Structural properties of the noncommutative KdV recursion operator

Abstract

Alternative sets of multisoliton solutions of some integrable KdV type equations via direct methods AIP Conf.The present work studies structural properties of the recursion operator of the noncommutative KdV equation. As the main result, it is proved that this operator is hereditary. The notion of hereditary operators was introduced by Fuchssteiner for infinite-dimensional integrable systems, building on classical concepts from differential topology. As an illustration for the consequences of this property, it is deduced that the flows of the noncommutative KdV hierarchy mutually commute. C 2011 American Institute of Physics. [doi:10.1063/1.3656271] I. INTRODUCTIONThe noncommutative Korteweg-de Vries equation (ncKdV) readswhere u = u(x, t) is a function with values in some (possibly noncommutative) Banach algebra A and {u, v} = uv + vu denotes the anticommutator. In the sequel we will consider (1) as an ordinary differential equationwhere u = u(t) is a point varying in an infinite-dimensional algebra F of A-valued functions depending on the space variable x ∈ R, and where K is the vector fieldOur main topic are the structural properties of the noncommutative KdV recursion operator KdV (u) = D 2 + 2A

KdV-type equations linked via Bäcklund transformations: Remarks and perspectives

“…A remarkable property to stress in the present contest is that, also in the non-Abelian case, hereditariness is preserved under Bäcklund transformations. Hence, proved the hereditariness of one recursion operator [52,10], the hereditariness of all the recursion operator of other non-Abelian equations linked to it follows, see [49,50,51,8,11].…”

A novel noncommutative KdV-type equation, its recursion operator, and solitons

“…The essence is to view an evolution equation as an ordinary differential equation with values in a suitable function space and to apply methods from differential topology and dynamical systems. We refer to [20] for an introduction to the general theory, and to [24] and the references therein for more details on applications relevant for the present article.…”

“…More precisely, recursion operators are derived, and it is explained why these operators are hereditary. Note that hereditariness is much harder to verify directly in the noncommutative setting [24]. This leads to hierarchies of commuting symmetries with the Cole-Hopf links extending to each level of the respective hierarchies.…”

Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart

“…Unfortunately its direct verification often requires involved computations, in particular in the non-Abelian case. The following proposition uses the Bäcklund links established in the present article to avoid computations by reducing the proof of hereditariness to the hereditariness of the non-Abelian KdV recursion operator, which is proved in [54].…”

“…implying that Φ(U ) is a strong symmetry (recursion operator in the sense of [43]) for the trivial member U t = U x of the non-Abelian KdV hierarchy [20]. Moreover, the main result in [54] is that Φ(U ) is hereditary. Hence Φ(U ) is a strong symmetry for all equations of the non-Abelian KdV hierarchy [20].…”