Normal forms of dispersive scalar Poisson brackets with two independent variables

2017/07/12 by Guido Carlet, Matteo Casati, Sergey Shadrin · 5 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Equivalence (formal languages) #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Poisson bracket #Poisson distribution #Pure mathematics #Scalar (mathematics) #Statistics #Triviality #Variable (mathematics) #Variables #math-ph #math.DG #math.MP #nlin.SI

paper · pdf · doi:10.1007/s11005-018-1076-x

published in Letters in Mathematical Physics 108(10), 2229-2253 (Springer Science+Business Media) · 19 pages

arxiv created 2017/07/12 · openalex publication_date 2018/03/26 · arxiv updated 2018/10/23 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05

Abstract

We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well-known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants.

Citations

• Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed
• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• On Deformation of Poisson Manifolds of Hydrodynamic Type
• Deformations of bi-Hamiltonian structures of hydrodynamic type
• Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets
• Central invariants revisited