Moving solitons in the discrete nonlinear Schrödinger equation

Details Moving solitons in the discrete nonlinear Schrödinger equation Moving solitons in the discrete nonlinear Schrödinger equation

2006-10-24

arxiv.org/abs/nlin/0610059v4

Phys. Rev. E 76, 036603 (2007)

Nonlinear Sciences

Pattern Formation and Solitons

19 pages, 7 figures; spelling errors corrected

Scientific paper

Using the method of asymptotics beyond all orders, we evaluate the amplitude of radiation from a moving small-amplitude soliton in the discrete nonlinear Schr\"odinger 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 travelling with virtually unchanged amplitude and speed.

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