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A new family of four-dimensional symplectic and integrable mappings

Author

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  • Capel, H.W.
  • Sahadevan, R.

Abstract

We investigate the generalisations of the Quispel, Roberts and Thompson (QRT) family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables. We assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with Poisson brackets of the Weyl type. All mappings satisfying these conditions are shown to be integrable either as four-dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double-discrete versions of the modified Korteweg–deVries (ΔΔMKdV) and sine-Gordon (ΔΔsG) equations or by reduction to two-dimensional mappings with one integral of the symmetric QRT family.

Suggested Citation

  • Capel, H.W. & Sahadevan, R., 2001. "A new family of four-dimensional symplectic and integrable mappings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 289(1), pages 86-106.
    • Handle: RePEc:eee:phsmap:v:289:y:2001:i:1:p:86-106
    DOI: 10.1016/S0378-4371(00)00314-9
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    References listed on IDEAS

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    1. Quispel, G.R.W. & Nijhoff, F.W. & Capel, H.W. & Van Der Linden, J., 1984. "Linear integral equations and nonlinear difference-difference equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 125(2), pages 344-380.
    2. Quispel, G.R.W. & Capel, H.W. & Papageorgiou, V.G. & Nijhoff, F.W., 1991. "Integrable mappings derived from soliton equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 173(1), pages 243-266.
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    Cited by:

    1. Sahadevan, R. & Capel, H.W., 2003. "Complete integrability and singularity confinement of nonautonomous modified Korteweg–de Vries and sine Gordon mappings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 330(3), pages 373-390.
    2. Iatrou, Apostolos & Roberts, John A.G., 2003. "Integrable mappings of the plane preserving biquadratic invariant curves III," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 326(3), pages 400-411.

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