Moving solitons in the discrete nonlinear Schrödinger equation

• Published in: Physical review. E, Statistical, nonlinear, and soft matter physics Statistical, nonlinear, and soft matter physics, 2007-09, Vol.76 (3 Pt 2), p.036603-036603, Article 036603
• Main Authors: Oxtoby, O F, Barashenkov, I V
• Format: Article
• Language: English
• ISSN: 1539-3755; 1550-2376
• DOI: 10.1103/physreve.76.036603

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.