A recursion operator for the universal hierarchy equation via Cartan’s method of equivalence

Abstract

Abstract:We apply Cartan's method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation. MSC:58H05, 58J70, 35A30, 37K05, 37K10

A simple construction of recursion operators for multidimensional dispersionless integrable systems

“…In particular, this method enabled us to find hitherto unknown recursion operators for the general heavenly equation (25) and for (35). Note that our approach, when applicable, is in general computationally less demanding than those of [35] and [40].…”

“…e.g. [1,13,26,31,35,39,40,41,47] and references therein. In fact, the overwhelming majority of integrable systems in four or more independent variable known to date, including those relevant for applications, like the (anti-)self-dual Yang-Mills equations or the (anti-)self-dual vacuum Einstein equations, is dispersionless, i.e., can be rewritten as first-order homogeneous quasilinear systems, cf.…”

“…There are several methods for construction of the ROs for the integrable dispersionless systems: the partner symmetry approach [26], the method based on adjoint action of the Lax operators [35], and the approach using the Cartan equivalence method [39,40,41].…”

Infinitely many nonlocal conservation laws for the ABC equation with $$A+B+C\ne 0$$ A + B + C ≠ 0

“…Later in [27] (cf. also [28,30,36,37] for related results) a recursion operator for (1) with A+B +C = 0 was found, and using the method of hydrodynamic reductions it was shown [5] that (1) is also integrable if A+B+C = 0. Note also that if A = B = C = 0 then (1) admits [12] a Lagrangian with the density −Au x u y u t /2.…”

Deformed cohomologies of symmetry pseudo-groups and coverings of differential equations

“…In [34] we propose an approach to the covering problem based on the technique of contact integrable extensions (cies) of the structure equations of the symmetry pseudo-groups, which is a generalization of the definition of integrable extension from [4, §6] for the case of more than two independent variables. Then in [36,37,38,39] the method of cies was applied to finding of coverings, Bäcklund transformations and recursion operators for a number of pdes.…”