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The Anisotropic Heisenberg spin chain and the nonlinear Schrödinger equation

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  • Quispel, G.R.W.
  • Capel, H.W.

Abstract

It is shown that the solutions of the continuous Anisotropic Heisenberg Spin Chain (AHSC) can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the Isotropic Heisenberg Spin Chain (IHSC) and the Nonlinear Schrödinger equation (NLS). An explicit expression is obtained for the Miura transformation which maps the solutions of the AHSC on solutions of the NLS. In the second part of the paper we investigate the similarity solutions of these partial differential equations which leads to ordinary differential equations of Painlevé type. As an application we discuss some new solutions of Painlevé IV.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:117:y:1983:i:1:p:76-102
    DOI: 10.1016/0378-4371(83)90022-5
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    1. Kennedy, Susan Estabrook, 1975. "The American Banking Community and New Deal Banking Reforms, 1933-1935. By Helen M. Burns. Westport, Conn., Greenwood Press, 1974. Pp. xiii + 203. $11.95," Business History Review, Cambridge University Press, vol. 49(2), pages 261-263, July.
    2. 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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