Algebraic Structure of Integrable Hierarchies

In [1-4] we used factorisation methods to derive (new) Miura maps between various hierarchies of isospectral flows. These maps usually have the Poisson property, so give natural co-ordinates in which some complicated Hamiltonian structures take simple form. We found a remarkable 2D extension of the Toda lattice (some nonlinear Klein-Gordon equations), giving its zero-curvature representation and its relationship with `modified Lax' equations, together with a Bäcklund transformation. This work naturally led to a generalisation of the 2D Toda lattice, associated with simple Lie algebras [5] and a classification of Nonlinear Schrödinger equations associated with symmetric spaces [7], with similar results for DNLS equations [8] and KdV like equations in [10]. This work was summarised in [12].
  1. A.P. Fordy and J. Gibbons, Some remarkable nonlinear transformations. Phys.Letts. A , 75, 325, 1980.
  2. A.P. Fordy and J. Gibbons, Factorization of operators I : Miura transformations. J.Math.Phys., 21 , 2508-10, 1980.
  3. A.P. Fordy and J. Gibbons. Integrable nonlinear Klein-Gordon equations and Toda lattices. Commun.Math.Phys., 77, 21-30, 1980.
  4. A.P. Fordy and J. Gibbons. Factorization of operators II. J.Math.Phys., 22, 1170-75, 1981.
  5. A.P. Fordy and J. Gibbons. Integrable nonlinear Klein-Gordon equations and simple Lie algebras. Proc. R. Ir. Acad., 83A, 33-45, 1983.
  6. A.P. Fordy, Projective representations and deformations of integrable systems. Proc. R. Ir. Acad., 83A, 75-93, 1983.
  7. A.P. Fordy and P.P. Kulish. Nonlinear Schrödinger equations and simple Lie algebras. Commun.Math.Phys., 89, 427-443, 1983.
  8. A.P. Fordy. Generalised derivative nonlinear Schrödinger equations and Hermitian symmetric spaces. J.Phys., A17, 1235-45, 1984.
  9. 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.
  10. C. Athorne and A.P. Fordy. Generalised KdV and MKdV equations associated with symmetric spaces. J.Phys., A20, 1377-86, 1987.
  11. C. Athorne and A.P. Fordy. Integrable equations in (2+1)-dimensions associated with Hermitian symmetric and homogeneous spaces. J.Math.Phys., 28, 2018-2024, 1987.
  12. A.P. Fordy, Equations associated with simple Lie algebras and symmetric spaces. In A.P. Fordy, editor, Soliton Theory : A Survey of Results , pages 315-337. MUP, Manchester, 1990.
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