Solitons in a Parametrically Driven Damped Discrete Nonlinear Schrödinger Equation

Abstract

We consider a parametrically driven damped discrete nonlinear Schrödinger (PDDNLS) equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental discrete bright solitons. We show that there are two types of onsite discrete soliton, namely onsite type I and II. We also show that there are four types of intersite discrete soliton, called intersite type I, II, III, and IV, where the last two types are essentially the same, due to symmetry. Onsite and intersite type I solitons, which can be unstable in the case of no dissipation, are found to be stabilized by the damping, whereas the other types are always unstable. Our further analysis demonstrates that saddlenode and pitchfork (symmetry-breaking) bifurcations can occur. More interestingly, the onsite type I, intersite type I, and intersite type III-IV admit Hopf bifurcations from which emerge periodic solitons (limit cycles). The continuation of the limit cycles as well as the stability of the periodic solitons are computed through the numerical continuation software Matcont. We observe subcritical Hopf bifurcations along the existence curve of the onsite type I and intersite type III-IV. Along the existence curve of the intersite type I we observe both supercritical and subcritical Hopf bifurcations.

Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein–Gordon equation and numerical comparisons

“…In our first simulation, we consider the fundamental site-centred discrete soliton of the dNLS equation, that has been considered before in, e.g., [9,10,11,12]. Such solutions will satisfy (3) with Ȧj = 0 and can be obtained rather straightforwardly using Newton's method.…”

“…The parametrically driven dNLS equation (3) was studied in [10,11], where it was shown that the parametric drive can change the stability of fundamental discrete solitons, i.e., it can destroy onsite solitons as well as restore the stability of intersite discrete solitons, both for bright and dark cases. In [12], breathers of (3), i.e., spatially localised solutions with periodically time varying |A j (τ )| emanating from Hopf bifurcations, were studied systematically.…”

Dissipative structures in a parametrically driven dissipative lattice: chimera, localized disorder, continuous-wave, and staggered state

“…In particular, the stability of discrete solitons in the parametrically driven DNLS equations, both conservative and dissipative ones, has been studied in Refs. [74][75][76][77][78].…”