Abstract
We introduce a natural class of multicomponent local Poisson structures \(\mathcal P_k + \mathcal P_1\), where \(\mathcal P_1\) is a local Poisson bracket of order one and \(\mathcal P_k\) is a homogeneous Poisson bracket of odd order \(k\) under the assumption that \(\mathcal P_k\) has Darboux coordinates (Darboux–Poisson bracket) and is nondegenerate. For such brackets, we obtain general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples), and provide the description of the Casimirs, using a purely algebraic procedure. In the two-component case, we completely classify such brackets up to a point transformation. From the description of Casimirs, we derive new Harry Dym (HD) hierarchies and new Hunter–Saxton (HS) equations for arbitrary number of components. In the two-component case, our HS equation differs from the well-known HS2 equation.
DOI 10.1134/S1061920822040100
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References
L. Fontanelli, P. Lorenzoni, M. Pedroni, and J. Zubelli, “Bi-Hamiltonian Aspects of a Matrix Harry Dym Hierarchy”, J. Math. Phys., 49 (2008), 092901.
M. Pedroni, V. Sciacca, and J. Zubelli, “The Bi-Hamiltonian Theory of the Harry Dym Equation”, Theoret. Math. Phys., 133 (2002), 1585–1597.
A. Bolsinov, A. Yu. Konyaev, and V. Matveev, “Applications of Nijenhuis Geometry III: Frobenius Pencils and Compatible Non-Homogeneous Poisson Structures”, preprint,.
M. Kruskal, Nonlinear Wave Equations, Dynamical Systems, Theory and Applications, Springer-Verb, Berlin, 1975.
P. C. Sabatier, “On Some Spectral Problems and Isospectral Evolutions Connected with the Classical String Problem. I, II”, Lett. Nuovo Cimento, 26 (1979), 477–486.
P. W. Doyle, “Differential Geometric Poisson Bivectors in One Space Variable”, J. Math. Phys., 34 (1993).
B. Dubrovin and S. Novikov, “Hamiltonian Formalism of One-Dimensional Systems of Hydrodynamic Type, and the Bogolyubov-Whitman Averaging Method”, Dokl. Akad. Nauk SSSR, 270:4 (1983), 781–785.
B. Dubrovin and S. Novikov, “Hydrodynamics of Weakly Deformed Soliton Lattices. Differential Geometry and Hamiltonian Theory”, Uspekhi Mat. Nauk, 44:6 (270) (1989), 29–98.
O. Mokhov, “Pencils of Compatible Metrics and Integrable Systems”, Russ. Math. Surv., 72:5 (2017), 889–93.
L. Landau, “Theory of the Superfluidity of Helium II”, Phys. Rev., 60 (1941).
G. Potemin, “On Third-Order Poisson Brackets of Differential Geometry”, Russ. Math. Surv., 52:3 (1997), 617–618.
A. V. Balandin and G. V. Potemin, “On Non-Degenerate Differential-Geometric Poisson Brackets of Third Order”, Russ. Math. Surv., 56:5 (2001), 976–977.
E. V. Ferapontov, M. V. Pavlov, and R. F. Vitolo, “Towards the Classification of Homogeneous Third-Order Hamiltonian Operators”, Internat. Math. Res. Notices, 2016:22 , 6829–6855.
O. Mokhov, “Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds and Integrable Equations”, Rev. Math. Math. Phys., (2008).
I. Gelfand and I. Dorfman, “Hamiltonian Operators and Algebraic Structures Related to Them”, Funct. Anal. Appl., 13:4 (1979), 248–262.
P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993.
James T. Ferguson, “Flat Pencils of Symplectic Connections and Hamiltonian Operators of Degree 2”, J. Geom. Phys., 58:4 , 468–486.
I. Gelfand, V. Retakh, and M. Shubin, “Fedosov Manifolds”, Adv. Math., 136:1 (1998), 104–140.
P. Lancaster and L. Rodman, “Canonical Forms for Hermitian Matrix Pairs Under Strict Equivalence and Congruence”, SIAM Review, 47 (2005), 407–443.
I. Kaygorodov, I. Rakhimov and Sh. K. Said Husain, “The Algebraic Classification of Nilpotent Associative Commutative Algebras”, J. Algebra Appl., 19:11 (2020), 2050220, 14 pp.
A. Balinskii and S. Novikov, “Poisson Brackets of Hydrodynamic Type, Frobenius Algebras and Lie Algebras”, Dokl. Akad. Nauk SSSR, 283:5 (1985), 1036–1039.
I. A. B. Strachan and B. M. Szablikowski, “Novikov Algebras and a Classification of Multicomponent Camassa–Holm Equations”, Stud. Appl. Math., 133:1 (2014), 84–117.
M. Antonowicz and A. Fordy, “Coupled Harry Dym Equations with Multi-Hamiltonian Structures”, J. Phys. A, 21:5 (1988), L269–L275.
B. Dubrovin, “Flat Pencils of Metrics and Frobenius Manifolds”, Integrable Systems and Algebraic Geometry (Kobe–Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, 47–72.
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.
Senyue Lou, Bao-Feng Feng, and Ruoxia Yao, “Multi-Soliton Solution to the Two-Component Hunter–Saxton Equation”, Wave Motion, 65 (2016), 17–28.
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Konyaev, A.Y. Geometry of Inhomogeneous Poisson Brackets, Multicomponent Harry Dym Hierarchies, and Multicomponent Hunter–Saxton Equations. Russ. J. Math. Phys. 29, 518–541 (2022). https://doi.org/10.1134/S1061920822040100
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DOI: https://doi.org/10.1134/S1061920822040100