January  2011, 29(1): 81-90. doi: 10.3934/dcds.2011.29.81

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

1. 

School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel

2. 

Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, 630090, Russian Federation

Received  October 2009 Revised  April 2010 Published  September 2010

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.
Citation: Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
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show all references

References:
[1]

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

[2]

Nonlin. Diff. Eq. and Appl., 16 (2009) 731-743.  Google Scholar

[3]

Nonlinearity, 7 (1994), 1169-1174. doi: doi:10.1088/0951-7715/7/4/005.  Google Scholar

[4]

Geom. and Funct. Analysis, 20 (2010), 357-367. doi: doi:10.1007/s00039-010-0069-4.  Google Scholar

[5]

Nonlinearity, 10 (1997), 925-930. doi: doi:10.1088/0951-7715/10/4/007.  Google Scholar

[6]

Monographs in Contemporary Mathematics, Consultants Bureau, New York, 2000.  Google Scholar

[7]

Dokl. Akad. Nauk SSSR, 270 (1983), 781-785; English transl. Soviet Math. Dokl., 27 (1983).  Google Scholar

[8]

Math Research Letters, 11 (2004), 715-722.  Google Scholar

[9]

Regular and Chaotic Dynamics, 4 (1999), 35-44. doi: doi:10.1070/rd1999v004n03ABEH000114.  Google Scholar

[10]

Funktsional. Anal, i Prilozhen., 13 (1979), 13-30. Google Scholar

[11]

Physica D, 8 (1983), 90-116. doi: doi:10.1016/0167-2789(83)90312-3.  Google Scholar

[12]

Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 994-1010.  Google Scholar

[13]

Birkhauser Verlag, Basel, 1990. Google Scholar

[14]

Comm. Math. Phys., 207 (1999), 641-663. doi: doi:10.1007/s002200050740.  Google Scholar

[15]

Geometric structures, oscillations, and initial-boundary value problems. Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 2000. Google Scholar

[16]

Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1048-1068; translation in Math. USSR-Izv., 37 (1991), 397-419.  Google Scholar

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