Bright Solitons in aPT-Symmetric Chain of Dimers

Abstract

We study the existence and stability of fundamental bright discrete solitons in a parity-time (PT )-symmetric coupler composed by a chain of dimers, that is modelled by linearly coupled discrete nonlinear Schrödinger equations with gain and loss terms. We use a perturbation theory for small coupling between the lattices to perform the analysis, which is then confirmed by numerical calculations. Such analysis is based on the concept of the so-called anti-continuum limit approach. We consider the fundamental onsite and intersite bright solitons. Each solution has symmetric and antisymmetric configurations between the arms. The stability of the solutions is then determined by solving the corresponding eigenvalue problem. We obtain that both symmetric and antisymmetric onsite mode can be stable for small coupling, on the contrary of the reported continuum limit where the antisymmetric solutions are always unstable. The instability is either due to the internal modes crossing the origin or the appearance of a quartet of complex eigenvalues. In general, the gain-loss term can be considered parasitic as it reduces the stability region of the onsite solitons. Additionally, we analyse the dynamic behaviour of the onsite and intersite solitons when unstable, where typically it is either in the form of travelling solitons or soliton blow-ups. arXiv:1607.05331v2 [nlin.PS]

Discrete solitons dynamics in $$\mathscr {PT}$$-symmetric oligomers with complex-valued couplings

“…The current model employs complex-valued coefficients in the vertical coupling between the parallel arrays, while the previous work [67] and [68] adopted purely imaginary and real-valued vertical coupling between the parallel arrays, respectively, which acts as the gain or loss in the system. Additionally, they also included the real-valued and purely imaginary phase-velocity mismatch between the horizontal cores in [67] and [68], respectively, which is absent in our current model.…”

“…In our previous work, we have considered the existence and linear stability of fundamental bright discrete solitons in PT -symmetric dimers with gain-loss terms [67], in a chain of charge-parity (C P)-symmetric dimers [68], and in a chain of PT -symmetric dimers with cubic-quintic nonlinearity [69]. The latter covers the snaking behavior in the bifurcation diagrams for the existence of standing localized solutions.…”

Topological edge solitons and their stability in a nonlinear Su-Schrieffer-Heeger model

“…3. The stability in the limit Λ → 0 can be established using perturbation theory, exploiting dimers of the system; see a similar problem considered in, e.g., [34]. The method, that has been standard by now, can be applied here to show that in that limit the edge soliton is stable.…”

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

“…A solution is unstable when there exists λ with Re(λ) > 0. However, if λ is a spectrum, so are −λ and ±λ [36]. A solution is therefore (linearly) stable only when Re(λ) = 0 for all λ, i.e.…”

“…In particular, when nonlinear dimers are put in arrays where elements with gain and loss are linearly coupled to the elements of the same type belonging to adjacent dimers, one can obtain a distinctive feature in the form of the existence of solutions localized in space as continuous families of their energy parameter [35]. The nonlinear localized solutions and their stability have been studied in [36][37][38] analytically and numerically (see also the references therein for localized solutions in systems of coupled nonlinear Schrödinger equations).…”