Multipeakons viewed as geodesics

• Autorzy: Cieślak T. , Gaczkowski M. , Kubkowski M. , Małogrosz M.

Identyfikatory

DOI

10.4064/ba8119-6-2017

• Języki publikacji: EN

Abstrakty

EN

We adress the problem of qualitative properties of multipeakons, particular solutions of the Camassa-Holm equation. Our approach makes use of the well-known fact that the evolution of multipeakons is governed by the geodesic motion of a particle on an N-dimensional surface whose metric tensor is given via the inverse matrix to the one defining the Hamiltonian. Our approach yields some properties of twopeakons in a very simple way. We classify initial shapes of twopeakons according to the occurrence of collision. Moreover we extend the class of matrices that are invertible for similar reasons to the one occurring in the Hamiltonian. We get exact formulas for the inverses.

Słowa kluczowe

EN

multipeakons; positive definite matrix; lagrangian

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2017
• Tom: Vol. 65, no. 2
• Strony: 153--164
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Cieślak T.

• Institute of Mathematics, Polish Academy of Sciences, 00-656 Warszawa, Poland

autor

Gaczkowski M.

• Institute of Mathematics, Polish Academy of Sciences, 00-656 Warszawa, Poland

autor

Kubkowski M.

• Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, 00-662 Warszawa, Poland

autor

Małogrosz M.

• Institute of Mathematics, Polish Academy of Sciences, 00-656 Warszawa, Poland

Bibliografia

• [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math. 60, Springer, 1989.
• [2] R. Beals, D. H. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Adv. Math. 140 (1998), 190-206.
• [3] P. Billingsley, Probability and Measure, Wiley, 1979.
• [4] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singapore) 5 (2007), 1-27.
• [5] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664.
• [6] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998), 303-328.
• [7] J. Eckhardt and A. Kostenko, An isospectral problem for global conservative multipeakon solutions of the Camassa-Holm equation, Comm. Math. Phys. 329 (2014), 893-918.
• [8] K. Grunert and H. Holden, The general peakon-antipeakon solution for the Camassa-Holm equation, J. Hyperbolic Differerential Equations 13 (2016), 353-380.
• [9] H. Holden and X. Raynaud, Global dissipative multipeakon solutions of the Camassa-Holm equation, Comm. Partial Differerential Equations 33 (2008), 2040-2063.
• [10] H. Holden and X. Raynaud, Global conservative multipeakon solutions of the Camassa-Holm equation, J. Hyperbolic Differerential Equations 4 (2007), 39-64.
• [11] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (2000), 1411-1433.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).