Analytic theory of narrow lattice solitons

TL;DR

This paper analytically investigates the stability of narrow lattice solitons, revealing how their stability depends on their position relative to the lattice and the problem's nonlinearity and dimension.

Contribution

It provides an analytical perturbation-based framework for understanding the stability of narrow lattice solitons, including the effects of lattice position and nonlinearity.

Findings

Solitons at lattice maxima are unstable and drift toward minima.

Stability at lattice minima depends on dimension and nonlinearity.

Lattice can stabilize or not affect soliton stability depending on the case.

Abstract

The profiles of narrow lattice solitons are calculated analytically using perturbation analysis. A stability analysis shows that solitons centered at a lattice (potential) maximum are unstable, as they drift toward the nearest lattice minimum. This instability can, however, be so weak that the soliton is ``mathematically unstable'' but ``physically stable''. Stability of solitons centered 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''.