Description

Classical works in Lie group analysis, e.g. important papers of S.Lie and A.V.Bäcklund are written in old German and somewhat old fashioned mathematical language. The present volume comprises translation into English of fundamental papers of S. Lie, A.V.Bäcklund and L.V. Ovsyannikov. I have selected here some of my favorite papers containing profound results significant for modern group analysis. The first paper imparts not only Lie's interesting view on the development of the general theory of differential equations but also contains Lie's theory of group invariant solutions. His second paper is dedicated to group classification of second-order linear partial differential equations in two variables and can serve as a concise practical guide to the group analysis of partial differential equations even today. The translation of Bäcklund's fundamental paper on non-existence of finite-order tangent transformations higher than first-order contains roots of the modern theory of Lie-Bäcklund transformation groups. Finally, Ovsyannikov's paper contains an essential development of the group classification of hyperbolic equations given in Lie's second paper. Moreover, it contains two proper invariants for hyperbolic equations discovered by Ovsyannikov.

contains Lie's (great historical overview of the attempts at solving differential equations!)

Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung [General Theory of Partial Differential Equations of Arbitrary Order]. Von Sophus Lie. Leipz. Berichte, Heft I, 1895, S. 53–128.

and

Arch. for Math., Bd.VI, Heft 3, S.328-368, Kristiania 1881.

S. Lie, A.V.Bäcklund and L.V. Ovsyannikov

Sophus Lie (1842-1899)

Created a new branch in mathematics - Lie groups, Lie algebras and group analysis of differential equations.

If the historical exposition given here is correct, I can claim that I have been the first to use the concept of groups in the integration theory of differential equations. (S. Lie, Paper I in this volume, Section 12).

Albert Victor Bäcklund (1845-1922)

His name is associated with Bäcklund transformations and Lie-Bäcklund transformation groups. Moreover, he developped the theory of characteristics for second-order partial differential equations with an arbitrary number of independent variables (for two variables, characteristics were known since Monge and Ampére).

Lev Vasilyevich Ovsyannikov (born 1919)

The spearhead in the restoration of group analysis of differential equations in the 1960s and establishing modern group analysis. He made a fundamental contribution to theory of invariant and partially invariant solutions, group classification of differential equations and fluid mechanics.