2010/07/22 by E. V. Ferapontov, A. V. Odesskiĭ, A. V. Odesskii +2
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Hamiltonian (control theory) #Hamiltonian mechanics #Hamiltonian system #Hodograph #Hypergeometric distribution #Hypergeometric function #Integrable system #Lie algebra #Mathematical analysis #Mathematical physics #Mathematics #Moduli space #Nonlinear Waves and Solitons #Phase space #Physics #Poisson bracket #Pure mathematics #Quantum mechanics #math.AP #msc:35L40 #msc:35L65 #msc:37K10 #nlin.SI
paper · pdf · doi:10.1063/1.3602081
Latex, 34 pages
arxiv created 2010/07/22 · openalex publication_date 2011/07/01 · arxiv updated 2015/05/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure, and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.