Moving embedded lattice solitons

Abstract

It was recently proved that solitons embedded in the spectrum of linear waves may exist in discrete systems, and explicit solutions for isolated unstable embedded lattice solitons (ELS) of a differentialdifference version of a higher-order NLS equation were found [Physica D 197 (2004) 86]. The discovery of these ELS gives rise to relevant questions such as the following: are there continuous families of ELS?, can ELS be stable?, is it possible for ELS to move along the lattice?, how do ELS interact?. The present work addresses these questions by showing that a novel equation (a discrete version of a complex modified KdV equation which includes next-nearest-neighbor couplings) has a two-parameter continuous family of exact ELS. These solitons can move with arbitrary velocities across the lattice, and the numerical simulations demonstrate that these ELS are completely stable. Moreover, the numerical tests show that these ELS are robust enough to withstand collisions, and the result of a collision is only a shift in the positions of the solitons. The model may apply to the description of a Bose-Einstein condensate with dipole-dipole interactions between the atoms, trapped in a deep optical-lattice potential. In nonlinear systems where solitons can exist, the propagation of small-amplitude linear waves, which obey the linearized version of the nonlinear equations, is possible too. However, for a soliton to exist, it is absolutely necessary that no resonances occur between the soliton and these linear waves. Otherwise, the soliton would decay due to an energy transfer towards the linear waves. Based on this no-resonance argument, it was frequently assumed that the solitons' internal frequencies could not be contained in the linear spectrum of the system, i.e., they could not lie within the band of frequencies permitted to linear waves. However, at the end of the nineties exceptions to this rule were found, and a special type of solitons were discovered, which do not resonate with linear waves, in spite of having frequencies immersed in the spectrum of these waves. In 1999 these peculiar solitary waves were given the name of embedded solitons (ES), and in the following years a number of models supporting ES were identified. Most of these models describe continuous systems. However, some examples of discrete ES were recently found too. These embedded lattice solitons (ELS) are isolated solutions which are stable against small perturbations in the linear approximation, but are nonlinearly unstable. The discovery of these isolated unstable ELS triggered the search for models admitting continuous families of stable ELS. In this article we present a novel differential-difference equation which has a two-parameter continuous family of exact ELS. These solitons can move with arbitrary velocities across the lattice, and the numerical simulations show that they are completely stable solutions.

Supertransmission channel for an intrinsic localized mode in a one-dimensional nonlinear physical lattice

“…27 More recently localized excitations have been shown to exist in some discrete nonlinear models where they are referred to as embedded lattice solitons (ELS). 17,28 The general result is that for either an ES or an ELS to exist there can be no resonance between it and the small amplitude linear waves that are also present.…”

Embedded soliton dynamics in the asymmetric array of Josephson junctions

“…If the inductive coupling between the junctions is small, the free fluxon motion becomes impossible. However, the possibility of radiationless propagation of solitary waves in discrete media for selected values of velocity has been reported in the number of papers [10][11][12][13][14][15][16][17][18][19][20]. Also, free propagation of the bound states of several discrete topological solitons has been reported by Peyrard and Kruskal more than 30 years ago [8].…”

Radiationless Higher-Order Embedded Solitons

“…Then, in 2004, it was found that ESs also exist in discrete systems, 18) thus implying that a new category of embedded lattice solitons (ELSs) should be added to the list of different solitons already known. These new ELSs were unstable solutions, but in 2006 it was found that stable ELSs also exist, 19) thus showing that there is a parallelism between the continuous ESs and discrete ELSs.…”