Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 53, Issue 2, 1996, Pages 1900-1906

  Olver, Peter J ^a   Rosenau, Philip ^b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000582346     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.53.1900     Document Type: Article

References (42)

1. P. Rosenau, Phys. Rev. Lett. 72, 123 (1994).
2. P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564 (1993).
3. M. Wadati, Y. H. Ichikawa and T. Shimizu, Prog. Theor. Phys. 64, 1959 (1980).
4. P. Rosenau (unpublished).
5. P. J. Olver, in Trends and Applications of Pure Mathematics to Mechanics, edited by P. G. Ciarlet and M. Roseau, Lecture Notes in Physics Vol. 195 (Springer-Verlag, New York, 1984), pp. 273–290.
6. R. Camassa and D. D. Holm, Phys. Rev. Lett. 71, 1661 (1993).
7. I. A. Kunin, Elastic Media with Microstructure I (Springer-Verlag, New York, 1982).
8. B. Fuchssteiner and A. S. Fokas, Physica 4D, 47 (1981).
9. B. Fuchssteiner, Prog. Theor. Phys. 65, 861 (1981).
10. A. S. Fokas, Acta Appl. Math. 39, 295 (1995).
11. F. Magri, J. Math. Phys. 19, 1156 (1978).
12. P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Graduate Texts in Mathematics Vol. 107 (Springer-Verlag, New York, 1993).
13. I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations (Wiley, New York, 1993).
14. See A. S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento 28, 299 (1980).
15. B. Fuchssteiner and A.S. Fokas, Physica 4D, 47 (1981).
16. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Stud. Appl. Math. 53, 249 (1974).
17. Here D=Dx denotes the derivative with respect to x.
18. A Casimir or distinguished functional is one whose variational derivative is annihilated by the Hamiltonian operator: J( δ H/ δ u)=0. It thus forms a conservation law for any Hamiltonian system having J as the Hamiltonian operator; see [11] and [12], for details.
19. We can, of course, use other linear combinations of D and D3. However, a simple rescaling reduces them to D +- D3.
20. Equation (9) (with the plus sign) is not well posed, but higher order members of its hierarchy are.
21. A. S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento 28, 299 (1980).
22. T. B. Benjamin, J. L. Bona and J. J. Mahoney, Philos. Trans. R. Soc. London 272, 47 (1972).
23. See Eq. (3.38) in [14] for an alternative, and unstudied, integrable perturbation of the BBM equation.
24. K. O. Abdullaev, I. L. Bogolubsky and V. G. Makhankov, Phys. Lett. 56A, 427 (1976).
25. P. J. Olver, Math. Proc. Cambridge Philos. Soc. 85, 143 (1979).
26. S. V. Duzhin and T. Tsujishita, J. Phys. A 17, 3267 (1984).
27. F. Calogero, Stud. Appl. Math. 79, 189 (1984).
28. J. K. Hunter and Y. Zheng, Physica D 79, 361 (1994).
29. Calogero's method of integration of (10), while certainly satisfying, masks its integrability in the Hamiltonian sense, and its hierarchical connection with the Harry Dym equation (11).
30. P. A. Clarkson, A. S. Fokas and M. J. Ablowitz, SIAM J. Appl. Math. 49, 1188 (1989).
31. See P. J. Olver and Y. Nutku, J. Math. Phys. 29, 1610 (1988) for applications of Hamiltonian triples in gas dynamics.
32. A. V. Mikhailov, A. B. Shabat, and V. V. Sokolov, in What is Integrability?, edited by V. E. Zakharov (Springer-Verlag, New York, 1990), pp. 115–184.
33. J. L. Bona and R. Smith, Math. Proc. Cambridge Philos. Soc. 79, 167 (1976).
34. P. J. Olver, Contemp. Math. 28, 231 (1984).
35. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
36. D. J. Kaup, Prog. Theor. Phys. 54, 396 (1975).
37. B. A. Kupershmidt, Commun. Math. Phys. 99, 51 (1985).
38. For simplicity, we just take the plus sign (compacton) version in this example.
39. P. Rosenau, Phys. Fluids 31, 1317 (1988).
40. P. Rosenau and J. L. Schwarzmeier, Phys. Lett. A 115, 75 (1986).
41. M. Ito, Phys. Lett. A 91, 335 (1982).
42. Alternatively, one can rewrite (49) as a system for the real and imaginary parts of the field variable u=v+iw.