Geodesic
Calogero–FranÇoise Flows and Periodic Peakons
Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 3, pp. 367-385 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The completely integrable Hamiltonian systems discovered by Calogero and FranÇoise contain the finite-dimensional reductions of the Camassa–Holm and Hunter–Saxton equations. We show that the associated spectral problem has the same form as that of the periodic discrete Camassa–Holm equation. The flow is linearized by the Abel map on a hyperelliptic curve. For two-particle systems, which correspond to genus-1 curves, explicit solutions are obtained in terms of the Weierstrass elliptic functions.
Keywords: finite-dimensional Hamiltonians, elliptic and hyperelliptic curves, Abel maps. 
@article{TMF_2002_133_3_a3,
     author = {R. Beals and D. H. Sattinger and J. Szmigielski},
     title = {Calogero{\textendash}Fran\c{C}oise {Flows} and {Periodic} {Peakons}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {367--385},
     year = {2002},
     volume = {133},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_133_3_a3/}
}
TY  - JOUR
AU  - R. Beals
AU  - D. H. Sattinger
AU  - J. Szmigielski
TI  - Calogero–FranÇoise Flows and Periodic Peakons
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2002
SP  - 367
EP  - 385
VL  - 133
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2002_133_3_a3/
LA  - ru
ID  - TMF_2002_133_3_a3
ER  - 
%0 Journal Article
%A R. Beals
%A D. H. Sattinger
%A J. Szmigielski
%T Calogero–FranÇoise Flows and Periodic Peakons
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2002
%P 367-385
%V 133
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2002_133_3_a3/
%G ru
%F TMF_2002_133_3_a3
R. Beals; D. H. Sattinger; J. Szmigielski. Calogero–FranÇoise Flows and Periodic Peakons. Teoretičeskaâ i matematičeskaâ fizika, Tome 133 (2002) no. 3, pp. 367-385. http://geodesic.mathdoc.fr/item/TMF_2002_133_3_a3/

[1] F. Calogero, Phys. Lett. A, 201 (1995), 306–310 | DOI | MR | Zbl

[2] R. Camassa, D. D. Holm, Phys. Rev. Lett., 71 (1993), 1661–1664 ; R. Camassa, D. D. Holm, J. M. Hyman, Adv. Appl. Mech., 31 (1994), 1–33 | DOI | MR | Zbl | DOI | Zbl

[3] F. Calogero, J. P. Françoise, J. Math. Phys., 37 (1996), 2863–2871 | DOI | MR | Zbl

[4] F. Calogero, Classical Many-Body Problems Amenable to Exact Treatments, Lect. Notes Phys., 66, Springer, New York, 2001 | DOI | MR

[5] R. Beals, D. H. Sattinger, J. Szmigielski, Appl. Anal., 78 (2001), 255–269 | DOI | MR | Zbl

[6] J. K. Hunter, R. Saxton, SIAM J. Appl. Math., 51 (1991), 1498–1521 ; J. K. Hunter, Y. Zheng, Physica D, 79 (1994), 361–386 | DOI | MR | Zbl | DOI | MR | Zbl

[7] R. Beals, D. H. Sattinger, J. Szmigielski, Adv. Math., 154 (2000), 229–257 | DOI | MR | Zbl

[8] A. Constantin, H. P. McKean, Commun. Pure Appl. Math., 52 (1999), 949–982 | 3.0.CO;2-D class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[9] M. S. Alber, R. Camassa, Y. Fedorov, D. D. Holm, J. E. Marsden, Phys. Lett. A, 264 (1999), 171–178 ; M. S. Alber, R. Camassa, M. E. Gekhtman, “On billiard weak solutions of nonlinear PDE's and Toda flows”, SIDE III-Symmetries and Integrability of Difference Equations, CRM Proc. Lecture Notes, 25, eds. D. Levi, O. Ragnisco, AMS, Providence, RI, 2000, 1–11 | DOI | MR | Zbl | DOI | MR | Zbl

[10] R. Beals, D. H. Sattinger, J. Szmigielski, Adv. Math., 140 (1998), 190–206 | DOI | MR | Zbl

[11] R. Beals, D. H. Sattinger, J. Szmigielski, Commun. Pure Appl. Math., 54 (2001), 91–106 ; O. Ragnisco, M. Bruschi, Physica A, 228 (1996), 150–159 | 3.0.CO;2-O class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl | DOI | MR

[12] E. T. Uitteker, Dzh. N. Vatson, Kurs sovremennogo analiza, V 2-kh chastyakh, Fizmatgiz, M., 1962