Exponentially Localized Solitons in 2 + 1 Dimensions

  • M. Boiti
  • J. Leon
  • L. Martina
  • F. Pempinelli
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)

Abstract

A new relevant result has been obtained recently [1,2,3]: the existence of two-dimensional solitons, exponentially decaying in all directions. Previously, the only known localized objects in 2+1 dimensions were the so-called lump-solutions, which decay algebraically at infinity and have no scattering properties. The use of the Bäcklund transformations (BT) and the non linear superposition formulae led BLMP [1,2,3,4] to discover the existence of exponentially localized solutions with the usual properties of scattering (they do not change velocity and form, but only have a position shift after interaction) for all the equations related to the hyperbolic version of the 2×2 Zakharov-Shabat spectral problem in the plane
$$\begin{array}{*{20}{c}} {({{\partial }_{x}} + {{\sigma }_{3}}{{\partial }_{y}} + Q)\psi = 0} & , & {Q = \left( {\begin{array}{*{20}{c}} 0 & {q(x,y)} \\ {r(x,y)} & 0 \\ \end{array} } \right).} \\ \end{array}$$
(1)

Keywords

Soliton 

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References

  1. [1]
    M. Boiti, J.J.-P. Léon, L. Martina and F. Pempinelli, Phys. Lett. A 132, 432 (1988).MathSciNetADSCrossRefGoogle Scholar
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    M. Boiti, J.J.-P. Léon, L. Martina and F. Pempinelli, “Localized solitons in the plane”, to appear in “Nonlinear evolution equations, integrability and spectral methods” Eds. A. Degasperis and A. Fordy, Manchester University Press (1989).Google Scholar
  3. [3]
    M. Boiti, J.J.-P. Léon and F. Pempinelli, “Multidimensional solitons and their spectral transforms”, preprint PM/88–44, submitted to Journal of Math. Phys.Google Scholar
  4. [4]
    M. Boiti, J.J.-P. Léon, L. Martina and F. Pempinelli, Journal of Physics A 21, 3611 (1988).MATHCrossRefGoogle Scholar
  5. [5]
    M. Boiti, B.G. Konopelchenko and F. Pempinelli, Inverse Problems 1, 33 (1985).MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • M. Boiti
    • 1
  • J. Leon
    • 2
  • L. Martina
    • 1
  • F. Pempinelli
    • 1
  1. 1.Dipartimento di Fisica dell’UniversitàSezione INFN di LecceLecceItaly
  2. 2.Laboratoire de Physique MathématiqueUSTLMontpellierFrance

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