On a Schwarzian PDE associated with the KdV hierarchy

Abstract

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Möbius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDE's for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the full Painlevé VI equation, i.e.

Difference schemes with point symmetries and their numerical tests

“…Then (57) is a Schwarzian O∆S. Schwarzian derivatives play a prominent role in the theory of integrable systems [16] and of dynamical systems [17,18].…”

The discrete and continuous Painlevé VI hierarchy and the Garnier systems

“…[4,5]), of a system of partial difference equations associated with the lattice KdV family. In subsequent papers [6,7] some more results on these systems were established, namely the existence of the Miura chain and the discovery of a novel Schwarzian PDE generating the entire (Schwarzian) KdV hierarchy of nonlinear evolution equations and whose similarity reduction is exactly the P VI equation, this being to our knowledge the first example of an integrable scalar PDE that reduces to full P VI with arbitrary parameters.In the present note we extend these results to multidimensional systems associated with higher-order generalisations of the P VI equation. Already in [3] we noted that the similarity reduction of the lattice KdV system could be generalised in a natural way to higher-order differential and difference equations without, however, clarifying in detail the nature of such equations.…”

“…[4,5]), of a system of partial difference equations associated with the lattice KdV family. In subsequent papers [6,7] some more results on these systems were established, namely the existence of the Miura chain and the discovery of a novel Schwarzian PDE generating the entire (Schwarzian) KdV hierarchy of nonlinear evolution equations and whose similarity reduction is exactly the P VI equation, this being to our knowledge the first example of an integrable scalar PDE that reduces to full P VI with arbitrary parameters.…”

“…In the case of the other Painleve´equa-tions, notably P VI , it has been less clear what to take as a starting point for the construction of its hierarchy. With the results of [3,6,7] we are now well-equipped to tackle this problem, and in the present paper we outline the basic construction of the equations in the discrete as well as the continuous P VI hierarchy. In fact, we shall demonstrate that the lattice KdV system can be naturally embedded in a multidimensional lattice system achieving the higher-order reductions by including more terms in the relevant similarity constraint that provokes the coupling between the various lattice directions.…”

“…Thus, we have here a large multidimensional system of equations with discrete (in terms of the variables n i ) as well as continuous (in terms of the parameters p i ) commuting flows, in terms of which compatible equations of three different types (partial difference, differential-difference and partial differential) figure in one and the same framework: the partial difference equations are precisely the lattice equations (2.2), the differential-difference equations are the relations (2.3), whilst for the partial differential equations in the scheme we refer to our recent paper [7]. Now, we turn to the issue of the symmetry reduction of the multidimensional lattice in the sense of [5].…”