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Multidimensional Nonhomogeneous Quasi-Linear Systems and Their Hamiltonian Structure (2401.10445v3)
Published 19 Jan 2024 in nlin.SI, math-ph, and math.MP
Abstract: In this paper, we investigate multidimensional first-order quasi-linear systems and find necessary conditions for them to admit Hamiltonian formulation. The insufficiency of the conditions is related to the Poisson cohomology of the admissible Hamiltonian operators. We present in detail the examples of two-dimensional, two-components systems of hydrodynamic type and of a real reduction of the 3-waves system.
- Poisson vertex algebras in the theory of Hamiltonian equations. Jpn. J. Math., 4(2):141–252, 2009.
- Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets. J. Geom. Phys., 114:404–419, 2017.
- Normal forms of dispersive scalar Poisson brackets with two independent variables. Lett. Math. Phys., 108(10):2229–2253, 2018.
- Matteo Casati. On deformations of multidimensional Poisson brackets of hydrodynamic type. Commun. Math. Phys., 335(2):851–894, 2015.
- Integrable Systems. I. In Dynamical Systems IV: Symplectic Geometry and Its Applications, pages 177–332. Springer, 2001.
- Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method. Dokl. Akad. Nauk SSSR, 270(4):781–785, 1983.
- Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. arXiv preprint math/0108160, 2001.
- Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1212+12 + 1 dimensions. J. Math. Phys., 52(7), 2011.
- Ezra Getzler. A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J., 111(3):535–560, 2002.
- Hamiltonian operators and ℓ*superscriptℓ\ell^{*}roman_ℓ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-coverings. J. Geom. Phys., 50(1-4):273–302, 2004.
- The Symbolic Computation of Integrability Structures for Partial Differential Equations. Texts & Monographs in Symbolic Computations. Springer International Publishing, 2018.
- Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys., 54(4):427–453, 2005.
- Oleg I Mokhov. Poisson brackets of Dubrovin-Novikov type (DN-brackets). Funct. Anal. Appl, 22(4):336–338, 1988.
- Oleg I Mokhov. Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems. Uspekhi Mat. Nauk, 53(3(321)):85–192, 1998.
- Oleg I Mokhov. The classification of nonsingular multidimensional Dubrovin-Novikov brackets. Funct. Anal. Appl., 42(1):33–44, 2008.
- Peter J Olver. On the Hamiltonian structure of evolution equations. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 88, pages 71–88. Cambridge University Press, 1980.
- Peter J Olver. Applications of Lie groups to differential equations, volume 107 of Graduate Texts in Mathematics. Springer Science & Business Media, 1993.
- S. P. Tsarëv. The Hamilton property of stationary and inverted equations in continuum mechanics and mathematical physics. Mat. Zametki, 46(1):105–111, 125, 1989.
- Pierandrea Vergallo. Non-homogeneous Hamiltonian structures for quasilinear systems. Boll. Unione Mat. Ital., pages 1–14, 2023.
- Alan Weinstein. The local structure of Poisson manifolds. J. Differ. Geom., 18(3):523–557, 1983.