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Linear integral equations and multicomponent nonlinear integrable systems II

Author

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

Abstract

In a previous paper (I) an extension of the direct linearization method was developed for obtaining solutions of multicomponent generalizations of integrable nonlinear partial differential equations (PDE's). The method is based on a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of the spectral parameter. In I the general framework has been presented, with as immediate application the direct linearization of multicomponent versions of the nonlinear Schrödinger equation and the (complex) modified Korteweg-de Vries equation. In the present paper we treat a varìety of examples of other multicomponent PDE's, and we also discuss Miura transformations and gauge equivalences. The examples include the direct linearization of multicomponent generalizations of the isotropic Heisenberg spin chain equation, the complex sine-Gordon equation, the Getmanov equation, the derivative non-linear Schrödinger equation and the massive Thirring model equations.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:160:y:1989:i:2:p:235-273
    DOI: 10.1016/0378-4371(89)90420-2
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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.
    2. Kobayashi, Y. & Ohkita, M. & Inoue, M., 1979. "On the use of an exponential function in approximation of elliptic integrals," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 21(2), pages 226-230.
    3. Kamran Asgharzadeh & G. F. Newell, 1978. "Optimal dispatching strategies for vehicles having exponentially distributed trip times," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 25(3), pages 489-509, September.
    4. Magaeva, Sneshana Dimitrova & Karaivanov, Stefan Jordanov, 1983. "Exergy analysis of processes of natural gas conversion and mazut gasification," Energy, Elsevier, vol. 8(7), pages 561-566.
    5. van der Linden, J. & Nijhoff, F.W. & Capel, H.W. & Quispel, G.R.W., 1986. "Linear integral equations and multicomponent nonlinear integrable systems I," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 137(1), pages 44-80.
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    Cited by:

    1. Xianguo, Geng, 1992. "A hierarchy of non-linear evolution equations its Hamiltonian structure and classical integrable system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 180(1), pages 241-251.

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