×

Poisson geometry of difference Lax operators and difference Galois theory. (English) Zbl 1322.37035

Bastos, M. Amélia (ed.) et al., Operator theory, operator algebras and applications. Selected papers based on the presentations at the workshop WOAT 2012, Lisbon, Portugal, September 11–14, 2012. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0815-6/hbk; 978-3-0348-0816-3/ebook). Operator Theory: Advances and Applications 242, 341-362 (2014).
The author studies the space of second-order Schrödinger differential operators on the line. Motivated by relations to non-linear equations of Lax type, he introduces a natural Poisson structure on the spaces of differential and difference operators which arise as compatibility conditions for a linear equation. His straightforward description is the identification of this space with the hyperplane in the dual of the Virasoro algebra. This space is precisely the phase space for the KdV hierarchy. He proves the existence of a unique Poisson structure on the space of difference connections which makes the gauge action a Poisson action. He extends this structure to the space of wave functions which are solutions of the Schrödinger equation. Remark that observables for KdV equations can be characterized as the subfield of invariants of the differential Galois group \(G= SL(2)\) which acts on wave functions. He also discusses the generalized Drinfeld-Sokolov theory for which the description of higher-order differential operators gives rise to the Hamiltonian reduction.
For the entire collection see [Zbl 1290.47001].

MSC:

37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
12H05 Differential algebra
35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDFBibTeX XMLCite
Full Text: DOI