On transverse stability of discrete line solitons

Abstract

We obtain sharp criteria for transverse stability and instability of line solitons in the discrete nonlinear Schrödinger 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.

15 pages, 5 figures

References in corpus (5)

• Controlling the transverse instability of dark solitons and nucleation of vortices by a potential barrier
• Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation
• X,Y,Z-Waves: Extended Structures in Nonlinear Lattices
• Observation of transverse instabilities in optically-induced lattices
• Transversely Stable Soliton Trains in Photonic Lattices

Cited by in corpus (5)

• Solitons in PT-symmetric ladders of optical waveguides
• Short-wave transverse instabilities of line solitons of the 2-D hyperbolic nonlinear Schrödinger equation
• Transverse instability of line solitons in massive Dirac equations
• Dynamics and stabilization of bright soliton stripes in the hyperbolic-dispersion nonlinear Schrödinger equation
• Exponential asymptotics for line solitons in two-dimensional periodic potentials