Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians

Journal of Physics A: Mathematical and General

Volumn 33, Issue 47, 2000, Pages 8409-8425

  Albert, Mark S ^a,b   Fedorov, Yuri N ^c  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF NOTRE DAME   (United States)

^c MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034367365     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/33/47/307     Document Type: Article

References (51)

1. Abenda S and Fedorov Yu 2000 On the weak Kowalewski-Painlevé property for hyperelliptically separable systems Acta Appl. Math. 60 137-78
2. Ablowitz M J and Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge University Press)
3. Ablowitz M J and Ma Y-C 1981 The periodic cubic Schrödinger equation Stud. Appl. Math. 65 113-58
4. Ablowitz M J and Segur H 1981 Solitons and the Inverse Scattering Transform (Philadelphia: SIAM)
5. Alber M S and Alber S J 1985 Hamiltonian formalism for finite-zone solutions of integrable equations C. R. Acad. Sci., Paris 301 777-81
6. Alber M S, Camassa R, Fedorov Yu, Holm D D and Marsden J E 1999 On billiard solutions of nonlinear PDE's Phys. Lett. A 264 171-8
7. _2000a The geometry of new classes of weak billiard solutions of nonlinear PDEs Preprint
8. Alber M S, Camassa R and Gekhtman M 2000b On billiard weak solutions of nonlinear PDE's and Toda flows CRM Proc. and Lecture Notes vol 25 pp 1-11
9. Alber M S, Camassa R, Holm D D and Marsden J E 1994 The geometry of peaked solitons and billiard solutions of a class of integrable PDE's Lett. Math. Phys. 32 137-51
10. _1995 On the link between umbilic geodesics and soliton solutions of nonlinear PDE's Proc. R. Soc. 450 677-92
11. Alber M S and Fedorov Yu 2000 Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians submitted
12. Alber M S, Luther G G and Marsden J E 1997 Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces Nonlinearity 10 223-41
13. Alber M S and Miller C 2001 On peakon solutions of the shallow water equation Appl. Math. Lett. 14 93-8
14. Alber S J 1979 Investigation of equations of Korteweg-de Vries type by the method of recurrence relations J. Lond. Math. Soc. 19 467-80 (in Russian)
15. Belokolos E D, Bobenko A I, Enol'sii V Z, Its A R and Matveev V B 1994 Algebra-Geometric Approach to Nonlinear Integrable Equations (Springer-Verlag Series in Nonlinear Dynamics) (Berlin: Springer)
16. Beals R, Sattinger D H and Szmigielski J 1999 Multi-peakons and a theorem of Stietjes Inverse Problems 15 L1-4
17. _2000 Multipeakons and the classical moment Adv. Math, at press
18. Calogero F 1995 An integrable Hamiltonian system Phys. Lett. A 201 306-10
19. Calogero F and Francoise J-P 1996 A completely integrable Hamiltonian system J. Math. Phys. 37 2863-71
20. Camassa R and Holm D D 1993 An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 1661-4
21. Camassa R, Holm D D and Hyman J M 1994 A new integrable shallow water equation Adv. Appl. Mech. 31 1-33
22. Cewen C 1990 Stationary Harry-Dym's equation and its relation with geodesics on ellipsoid Acta Math. Sin. 6 35-41
23. Clebsch A and Gordan P 1866 Theorie der Abelschen Funktionen (Leipzig: Teubner)
24. Comessatti A 1924 Sulle varietá abeliane reale I Ann. Math. Pure Appl. 2 67-106
25. Dmitrieva L A 1993a Finite-gap solutions of the Harry Dym equation Phys. Lett. A 182 65-70
26. _1993b The higher-times approach to multisoliton solutions of the Harry Dym equation J. Phys. A: Math. Gen. 26 6005-20
27. Dubrovin B 1981 Theta functions and nonlinear equations Russ. Math. Surv. 36 11-92
28. Dubrovin B A, Novikov S P and Krichever I M 1985 Integrable Systems. I. Itogi Nauki i Tekhniki. Sovr. Probt. Mat. Fund. Naprav. 4, VINITI (Moscow) (Engl. Transl. 1989 Encyclopaedia of Mathematical Sciences vol 4 (Berlin: Springer))
29. Dubrovin B A, Novikov S P and Krichever I M 1985 Integrable Systems. I. Itogi Nauki i Tekhniki. Sovr. Probt. Mat. Fund. Naprav. 4, VINITI (Moscow) (Engl. Transl. 1989 Encyclopaedia of Mathematical Sciences vol 4 (Berlin: Springer))
30. Ercolani N 1989 Generalized theta functions and homoclinic varieties Proc. Symp. Pure Appl. Math. 49 87-100
31. Fedorov Yu 1999 Classical integrable systems and billiards related to generalized Jacobians Acta Appl. Math 55 151-201
32. Gagnon L, Harnad J, Huitubise J and Winternitz P 1985 Abelian integrals and the reduction method for an integrable Hamiltonian system J. Math. Phys. 26 1605-12
33. Gavrilov L 2000 Generalized Jacobians of spectral curves and completely integrable systems Math. Z. at press
34. Gunning R 1972 Lectures on Riemann Surfaces. Jacobi Varieties (Princeton, NJ: Princeton University Press)
35. Hunter J K and Zheng Y X 1994 On a completely integrable nonlinear hyperbolic variational equation Physica D 79 361-86
36. Its A R and Kotlyarov V P 1976 Explicit formulas for solutions of a nonlinear Schrödinger equation Dokl Akad. Nauk A (Ukrain) 11 965-8 (in Russian)
37. Lax P 1975 Periodic solutions of the KdV equations Commun. Pure Appl. Math. 28 141-88
38. Li Y A and Olver P J 1998 Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system Discrete Continuous Dyn. Syst. 4 159-91
39. Markushevich A I 1977 Theory of Functions of a Complex Variable (New York: Chelsea)
40. McKean H P 1979 Integrable systems and algebraic curves Global Analysis (Springer Lecture Notes in Mathematics vol 775) pp 83-200
41. McKean H P and Constantin A 1999 A shallow water equation on the circle Commun. Pure Appl. Math. 52 949-82
42. Mumford D 1983 Tata Lectures on Theta II (Basle: Birkhauser)
43. Newell A C 1985 Solitons in Mathematical Physics (Philadelphia, PA: SIAM)
44. Novikov D P 1999 Algebraic geometric solutions of the Harry Dym equations Siber. Math. J. 40 136-40
45. Novikov S P 1974 A periodic problem for the Korteweg-de Vries equation Funct. Anal. Appl. 8 54-66
46. Novikov S P, Manakov S V, Pitaevskiǐ L P and Zakharov V E 1984 Theory of Solitons. The Inverse Scattering Method (Contemporary Soviet Mathematics. Consultants Bureau) (New York: Plenum) (translated from Russian)
47. Previato E 1985 Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation Duke Math. J. 52 329-77
48. Ragnisco O and Bruschi M 1996 Peakons, r-matrix and Toda lattice Physica A 228 150-9
49. Suris Y B 1996 A discrete time peakons lattice Phys. Lett. A 217 321-9
50. Vanhaecke P 1995 Integrable systems and symmetric products of algebraic curves Math. Z, 40 143-72
51. Weierstrass K 1878 Über die geodätischen Linien auf dem dreiachsigen ellipsoid Math. Werke I 257-66