On transverse stability of discrete line solitons

TL;DR

This paper establishes precise criteria for the transverse stability and instability of line solitons in discrete nonlinear Schrödinger equations on 1D and 2D lattices, supported by analytical and numerical evidence.

Contribution

It provides the first sharp stability criteria for line solitons near the anti-continuum limit in discrete lattices, including explicit eigenvalue asymptotics.

Findings

Fundamental line solitons are transversely stable on 2D lattices when bifurcating from the X point.

Fundamental line solitons are transversely unstable on 1D lattices for both signs of transverse dispersion.

Analytical eigenvalue asymptotics match numerical results perfectly.

Abstract

We obtain sharp criteria for transverse stability and instability of line solitons in the discrete nonlinear Schr\"{o}dinger equations on one- and two-dimensional lattices near the anti-continuum limit. On a two-dimensional lattice, the fundamental line soliton is proved to be transversely stable (unstable) when it bifurcates from the $X$ ($\Gamma$) point of the dispersion surface. On a one-dimensional (stripe) lattice, the fundamental line soliton is proved to be transversely unstable for both signs of transverse dispersion. If this transverse dispersion has the opposite sign to the discrete dispersion, the instability is caused by a resonance between isolated eigenvalues of negative energy and the continuous spectrum of positive energy. These results hold for both focusing and defocusing nonlinearities via a staggering transformation. When the line soliton is transversely unstable, asymptotic expressions for unstable eigenvalues are also derived. These analytical results are compared with numerical results, and perfect agreement is obtained.