On a Schwarzian PDE associated with the KdV Hierarchy
نویسندگان
چکیده
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. with four arbitary parameters.
منابع مشابه
ar X iv : n lin / 0 10 50 23 v 2 [ nl in . S I ] 2 J ul 2 00 1 Menelaus ’ theorem , Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, ...
متن کاملar X iv : n lin / 0 10 50 23 v 1 [ nl in . S I ] 9 M ay 2 00 1 Menelaus ’ theorem , Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, ...
متن کاملMoving Frames, Geometric Poisson Brackets and the Kdv-schwarzian Evolution of Pure Spinors
In this paper we describe a non-local moving frame along a curve of pure spinors in O(2m, 2m)/P , and its associated basis of differential invariants. We show that the space of differential invariants of Schwarzian-type define a Poisson submanifold of the spinor Geometric Poisson brackets. The resulting restriction is given by a decoupled system of KdV Poisson structures. We define a generaliza...
متن کاملRemarks on Kdv-type Flows on Star-shaped Curves
We study the relation between the centro-affine geometry of starshaped planar curves and the projective geometry of parametrized maps into RP. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwa...
متن کاملSearching for integrable lattice maps using factorization
We analyze the factorization process for lattice maps, searching for integrable cases. The maps were assumed to be at most quadratic in the dependent variables, and we required minimal factorization (one linear factor) after 2 steps of iteration. The results were then classified using algebraic entropy. Some new models with polynomial growth (strongly associated with integrability) were found. ...
متن کامل