This article is cited in 97 papers

Positons: Slowly Decreasing Analogues of Solitons

V. B. Matveev<sup>abc

a</sup> St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
^b Max Planck Institute for Mathematics
^c Université de Bourgogne

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).

DOI: 10.4213/tmf1946

 English version: Theoretical and Mathematical Physics, 2002, 131:1, 483–497

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