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Positons: Slowly Decreasing Analogues of Solitons

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Abstract

We present an introduction to positon theory, almost never covered in the Russian scientific literature. Positons are long-range analogues of solitons and are slowly decreasing, oscillating solutions of nonlinear integrable equations of the KdV type. Positon and soliton–positon solutions of the KdV equation, first constructed and analyzed about a decade ago, were then constructed for several other models: for the mKdV equation, the Toda chain, the NS equation, as well as the sinh-Gordon equation and its lattice analogue. Under a proper choice of the scattering data, the one-positon and multipositon potentials have a remarkable property: the corresponding reflection coefficient is zero, but the transmission coefficient is unity (as is known, the latter does not hold for the standard short-range reflectionless potentials).

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Authors and Affiliations

  1. St. Petersburg Branch, Steklov Mathematical Institute, RAS, St. Petersburg, Russia

    V. B. Matveev

  2. Laboratoire Gevrey Mathématique Physique, Université de Bourgogne, Dijon Cedex, France

    V. B. Matveev

  3. Max-Planck-Institut für Mathematik, Bonn, Germany

    V. B. Matveev

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Matveev, V.B. Positons: Slowly Decreasing Analogues of Solitons. Theoretical and Mathematical Physics 131, 483–497 (2002). https://doi.org/10.1023/A:1015149618529

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