Snakes and ghosts in a parity-time-symmetric chain of dimers

Abstract

We consider linearly coupled discrete nonlinear Schrödinger equations with gain and loss terms and with a cubic-quintic nonlinearity. The system models a parity-time ($\mathcal{P}\mathcal{T}$)-symmetric coupler composed by a chain of dimers. Particularly we study site-centered and bond-centered spatially-localized solutions and present that each solution has a symmetric and antisymmetric configuration between the arms. When a parameter is varied, the resulting bifurcation diagrams for the existence of standing localized solutions have a snaking behaviour. The critical gain/loss coefficient above which the $\mathcal{P}\mathcal{T}-$symmetry is broken corresponds to the condition when bifurcation diagrams of symmetric and antisymmetric states merge. Past the symmetry breaking, the system no longer has time-independent states. Nevertheless, equilibrium solutions can be analytically continued by defining a dual equation that leads to so-called ghost states associated with growth or decay, that are also identified and examined here. We show that ghost localized states also exhibit snaking bifurcation diagrams. We analyse the width of the snaking region and provide asymptotic approximations in the limit of strong and weak coupling where good agreement is obtained.

arXiv commentTo appear in PRE

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Cited by in corpus (8)

• Spontaneous symmetry breaking and ghost states supported by the fractional nonlinear Schrödinger equation with focusing saturable nonlinearity and PT-symmetric potential
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• Isolas of multi-pulse solutions to lattice dynamical systems
• Snakes in square, honeycomb, and triangular lattices
• Snaking bifurcations of localized patterns on ring lattices
• Analysis of multistability in discrete quantum droplets and bubbles
• Higher-dimensional soliton generation, stability and excitations of the PT-symmetric nonlinear Schrödinger equations
• From Snaking to Isolas: A One-Active-Site Approximation in Discrete Optical Cavities