Solitons in a parametrically driven damped discrete nonlinear Schr\"odinger equation

TL;DR

This paper investigates the existence, stability, and bifurcations of discrete bright solitons in a parametrically driven damped discrete nonlinear Schrödinger equation, revealing stabilization effects of damping and emergence of periodic solitons.

Contribution

It provides a comprehensive analysis of various soliton types, their stability, and bifurcation phenomena in the PDDNLS equation, including the discovery of Hopf bifurcations leading to periodic solitons.

Findings

Onsite and intersite type I solitons can be stabilized by damping.

Multiple bifurcation types, including saddle-node, pitchfork, and Hopf, are identified.

Periodic solitons emerge from Hopf bifurcations, with both subcritical and supercritical cases observed.

Abstract

We consider a parametrically driven damped discrete nonlinear Schr\"odinger (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 saddle-node 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.