Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians

Inverse Problems

Volumn 17, Issue 4, 2001, Pages 1017-1042

  Alber, 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: 0035420587     PISSN: 02665611     EISSN: None     Source Type: Journal    
DOI: 10.1088/0266-5611/17/4/329     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 Segur H 1981 Solitons and the Inverse Scattering Transform (Philadelphia: SIAM)
3. Adler M 1979 On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations Invent. Math. 50 219-48
4. Alber M S and Alber S J 1985 Hamiltonian formalism for finite-zone solutions of integrable equations C. R. Acad. Sci., Paris 1 301 777-81
5. Alber M S, Camassa R, Fedorov F, Holm D D and Marsden J E 1999 On billiard solutions of nonlinear PDE's Phys. Lett. A 264 171-8
6. _ 2000a The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type submitted
7. Alber M S, Camassa R, Holm D D and Marsden J E 1994 The geometry of peaked solitons and piecewise solutions of a class of integrable pde's Lett. Math. Phys. 32 137-51
8. _ 1995 On the link between umbilic geodesics and soliton solutions of nonlinear PDE's Proc. R. Soc. 450 677-92
9. _ 1997 Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces Nonlinearity 10 223-41
10. Alber M S and Fedorov Yu 2000 Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians J. Phys. A: Math. Gen. 33 8409-25
11. Alber M S, Luther G G and Miller C 2000b On soliton-type solutions of the equations, associated with N-component systems J. Math. Phys. 41 284-316
12. Alber M S and Miller C 2001 On peakon solutions of the shallow water equation Appl. Math. Lett. 14 93-8
13. Alber S J 1979 Investigation of equations of Korteweg-de Vries type by the method of recurrence relations J. Lond. Math. Soc. 2 19 467-80 (in Russian)
14. Antonowicz M and Fordy A P 1989 Factorization of energy dependent Schrödinger operators: Miura maps and modified systems Commun. Math. Phys. 124 465-86
15. Antonowicz M, Fordy A P and Liu Q P 1991 Energy dependent third-order Lax operators Nonlinearity 4 669-84
16. Beals R, Sattinger D and Szmigielski J 1999 Multi-peakons and a theorem of Stietjes Inverse Problems 15 L1-4
17. _ 2000 Multipeakons and the classical moment problem Adv. Math. 154 229-57
18. Beals W, Sattinger D and Szmigielski J 1998 Acoustic scattering and the extended Kerteweg-de Vries hierarchy Adv. Math. 140 190-206
19. Belokolos E D, Bobenko A I, Enol'sii V Z, Its A R and Matveev V B 1994 Algebro-Geometric Approach to Nonlinear Integrable Equations (Springer-Verlag Series in Nonlinear Dynamics)
20. Calogero F 1995 An integrable Hamiltonian system Phys. Lett. A 201 306-10
21. Calogero F and Degasperis A 1982 Spectral transform and solitons Tools to solve and investigate nonlinear evolution equations Studies in Mathematics and its Applications 13 (Lecture Notes in Computer Science vol 144) vol 1 (Amsterdam: North-Holland)
22. Calogero F and Francoise J-P 1996 A completely integrable Hamiltonian system J. Math. Phys. 37 2863-71
23. Camassa R 2000 Characteristic variables for a completely integrable shallow water equation Nonlinearity, Integrability and All That: Twenty Years After NEEDS '79 ed M Boiti et al (Singapore: World Scientific)
24. Camassa R and Holm D D 1993 An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 1661-4
25. Camassa R, Holm D D and Hyman J M 1994 A new integrable shallow water equation Adv. Appl. Mech. 31 1-33
26. Cewen C 1990 Stationary Harry-Dym's equation and its relation with geodesics on ellipsoid Acta Math. Sin. 6 35-41
27. Clebsch A and Gordan P 1866 Theorie der Abelschen Funktionen (Leipzig: Teubner)
28. Comessatti A 1924 Sulle varietá Abeliane reale I Ann. Math. Pure Appl. 2 67-106
29. Dmitrieva L A 1993 Finite-gap solutions of the Harry-Dym equation Phys. Lett. A 182 65-70
30. Dubrovin B A 1981 Theta-functions and nonlinear equations Russ. Math. Surv. 36 11-92
31. Dubrovin B A, Novikov S P and Krichiver I M 1985 Integrable systems I Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fund. Naprav. 4 (Moscow: Viniti) (Engl. Transl. 1989 Encyclopaedia of Mathematical Sciences vol 4 (Berlin: Springer))
32. Dubrovin B A, Novikov S P and Krichiver I M 1985 Integrable systems I Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fund. Naprav. 4 (Moscow: Viniti) (Engl. Transl. 1989 Encyclopaedia of Mathematical Sciences vol 4 (Berlin: Springer))
33. Ercolani N 1989 Generalized theta functions and homoclinic varieties Proc. Symp. Pure Appl. Math. 49 87-100
34. Fedorov Yu 1999 Classical integrable systems and billiards related to generalized Jacobians Acta Appl. Math. 55 151-201
35. Gagnon L, Harnad J, Hurtubise J and Winternitz P 1985 Abelian integrals and the reduction method for an integrable Hamiltonian system J. Math. Phys. 26 1605-12
36. Gavrilov L 1999 Generalized Jacobians of spectral curves and completely integrable systems Math. Z. 230 487-508
37. Gunning R 1972 Lectures on Riemann Surfaces. Jacobi Varieties (Princeton, NJ: Princeton University Press)
38. Hunter J K and Zheng Y X 1994 On a completely integrable nonlinear hyperbolic variational equation Physica D 79 361-86
39. Its A R and Kotrlyarov V P 1976 Explicit formulas for solutions of a nonlinear Schrödinger equation Dokl. Akad. Nauk Ukr. SSR A 11 965-8 (in Russian)
40. Kozlov V and Treschev D 1992 Billiards. A Generic Introduction to Systems with Impacts (Am. Math. Soc. Transl. Ser. 2 160) (Providence, RI: AMS)
41. 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
42. Markushevich A I 1977 Theory of Functions of a Complex Variable 2nd Engl. edn, Vol I, II, III, transl. and ed R A Silverman (New York: Chelsea)
43. McKean H P 1979 Integrable systems and algebraic curves Global Analysis (Springer Lecture Notes in Mathematics vol 775) pp 83-200
44. McKean H P and Constantin A 1999 A shallow water equation on the circle Commun. Pure Appl. Math. LII 949-82
45. Mumford D 1984 Tata Lectures on Theta II. Jacobian Theta Functions and Differential Equations (Prog. Math. 43) (Boston. MA: Birkhauser)
46. Novikov D P 1999 Algebraic-geometric solutions of the Harry-Dym equation Siber. Math. J. 40 136-40
47. Novikov S P, Manakov S V, Pitaevskiǐ L P and Zakharov V E 1984 Theory of Solitons. The Inverse Scattering Method (New York: Consultants Bureau) (translated from the Russian)
48. Previato E 1985 Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation Duke Math. J. 52 329-77
49. Serre J-P 1959 Groupes Algèbriques et Corps de Classes (Paris: Hermann)
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 1 257-66