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Bäklund transformations and singular integral equations

Author

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  • Quispel, G.R.W.
  • Nijhoff, F.W.
  • Capel, H.W.
  • der Linden, J.van

Abstract

A systematic method for deriving Bäcklund transformations for singular (linear) integral equations is presented. The method leads in a natural way to the Bäcklund transformations for the corresponding (integrable) nonlinear partial differential equations. By repeated use of the method a hierarchy of singular integral equations and Bäcklund transformations can be derived, corresponding to so-called multi-modified partial differential equations. Specific examples include the Korteweg-de Vries equation and the first, second, and third modified Korteweg-de Vries equation; the Nonlinear Schrödinger equation and the Anisotropic Heisenberg Spin Chain; the sine-Gordon equation and the modified sine-Gordon equation.

Suggested Citation

  • Quispel, G.R.W. & Nijhoff, F.W. & Capel, H.W. & der Linden, J.van, 1984. "Bäklund transformations and singular integral equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 123(2), pages 319-359.
    • Handle: RePEc:eee:phsmap:v:123:y:1984:i:2:p:319-359
    DOI: 10.1016/0378-4371(84)90159-6
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    References listed on IDEAS

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    1. Nijhoff, F.W. & Van Der Linden, J. & Quispel, G.R.W. & Capel, H.W. & Velthuizen, J., 1982. "Linearization of the nonlinear Schrödinger equation and the isotropic Heisenberg spin chain," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 116(1), pages 1-33.
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    1. Nijhoff, F.W. & Quispel, G.R.W. & Van Der Linden, J. & Capel, H.W., 1983. "On some linear integral equations generating solutions of nonlinear partial differential equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(1), pages 101-142.
    2. van der Linden, J. & Capel, H.W. & Nijhoff, F.W., 1989. "Linear integral equations and multicomponent nonlinear integrable systems II," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 160(2), pages 235-273.
    3. Quispel, G.R.W. & Capel, H.W., 1983. "The Anisotropic Heisenberg spin chain and the nonlinear Schrödinger equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 117(1), pages 76-102.

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