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Systems of conservation laws with third-order Hamiltonian structures

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Abstract

We investigate n-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in \(\mathbb {P}^{n+2}\) satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space W of dimension \(n+2\), classify n-tuples of skew-symmetric 2-forms \(A^{\alpha } \in \varLambda ^2(W)\) such that

$$\begin{aligned} \phi _{\beta \gamma }A^{\beta }\wedge A^{\gamma }=0, \end{aligned}$$

for some non-degenerate symmetric \(\phi \).

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Acknowledgements

We thank R. Chirivì, N. Hitchin, A. King, J.S. Krasil’shchik, L. Manivel, and A.M. Verbovetsky for clarifying discussions. We acknowledge financial support from GNFM of the Istituto Nazionale di Alta Matematica, the Istituto Nazionale di Fisica Nucleare by IS-CSN4 Mathematical Methods of Nonlinear Physics, and the Dipartimento di Matematica e Fisica “E. De Giorgi” of the Università del Salento. MVP’s work was partially supported by the grant of the Presidium of RAS ‘Fundamental Problems of Nonlinear Dynamics’.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, UK

    Evgeny V. Ferapontov

  2. Lebedev Physical Institute of Russian Academy of Sciences, Leninskij Prospekt 53, Moscow, Russia, 119991

    Maxim V. Pavlov

  3. Department of Mathematics and Physics “E. De Giorgi”, University of Salento, Lecce, Italy

    Raffaele F. Vitolo

  4. Sezione INFN di Lecce, Lecce, Italy

    Raffaele F. Vitolo

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  1. Evgeny V. Ferapontov
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  2. Maxim V. Pavlov
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  3. Raffaele F. Vitolo
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Correspondence to Raffaele F. Vitolo.

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To Professor Boris Konopelchenko on the occasion of his 70th birthday.

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Ferapontov, E.V., Pavlov, M.V. & Vitolo, R.F. Systems of conservation laws with third-order Hamiltonian structures. Lett Math Phys 108, 1525–1550 (2018). https://doi.org/10.1007/s11005-018-1054-3

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  • DOI: https://doi.org/10.1007/s11005-018-1054-3

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