Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 015, 34 pages      arXiv:2210.12771      https://doi.org/10.3842/SIGMA.2023.015

Stationary Flows Revisited

Allan P. Fordy a and Qing Huang b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) School of Mathematics, Center for Nonlinear Studies, Northwest University, Xi'an 710069, P.R. China

Received October 28, 2022, in final form March 08, 2023; Published online March 29, 2023

Abstract
In this paper we revisit the subject of stationary flows of Lax hierarchies of a coupled KdV class. We explain the main ideas in the standard KdV case and then consider the dispersive water waves (DWW) case, with respectively 2 and 3 Hamiltonian representations. Each Hamiltonian representation gives us a different form of stationary flow. Comparing these, we construct Poisson maps, which, being non-canonical, give rise to bi-Hamiltonian representations of the stationary flows. An alternative approach is to use the Miura maps, which we do in the case of the DWW hierarchy, which has two ''modifications''. This structure gives us 3 sequences of Poisson related stationary flows. We use the Poisson maps to build a tri-Hamiltonian representation of each of the three stationary hierarchies. One of the Hamiltonian representations allows a multi-component squared eigenfunction expansion, which gives $N$ degrees of freedom Hamiltonians, with first integrals. A Lax representation for each of the stationary flows is derived from the coupled KdV matrices. In the case of 3 degrees of freedom, we give a generalisation of our Lax matrices and Hamiltonian functions, which allows a connection with the rational Calogero-Moser (CM) system. This gives a coupling of the CM system with other potentials, along with a Lax representation. We present the particular case of coupling one of the integrable Hénon-Heiles systems to CM.

Key words: KdV hierarchy; stationary flows; bi-Hamiltonian; complete integrability; Hénon-Heiles; Calogero-Moser.

pdf (551 kb)   tex (34 kb)  

References

  1. Antonowicz M., Fordy A.P., Coupled KdV equations with multi-Hamiltonian structures, Phys. D 28 (1987), 345-357.
  2. Antonowicz M., Fordy A.P., Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys. 124 (1989), 465-486.
  3. Antonowicz M., Fordy A.P., Wojciechowski S., Integrable stationary flows: Miura maps and bi-Hamiltonian structures, Phys. Lett. A 124 (1987), 143-150.
  4. Baker S., Enolskii V.Z., Fordy A.P., Integrable quartic potentials and coupled KdV equations, Phys. Lett. A 201 (1995), 167-174, arXiv:hep-th/9504087.
  5. Ballesteros A., Enciso A., Herranz F.J., Ragnisco O., Superintegrability on $N$-dimensional curved spaces: central potentials, centrifugal terms and monopoles, Ann. Physics 324 (2009), 1219-1233, arXiv:0812.1882.
  6. Bogojavlenskii O.I., Novikov S.P., The connection between the Hamiltonian formalisms of stationary and nonstationary problems, Funct. Anal. Appl. 10 (1976), 8-11.
  7. Calogero F., Solution of the one-dimensional $N$-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.
  8. Cariglia M., Hidden symmetries of dynamics in classical and quantum physics, Rev. Modern Phys. 86 (2014), 1283-1333, arXiv:math-ph/1411.1262.
  9. Chanu C., Degiovanni L., Rastelli G., Superintegrable three-body systems on the line, J. Math. Phys. 49 (2008), 112901, 10 pages, arXiv:0802.1353.
  10. Dorizzi B., Grammaticos B., Hietarinta J., Ramani A., Schwarz F., New integrable three-dimensional quartic potentials, Phys. Lett. A 116 (1986), 432-436.
  11. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666-5676.
  12. Fordy A.P., The Hénon-Heiles system revisited, Phys. D 52 (1991), 204-210.
  13. Fordy A.P., Huang Q., Integrable and superintegrable extensions of the rational Calogero-Moser model in three dimensions, J. Phys. A 55 (2022), 225203, 36 pages, arXiv:2111.15659.
  14. Hietarinta J., A search for integrable two-dimensional Hamiltonian systems with polynomial potential, Phys. Lett. A 96 (1983), 273-278.
  15. Horwood J.T., Higher order first integrals in classical mechanics, J. Math. Phys. 48 (2007), 102902, 18 pages.
  16. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), 423001, 97 pages, arXiv:math-ph/1309.2694.
  17. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
  18. Perelomov A.M., Integrable systems of classical mechanics and Lie algebras, Vol. I, Birkhäuser, Basel, 1990.
  19. Ramani A., Dorizzi B., Grammaticos B., Painlevé conjecture revisited, Phys. Rev. Lett. 49 (1982), 1539-1541.
  20. Reyman A.G., Semenov-Tian-Shansky M.A., Compatible Poisson structures for Lax equations: an $r$-matrix approach, Phys. Lett. A 130 (1988), 456-460.
  21. Wojciechowski S., Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983), 279-281.
  22. Wojciechowski S., Integrability of one particle in a perturbed central quartic potential, Phys. Scripta 31 (1985), 433-438.

Previous article  Next article  Contents of Volume 19 (2023)