Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Richness or semi-Hamiltonicity of quasi-linear systems that are not in evolution form

Author: Misha Bialy
Journal: Quart. Appl. Math. 71 (2013), 787-796
MSC (2000): Primary 35L65, 35L67, 70H06
DOI: https://doi.org/10.1090/S0033-569X-2013-01327-3
Published electronically: August 30, 2013
MathSciNet review: 3136996
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Abstract: The aim of this paper is to consider strictly hyperbolic quasi-linear systems of conservation laws which appear in the form $ A(u)u_{x}+B(u)u_{y}=0.$ If one of the matrices $ A(u), B(u)$ is invertible, then this system is in fact in the form of evolution equations. However, it may happen that traveling along characteristics one moves from the ``chart'' where $ A(u)$ is invertible to another ``chart'' where $ B(u)$ is invertible. We propose a new condition of richness or semi-Hamiltonicity for such a system that is ``chart''-independent. This new condition enables one to perform the blow-up analysis along characteristic curves for all times, not passing from one ``chart'' to another. This opens a possibility to use this theory for geometric problems as well as for stationary solutions of 2D+1 systems. We apply the results to the problem of polynomial integral for geodesic flows on the 2-torus.

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  • [1] Misha Bialy, On periodic solutions for a reduction of Benney chain, NoDEA Nonlinear Differential Equations Appl. 16 (2009), no. 6, 731-743. MR 2565284 (2011d:35302), https://doi.org/10.1007/s00030-009-0032-y
  • [2] Misha Bialy and Andrey E. Mironov, Rich quasi-linear system for integrable geodesic flows on 2-torus, Discrete Contin. Dyn. Syst. 29 (2011), no. 1, 81-90. MR 2725282 (2011j:37104), https://doi.org/10.3934/dcds.2011.29.81
  • [3] M. Bialy, A. Mironov. Qubic and Quartic integrals for geodesic flow on 2-torus via system of Hydrodynamic type. Preprint arXiv:1101.3449, 2011 MR 2725282 (2011j:37104)
  • [4] A. V. Bolsinov and A. T. Fomenko, Integrable geodesic flows on two-dimensional surfaces, Monographs in Contemporary Mathematics, Consultants Bureau, New York, 2000. MR 1771493 (2003a:37072)
  • [5] B. A. Dubrovin and S. P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat. Nauk 44 (1989), no. 6(270), 29-98, 203 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 6, 35-124. MR 1037010 (91g:58109), https://doi.org/10.1070/RM1989v044n06ABEH002300
  • [6] Holger R. Dullin and Vladimir S. Matveev, A new integrable system on the sphere, Math. Res. Lett. 11 (2004), no. 5-6, 715-722. MR 2106237 (2005j:37094)
  • [7] E. V. Ferapontov and D. G. Marshall, Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor, Math. Ann. 339 (2007), no. 1, 61-99. MR 2317763 (2008f:37146), https://doi.org/10.1007/s00208-007-0106-2
  • [8] V. N. Kolokoltsov, Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial with respect to velocities, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 994-1010, 1135 (Russian). MR 675528 (84d:58064)
  • [9] Helge Kristian Jenssen and Irina A. Kogan, Extensions for systems of conservation laws, Comm. Partial Differential Equations 37 (2012), no. 6, 1096-1140. MR 2924467, https://doi.org/10.1080/03605302.2011.626827
  • [10] A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. Translated from the Russian by A. G. Reyman [A. G. Reĭman]. MR 1048350 (91g:58127)
  • [11] D. Serre. Systems of Conservation Laws. vol. 2, Cambridge University Press, 1999. MR 1775057 (2001c:35146)
  • [12] Sevennec, B. Geometrie des systemes de lois de conservation, vol. 56, Memoires, Soc. Math. de France, Marseille, 1994.
  • [13] S. P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 5, 1048-1068 (Russian); English transl., Math. USSR-Izv. 37 (1991), no. 2, 397-419. MR 1086085 (92b:58109)

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Additional Information

Misha Bialy
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel
Email: bialy@post.tau.ac.il

DOI: https://doi.org/10.1090/S0033-569X-2013-01327-3
Keywords: Rich, conservation laws, genuine nonlinearity, blow-up, systems of hydrodynamic type
Received by editor(s): March 26, 2012
Published electronically: August 30, 2013
Additional Notes: Partially supported by ISF grant 128/10
Article copyright: © Copyright 2013 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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