
Group Analysis of Differential Equations and Integrable Systems − 2018
Pavel Bibikov
(Institute of Control Sciences, Moscow, Russia)
On Lie problem in PDEs and effective classification of equations
$u_{xy} = F(x, y, u)$ with algebraic right hand sides
Abstract:
The problem of studying the action of contact or point
transformations on various classes of differential equations is one
of the most important problems in mathematics and has been studied
by many prominent mathematicians, including S. Lie, E. Cartan, etc.
The main methods in their works are based on ideas of the geometry
of differential equations, which has a long history and is
extensively studied nowadays. While studying the class of
secondorder ordinary differential equations cubic in the first
variable, Lie set the problem of finding differential invariants and
classifying differential equations of the form $y'' = F(x, y)$ with
respect to the pseudogroup of point transformations. Despite
considerable efforts, Lie could not solve this problem; he only
mentioned that there are no differential invariants of order $<4$.
The solution of this problem has been recently obtained by P.
Bibikov.
In this talk we discuss the generalization of the Lie problem in
case of partial differential equations. Namely, we consider the
equations $u_{xy}=F(x,y,u)$. For such equations we calculate the
field of differential invariants and obtain the criterion for the
local equivalence of two such equations. Finally, we consider the
equations with algebraic right hand side and provide the effective
analog of this criterion of equivalence for such equations.
The author is supported by RFBN, grant 16\_01\_60018 mol\_a\_dk.

