On recursion operators for symmetries of the Pavlov-Mikhalev equation

TL;DR

This paper explores recursion operators for symmetries of an extended Pavlov-Mikhalev equation, constructing and analyzing their properties, including hereditary and compatibility features, and introduces new degenerate operators called queer operators.

Contribution

It constructs and studies new recursion operators for a two-component extension of the Pavlov-Mikhalev equation, revealing their algebraic and geometric properties.

Findings

Constructed two recursion operators and analyzed their properties.

Established hereditary and compatibility properties of these operators.

Discovered twelve new degenerate 'queer' operators and discussed their significance.

Abstract

In geometry of nonlinear partial differential equations, recursion operators that act on symmetries of an equation $\mathcal{E}$ are understood as B\"{a}cklund auto-transformations of the equation $\mathcal{TE}$ tangent to $\mathcal{E}$. We apply this approach to a natural two-component extension of the 3D Pavlov-Mikhalev equation \begin{equation*} u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}. \end{equation*} We describe the Lie algebra of symmetries for this extension, construct two recursion operators (one of them was known earlier) and find their action. We also establish the hereditary property of these operators as well as their compatibility (in the sense of the Fr\"{o}licher-Nijenhuis bracket). We find also twelve additional operators which are degenerate in a sense (we call them \emph{queer}) and discuss their properties. In the concluding part, a geometrical background of two-component conservation laws for multi-dimensional equations is exposed together with its relations to differential coverings.