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

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 second-order 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.