Hamiltonian Theory

An important idea introduced within soliton theory was that of a bi-Hamiltonian system. F. Magri and his collaborators developed a geometric theory of integrability around the notion of compatible Poisson brackets.

In a series of papers [2,4-8] we constructed a large class of coupled KdV (and related) equations, which are isospectral to an energy-dependent Schrödinger operator and which possess N+1 compatible, local Hamiltonian structures (in the N-component case). A large number of previously known, but disparate, examples were found to be special cases of these systems. We also found a chain of N Miura maps and modified systems, which have a beautiful algebraic structure. This work was summarised in [9]. These results were further extended to general energy-dependent zero-curvature equations [10], with an application to nonlinear optics [12], and to energy-dependent third order operators (generalised Boussinesq equations) in [13]. The latter was part of the thesis of Q.P.Liu [14]. Multi-component Schwarzian KdV hierarchies were presented in [15].

Another theme is the relationship between the Poisson brackets for a nonlinear evolution equation (PDE) and those for its stationary flow (an ODE, which often takes finite dimensional Hamiltonian form). In [1] we studied the stationary flows of the NLS equations. In [3] we used the Miura map (when restricted to the stationary manifold) to construct second (non-canonical) Poisson brackets for stationary members of the KdV and MKdV hierarchies. A different approach, with applications to many more hierarchies was developed in the thesis of Simon Harris and is presented in [17,21,22]. By writing the PDE in x - t reversed form and considering the zero curvature representation, we give a systematic construction of the Hamiltonian structures, giving a clear relationship between the Poisson brackets for the PDE and those of its stationary flows. Another approach to the same problem was presented in [11,17,18], with many more results in Sarah Baker's thesis [19].

In recent years I became interested in integrable "systems of hydrodynamic type". In [20] we gave a connection between separable systems of Stäckel type and a class of systems of "hydrodynamic type" and used this to derive the general solution of the latter. In [23,24] we generalised these results to a class of non-homogeneous hydrodynamic systems which can be associated with a pair of quadratic Hamiltonians with terms which are linear in momenta. These systems possess some intriguing properties, which are close to (but do not imply) complete integrability, although some particular cases are integrable. The classification of such systems is rather difficult, but partial results are given in the thesis of Kav Aujla [25]. The question of recursion operators and higher order symmetries for systems of hydrodynamic type is considered in [26]. The Poisson brackets of systems of hydrodynamic type have interesting geometric properties. The problem of compatible Poisson brackets of hydrodynamic type is considered in [27].

  1. A.P. Fordy, S. Wojciechowski, and I. Marshall. A family of integrable quartic potentials related to symmetric spaces. Phys.Letts. A, 113 , 395-4OO, 1986.
  2. M. Antonowicz and A.P. Fordy, A family of completely integrable multi-Hamiltonian systems. Phys.Letts.A, 122, 95-99, 1987.
  3. M. Antonowicz, A.P. Fordy, and S. Wojciechowski, Integrable stationary flows : Miura maps and bi-Hamiltonian structures. Phys.Letts.A , 124 , 143-50, 1987.
  4. M. Antonowicz and A.P. Fordy, Coupled KdV equations with multi-Hamiltonian structures. Physica , 28D, 345-57, 1987.
  5. M. Antonowicz and A.P. Fordy, Coupled Harry Dym equations with multi-Hamiltonian structures. J.Phys., A21, L269-75, 1988.
  6. A.P. Fordy, A.G. Reyman, and M. Semenov-Tian-Shansky, Classical r-matrices and compatible Poisson brackets for coupled KdV systems. Lett.Math.Phys., 17, 25-9, 1989.
  7. M. Antonowicz and A.P. Fordy, Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems. Commun.Math.Phys., 124, 465-86, 1989.
  8. M. Antonowicz and A.P. Fordy, Super-extensions of energy dependent Schrödinger operators. Commun.Math.Phys., 124, 487-500, 1989.
  9. M. Antonowicz and A.P. Fordy. Hamiltonian structures of nonlinear evolution equations. In A.P. Fordy, editor, Soliton Theory : A Survey of Results, pages 273-312. MUP, Manchester, 1990.
  10. A.P. Fordy, Isospectral flows : Their Hamiltonian structures, Miura maps and master symmetries. In P.J. Olver and D.L. Sattinger, editors, Solitons in Physics, Mathematics, and Nonlinear Optics, pages 97-121. Springer, NY, 1990.
  11. A.P. Fordy, The Hénon-Heiles system revisited. Physica, 52D, 201-210, 1991.
  12. A.P. Fordy and D.D. Holm, A tri-Hamiltonian formulation of the self-induced transparency equations. Phys.Letts.A, 160, 143-148, 1991.
  13. M. Antonowicz, A.P. Fordy, and Q.P. Liu, Energy-dependent third-order Lax operators. Nonlinearity, 4, 669-84, 1991.
  14. Q.P. Liu, Hamiltonian structures for integrable nonlinear evolution equations, Ph.D. Thesis, University of Leeds, 1991.
  15. M. Antonowicz and A.P. Fordy, Multi-component Schwarzian KdV hierarchies. Rep.Math.Phys., 32, 223-233, 1993.
  16. S.D. Harris, Integrable nonlinear evolution equations and their stationary flows, Ph.D. Thesis, University of Leeds, 1994.
  17. A.P. Fordy, Stationary flows : Hamiltonian structures and canonical transformations. Physica, 87D, 20-31, 1995.
  18. S. Baker, V.Z. Enolskii, and A.P. Fordy, Integrable quartic potentials and coupled KdV equations. Phys.Letts.A, 201, 167-74, 1995.
  19. S. Baker, Squared eigenfunction representations of integrable hierarchies, Ph.D. Thesis, University of Leeds, 1995.
  20. E.V. Ferapontov and A.P. Fordy, Separable Hamiltonians and integrable systems of hydrodynamic type. J. Geom. and Phys., 21, 169-82, 1997.
  21. A.P. Fordy and S.D. Harris, Hamiltonian flows on stationary manifolds, In A.S. Fokas and I.M. Gelfand, editors, Methods and Applications of Analysis, 4, 212-38, 1997.
  22. A.P. Fordy and S.D. Harris, Hamiltonian structures in stationary manifold co-ordinates, In A.S. Fokas and I.M. Gelfand, editors, Papers dedicated to the memory of Irene Dorfman, pages . Birkhauser, 1997.
  23. E.V. Ferapontov and A.P. Fordy, Nonhomogeneous systems of hydrodynamic type, related to quadratic Hamiltonians with electromagnetic term. Physica D , 108, 350-64, 1997.
  24. E.V. Ferapontov and A.P. Fordy, Commuting quadratic {Hamiltonians} with velocity dependent potentials. Rep.Math.Phys. , 44, 71-80, 1999.
  25. K.S. Aujla, Hamiltonian systems and related equations of hydrodynamic type, Ph.D. Thesis, University of Leeds, 1999.
  26. A.P. Fordy and T.B. G{\" u}rel, A new construction of recursion operators for systems of hydrodynamic type. Theor.Math.Phys. , 122, 29-38, 2000.
  27. A.P. Fordy and O.I. Mokhov, On a special class of compatible {Poisson} structures of hydrodynamic type. Physica D , 152-3, 475-490, 2001.
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