2D-Schrödinger operator, \((2+1)\) evolution systems and new reductions, 2D-Burgers hierarchy and inverse problem data. (English. Russian original) Zbl 1231.35199

Grinevich, P. G.; Mironov, A. E.; Novikov, S. P.

Russ. Math. Surv. 65, No. 3, 580-582 (2010); translation from Uspekhi Mat. Nauk. 65, No. 3, 195-196 (2010).

The paper discusses integrable systems in dimension (2+1) associated with the Lax pair operators in the form \(L=\partial_x\partial_y +G\partial_y + S\) and \(H=\partial_x^2+\partial_y^2+F\partial_y +A\) (the so-called GKMMN systems) and the related problems such as real reductions, reductions to the 2D-Burgers hierarchies, structure of the manifold of the Bloch-Floquet eigenfunctions, and associated algebro-geometric inverse spectral problems.

Reviewer: Dmitry Shepelsky (Kharkov)

MSC:

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.)
35R30	Inverse problems for PDEs

Keywords:

Lax pair; Burgers-type reduction; algebro-geometric inverse spectral problem; Baker-Akhiezer function