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Rich quasi-linear system for integrable geodesic flows on 2-torus

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  • Consider a Riemannian metric on two-torus. We prove that the question of existence of polynomial first integrals leads naturally to a remarkable system of quasi-linear equations which turns out to be a Rich system of conservation laws. This reduces the question of integrability to the question of existence of smooth (quasi-) periodic solutions for this Rich quasi-linear system.
    Mathematics Subject Classification: Primary: 37J35, 37K05, 37K10; Secondary: 35L67.

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  • [1]

    V. I. Arnold, "Mathematical Methods of Classical Mechanics," (Graduate Texts in Mathematics v. 60), Springer.

    [2]

    M. Bialy, On periodic solutions for a reduction of Benney chain, Nonlin. Diff. Eq. and Appl., 16 (2009) 731-743.

    [3]

    M. Bialy, Polynomial integrals for a Hamiltonian system and breakdown of smooth solutions for quasi-linear equations, Nonlinearity, 7 (1994), 1169-1174.doi: doi:10.1088/0951-7715/7/4/005.

    [4]

    M. Bialy, Integrable geodesic flows on surfaces, Geom. and Funct. Analysis, 20 (2010), 357-367.doi: doi:10.1007/s00039-010-0069-4.

    [5]

    M. Bialy, Hamiltonian form and infinity many conservation laws for a quasiliner system, Nonlinearity, 10 (1997), 925-930.doi: doi:10.1088/0951-7715/10/4/007.

    [6]

    A. V. Bolsinov and A. T. Fomenko, "Integrable Geodesic Flows on two Dimensional Surfaces," Monographs in Contemporary Mathematics, Consultants Bureau, New York, 2000.

    [7]

    B. A. Dubrovin and S. P. Novikov, Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov- Whitham averaging method, Dokl. Akad. Nauk SSSR, 270 (1983), 781-785; English transl. Soviet Math. Dokl., 27 (1983).

    [8]

    H. R. Dullin and V. Matveev, A new integrable system on the sphere, Math Research Letters, 11 (2004), 715-722.

    [9]

    H. R. Dullin, V. Matveev and P. Topalov, On integrals of third degree in momenta, Regular and Chaotic Dynamics, 4 (1999), 35-44.doi: doi:10.1070/rd1999v004n03ABEH000114.

    [10]

    I. M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and related algebraic structures, Funktsional. Anal, i Prilozhen., 13 (1979), 13-30.

    [11]

    L. S. Hall, A theory of exact and approximate configuration invariants, Physica D, 8 (1983), 90-116.doi: doi:10.1016/0167-2789(83)90312-3.

    [12]

    V. N. Kolokoltsov, Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 994-1010.

    [13]

    A. M. Perelomov, "Integrable Systems of Classical Mechanics and Lie Algebras," Vol. I, Birkhauser Verlag, Basel, 1990.

    [14]

    E. N. Selivanova, New examples of Integrable Conservative Systems on $S^2$ and the Case of Goryachev-Chaplygin, Comm. Math. Phys., 207 (1999), 641-663.doi: doi:10.1007/s002200050740.

    [15]

    D. Serre, "Systems of Conservation Laws," Vol. 2, Geometric structures, oscillations, and initial-boundary value problems. Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 2000.

    [16]

    S. P. Tsarev, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419.

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