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Moving solitons in the discrete nonlinear Schrödinger equation

O. F. Oxtoby* and I. V. Barashenkov†

  • Department of Maths and Applied Maths, University of Cape Town, Rondebosch 7701, South Africa

  • *Oliver.Oxtoby@uct.ac.za
  • †Igor.Barashenkov@uct.ac.za; igor@odette.mth.uct.ac.za
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Phys. Rev. E 76, 036603 – Published 13 September, 2007

DOI: https://doi.org/10.1103/PhysRevE.76.036603

Abstract

Using the method of asymptotics beyond all orders, we evaluate the amplitude of radiation from a moving small-amplitude soliton in the discrete nonlinear Schrödinger equation. When the nonlinearity is of the cubic type, this amplitude is shown to be nonzero for all velocities and therefore small-amplitude solitons moving without emitting radiation do not exist. In the case of a saturable nonlinearity, on the other hand, the radiation is found to be completely suppressed when the soliton moves at one of certain isolated “sliding velocities.” We show that a discrete soliton moving at a general speed will experience radiative deceleration until it either stops and remains pinned to the lattice or—in the saturable case—locks, metastably, onto one of the sliding velocities. When the soliton’s amplitude is small, however, this deceleration is extremely slow; hence, despite losing energy to radiation, the discrete soliton may spend an exponentially long time traveling with virtually unchanged amplitude and speed.

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Phys. Rev. E 76, 036603– Published 13 September, 2007
  • Received 1 November 2006

© 2007 American Physical Society

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