Analytic theory of narrow lattice solitons

Abstract

The profiles of narrow lattice solitons are calculated analytically using perturbation analysis. A stability analysis shows that solitons centred at a lattice (potential) maximum or saddle point are unstable, as they drift towards the nearest lattice minimum. This instability can, however, be so weak that the soliton is 'mathematically unstable' but 'physically stable'. Stability of solitons centred at a lattice minimum depends on the dimension of the problem and on the nonlinearity. In the subcritical and supercritical cases, the lattice does not affect the stability, leaving the solitons stable and unstable, respectively. In contrast, in the critical case (e.g. a cubic nonlinearity in two transverse dimensions), the lattice stabilizes the (previously unstable) solitons. The stability in this case can be so weak, however, that the soliton is 'mathematically stable' but 'physically unstable'.

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

“…We note that all of the above hold more generally for NLS equations with a nonlinear potential [14,15] and for narrow solitons of a linear potential [43]. The rest of the paper is organized as follows.…”

“…For example, in the case of a periodic lattice, narrow solitary waves are affected, to leading order, by the local changes of the potential near the soliton center [16,14,15,43], whereas wide solitary waves are affected by the potential average over a single period [14,15].…”

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

“…Later, in [47] (Theorem 3.1; see also Theorem 6 of [48]) it was shown that in the presence of a linear potential which is bounded below and decaying at infinity, solitons are stable if in addition, L + also has no zero eigenvalue(s). A related treatment was given to the narrow soliton (semi-classical limit) subcritical nonlinearity case in [35,44,49] and for solitons in spatially varying nonlinear potentials in [33,34].…”

“…In this formulation, the spectral condition on n − (L + ) and the slope condition are coupled, see detailed discussion in [35]. The formulation of Theorem III.1 is a more refined and stronger statement.…”

“…In addition, they presented a quantitative approach for prediction of the stability or instability strength. Specifically, these papers considered the cases of a one-dimensional nonlinear lattice [33], a twodimensional nonlinear lattice [34], a one-dimensional linear delta-function potential [36] and narrow solitons in a linear lattice [35].…”