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Integrable mappings derived from soliton equations

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  • Quispel, G.R.W.
  • Capel, H.W.
  • Papageorgiou, V.G.
  • Nijhoff, F.W.

Abstract

We derive a hierarchy of ibtegrable mappings (integrable ordinary difference equations) corresponding to solutions of the initial-value problem of an integrable partial difference equation with periodic initial data. For each n ϵ N this hierarchy contains at least one integrable mapping Rn→Rn. The integrals of these mappings are constructed using the Lax pair of the underlying partial difference equation. Our approach is illustrated for the integrable partial difference analogues of the sine-Gordon and the (modified) Korteweg-de Vries equations.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:173:y:1991:i:1:p:243-266
    DOI: 10.1016/0378-4371(91)90258-E
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    Citations

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    Cited by:

    1. Byrnes, G.B. & Haggar, F.A. & Quispel, G.R.W., 1999. "Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 272(1), pages 99-129.
    2. 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.
    3. Komineas, Stavros & Vrahatis, Michael N. & Bountis, Tassos, 1994. "2D universality of period-doubling bifurcations in 3D conservative reversible mappings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(2), pages 218-233.
    4. 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.
    5. 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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