-posed initial value problem and thereby generalize the
classical results of Janet to arbitrary involutive divisions. This
formulation makes it possible, among other things, to reveal the
structure of arbitrariness in general solution. In particular, for
determining systems occurring in Lie symmetry analysis it allows one
to find the size of symmetry groups, that has been shown by F.Schwarz
for Janet division.