Multidimensional Nonhomogeneous Quasi-Linear Systems and Their Hamiltonian Structure

Xin Hu ; Matteo Casati

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

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.

Détail

Keywords: Hamiltonian structures, quasilinear systems, non-homogeneous operators.

Bibliographie

[1] Ablowitz M.J., Haberman R., “Nonlinear evolution equations – two and three dimensions”, Phys. Rev. Lett., 35 (1975), 1185–1188 | DOI | MR

[2] Barakat A., De Sole A., Kac V.G., “Poisson vertex algebras in the theory of Hamiltonian equations”, Jpn. J. Math., 4 (2009), 141–252, arXiv: 0907.1275 | DOI | MR | Zbl

[3] Carlet G., Casati M., Shadrin S., “Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets”, J. Geom. Phys., 114 (2017), 404–419, arXiv: 1512.05744 | DOI | MR | Zbl

[4] Carlet G., Casati M., Shadrin S., “Normal forms of dispersive scalar Poisson brackets with two independent variables”, Lett. Math. Phys., 108 (2018), 2229–2253, arXiv: 1707.03703 | DOI | MR | Zbl

[5] Carlet G., Posthuma H., Shadrin S., “Bihamiltonian cohomology of KdV brackets”, Comm. Math. Phys., 341 (2016), 805–819, arXiv: 1406.5595 | DOI | MR | Zbl

[6] Casati M., “On deformations of multidimensional Poisson brackets of hydrodynamic type”, Comm. Math. Phys., 335 (2015), 851–894, arXiv: 1312.1878 | DOI | MR | Zbl

[7] Conn J.F., “Normal forms for smooth Poisson structures”, Ann. of Math., 121 (1985), 565–593 | DOI | MR | Zbl

[8] Dubrovin B.A., Krichever I.M., Novikov S.P., “Integrable systems. I”, Dynamical Systems, IV, Encyclopaedia Math. Sci., 4, Springer, Berlin, 2001, 177–332 | DOI | MR

[9] Dubrovin B.A., Novikov S.P., “Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov–Whitham averaging method”, Dokl. Akad. Nauk SSSR, 270 (1983), 781–785 | MR | Zbl

[10] Dubrovin B.A., Zhang Y., Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, arXiv: math.DG/0108160

[11] Ferapontov E.V., Odesskii A.V., Stoilov N.M.,, “Classification of integrable two-component Hamiltonian systems of hydrodynamic type in $2+1$ dimensions”, J. Math. Phys., 52 (2011), 073505, 28 pp., arXiv: 1007.3782 | DOI | MR | Zbl

[12] Getzler E., “A Darboux theorem for Hamiltonian operators in the formal calculus of variations”, Duke Math. J., 111 (2002), 535–560, arXiv: math/0002164 | DOI | MR | Zbl

[13] Ginzburg V.L., Weinstein A., “Lie–Poisson structure on some Poisson Lie groups”, J. Amer. Math. Soc., 5 (1992), 445–453 | DOI | MR | Zbl

[14] Kersten P., Krasil'shchik I., Verbovetsky A., “Hamiltonian operators and $l^*$-coverings”, J. Geom. Phys., 50 (2004), 273–302, arXiv: math/0304245 | DOI | MR | Zbl

[15] Krasil'shchik I.S., Vinogradov A.M., “Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations: Symmetries of partial differential equations, Part I”, Acta Appl. Math., 15 (1989), 161–209 | DOI | MR | Zbl

[16] Lichnerowicz A., “Les variétés de Poisson et leurs algèbres de Lie associées”, J. Differential Geometry, 12 (1977), 253–300 | DOI | MR | Zbl

[17] Liu S.-Q., Zhang Y., “Deformations of semisimple bihamiltonian structures of hydrodynamic type”, J. Geom. Phys., 54 (2005), 427–453, arXiv: math.DG/0405146 | DOI | MR | Zbl

[18] Mokhov O.I., “Poisson brackets of Dubrovin–Novikov type (DN-brackets)”, Funct. Anal. Appl., 22 (1988), 336–338 | DOI | MR | Zbl

[19] Mokhov O.I., “Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems”, Russian Math. Surveys, 53 (1998), 515–622 | DOI | MR | Zbl

[20] Mokhov O.I., “The classification of nonsingular multidimensional Dubrovin–Novikov brackets”, Funct. Anal. Appl., 42 (2008), 33–44, arXiv: math.DG/0611785 | DOI | MR | Zbl

[21] Mokhov O.I., Ferapontov E.V., “Hamiltonian pairs generated by skew-symmetric Killing tensors on spaces of constant curvature”, Funct. Anal. Appl., 28 (1994), 123–125 | DOI | MR | Zbl

[22] Olver P.J., “On the Hamiltonian structure of evolution equations”, Math. Proc. Cambridge Philos. Soc., 88 (1980), 71–88 | DOI | MR | Zbl

[23] Olver P.J., Applications of Lie groups to differential equations, Grad. Texts Math., 107, 2nd ed., Springer, New York, 1993 | DOI | MR | Zbl

[24] Tsarëv S.P., “The Hamilton property of stationary and inverted equations in continuum mechanics and mathematical physics”, Math. Notes, 46 (1989), 569–573 | DOI | MR | Zbl

[25] Vergallo P., “Quasilinear systems of first order PDEs with nonlocal Hamiltonian structures”, Math. Phys. Anal. Geom., 25 (2022), 26, 19 pp., arXiv: 2203.13554 | DOI | MR | Zbl

[26] Vergallo P., “Non-homogeneous Hamiltonian structures for quasilinear systems”, Boll. Unione Mat. Ital., 17 (2024), 513–526, arXiv: 2301.10713 | DOI | MR

[27] Vergallo P., Vitolo R., “Homogeneous Hamiltonian operators and the theory of coverings”, Differential Geom. Appl., 75 (2021), 101713, 16 pp., arXiv: 2007.15294 | DOI | MR

[28] Weinstein A., “The local structure of Poisson manifolds”, J. Differential Geom., 18 (1983), 523–557 | DOI | MR | Zbl