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On billiard solutions of nonlinear PDEs. (English) Zbl 0944.37032

Summary: This Letter presents some special features of a class of integrable PDEs admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discontinuities at peaks in their profiles. A connection is established between the peak locations and finite dimensional billiard systems moving inside n-dimensional quadrics under the field of Hooke potentials. Points of reflection are described in terms of theta-functions and are shown to correspond to the location of peak discontinuities in the PDEs weak solutions. The dynamics of the peaks is described in the context of the algebraic-geometric approach to integrable systems.

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
35Q53 KdV equations (Korteweg-de Vries equations)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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[1] Camassa, R.; Holm, D. D., Phys. Rev. Lett., 71, 1661 (1993)
[2] Camassa, R.; Holm, D. D.; Hyman, J. M., Adv. Appl. Mech., 31, 1 (1994)
[3] Calogero, F., Phys. Lett. A, 201, 306 (1995)
[4] Calogero, F.; Francoise, J.-P., J. Math. Phys., 37, 2863 (1996)
[5] Ragnisco, O.; Bruschi, M., Physica A, 228, 150 (1996)
[6] Suris, Y. B., Phys. Lett. A, 217, 321 (1996)
[7] M.S. Alber, R. Camassa, M. Gekhtman, CRM Proc. & Lecture Notes AMS to appear, 1999.; M.S. Alber, R. Camassa, M. Gekhtman, CRM Proc. & Lecture Notes AMS to appear, 1999.
[8] Antonowicz, M.; Fordy, A. P., Phys. Lett. A, 122, 95 (1987)
[9] Antonowicz, M.; Fordy, A. P., Physica D, 28, 345 (1987)
[10] Manas, M.; Alonso, L. M.; Medina, E., J. Geom. Phys., 29, 13 (1999)
[11] Alber, M. S.; Camassa, R.; Holm, D. D.; Marsden, J. E., Lett. Math. Phys., 32, 137 (1994)
[12] Alber, M. S.; Camassa, R.; Holm, D. D.; Marsden, J. E., Proc. Roy. Soc. A, 450, 677 (1995)
[13] M.S. Alber, R. Camassa, Yu. N. Fedorov, D.D. Holm, J.E. Marsden, preprint 1999, submitted.; M.S. Alber, R. Camassa, Yu. N. Fedorov, D.D. Holm, J.E. Marsden, preprint 1999, submitted.
[14] M.D. Kruskal, Nonlinear wave equations, in: J. Moser (Ed.) Dynamical Systems, Theory and Applications, Lect. Notes in Phys, vol. 38, Springer, New York, 1975.; M.D. Kruskal, Nonlinear wave equations, in: J. Moser (Ed.) Dynamical Systems, Theory and Applications, Lect. Notes in Phys, vol. 38, Springer, New York, 1975.
[15] Cewen, C., Acta Math. Sinica, 6, 35 (1990)
[16] Hunter, J. K.; Zheng, Y. X., Physica D, 79, 361 (1994)
[17] Rauch-Wojciechowski, S., Chaos, Solitons & Fractals, 5, 2235 (1995)
[18] C.G.J. Jacobi, Vorlesungen über Dynamik, Gesamelte Werke. Berlin: Supplementband, 1884.; C.G.J. Jacobi, Vorlesungen über Dynamik, Gesamelte Werke. Berlin: Supplementband, 1884.
[19] C.G.J. Jacobi, Solution nouvelle d’un probleme fondamental de geodesie. Berlin: Gesamelte Werke Bd. 2, 1884.; C.G.J. Jacobi, Solution nouvelle d’un probleme fondamental de geodesie. Berlin: Gesamelte Werke Bd. 2, 1884.
[20] K. Weierstrass, Mathematische Werke I (1878) 257.; K. Weierstrass, Mathematische Werke I (1878) 257. · JFM 25.0049.01
[21] Knörrer, H., J. Reine Angew. Math., 334, 69 (1982)
[22] Moser, J.; Veselov, A., Commun. Math. Phys., 139, 217 (1991)
[23] V.V. Kozlov, D.V. Treschev, Billiards, a Genetic Introduction to Systems with Impacts, AMS Translations of Math. Monographs, vol. 89, AMS, New York, 1991.; V.V. Kozlov, D.V. Treschev, Billiards, a Genetic Introduction to Systems with Impacts, AMS Translations of Math. Monographs, vol. 89, AMS, New York, 1991.
[24] Fedorov, Yu., Amer. Math. Soc. Transl., 168, 173 (1995)
[25] Fedorov, Yu., Acta Appl. Math., 55, 151 (1999)
[26] M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.; M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. · Zbl 0472.35002
[27] Ablowitz, M. J.; Ma, Y-C., Studies in Appl. Math., 65, 113 (1981)
[28] Alber, M. S.; Alber, S. J., C.R. Acad. Sci. Paris Ser. I Math., 301, 77 (1985)
[29] M.S. Alber, C. Miller, Appl. Math. Lett. (1999), to appear.; M.S. Alber, C. Miller, Appl. Math. Lett. (1999), to appear.
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