Graded differential equations and their deformations: A computational theory for recursion operators

Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations

“…In accordance with [10], recursion operators are Bäcklund auto-transformations for the tangent covering of the equation under study. To construct recursion operators we use the technique of [25], see also [15,18,19].…”

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

“…for unknown 1-forms ζ q , q ∈ {1, ..., Q}, µ ρ , ρ ∈ {1, ..., R}, and unknown functions V ǫ , ǫ ∈ {1, ..., S} with some Q, R, S ∈ N. The coefficients D κ ρr , ..., K ǫ q in equations (20), (21) are supposed to be functions of U λ and V κ . definition 1.…”

“…definition 1. The system (20), (21) is called an integrable extension of the system (11), (12), if equations (20), (21), (11), (12) together meet the involutivity conditions and the compatibility conditions…”

“…Equations (22) give an over-determined system of pdes for the coefficients D κ ρr , ..., K ǫ q in equations (20), (21). Suppose this system is satisfied.…”

“…Typically, recursion operators are considered as integro-differential operators, [37,12,56,10,9,11,47,1,38,16,48,46,55], although this interpretation is accompanied by a number of difficulties, e.g., discussed in [14,23]. Another definition is proposed in [40,14,19,20] (see also [44] and references therein) and developed in [27,48,32,28,29,31,30]. It treats recursion operators as the Bäcklund autotransformations of the tangent (or linearized) coverings of the pdes.…”