Nondecaying linear and nonlinear modes in a periodic array of spatially localized dissipations

Abstract

We demonstrate the existence of extremely weakly decaying linear and nonlinear modes (i.e. modes immune to dissipation) in the one-dimensional periodic array of identical spatially localized dissipations, where the dissipation width is much smaller than the period of the array. We consider wave propagation governed by the one-dimensional Schrödinger equation in the array of identical Gaussian-shaped dissipations with three parameters, the integral dissipation strength ${\Gamma }_0$, the width $\sigma$ and the array period $d$. In the linear case, setting $\sigma \to 0$, while keeping ${\Gamma }_0$ fixed, we get an array of zero-width dissipations given by the Dirac delta-functions, i.e. the complex Kroning-Penney model, where an infinite number of nondecaying modes appear with the Bloch index being either at the center, $k=0$, or at the boundary, $k=\pi /d$, of an analog of the Brillouin zone. By using numerical simulations we confirm that the weakly decaying modes persist for $\sigma$ such that $\sigma /d\ll 1$ and have the same Bloch index. The nondecaying modes persist also if a real-valued periodic potential is added to the spatially periodic array of dissipations, with the period of the dissipative array being multiple of that of the periodic potential. We also consider evolution of the soliton-shaped pulses in the nonlinear Schrödinger equation with the spatially periodic dissipative lattice and find that when the pulse width is much larger than the lattice period and its wave number $k$ is either at the center, $k=2\pi /d$, or at the boundary, $k=\pi /d$, a significant fraction of the pulse escapes the dissipation forming a stationary nonlinear mode with the soliton shaped envelope and the Fourier spectrum consisting of two peaks centered at $k$ and $-k$.

arXiv comment20 pages, 11 figures (best viewed in color). Accepted to Journal of Physics A: Mathematical and Theoretical

References in corpus (11)

• Bright gap solitons of atoms with repulsive interaction
• Controlling the dynamics of an open many-body quantum system with localized dissipation
• Nonlinear localized waves in a periodic medium
• Dissipation-induced coherent structures in Bose-Einstein condensates
• Photonic crystals of coated metallic spheres
• Nonlinear Band Structure in Bose Einstein Condensates: The Nonlinear Schrödinger Equation with a Kronig-Penney Potential
• Macroscopic Zeno effect and stationary flows in nonlinear waveguides with localized dissipation
• Control of the Bose-Einstein condensate by dissipation. Nonlinear Zeno effect
• Soliton-radiation coupling in the parametrically driven, damped nonlinear Schrödinger equation
• A Scanning Electron Microscope for Ultracold Atoms
• State engineering of Bose-Einstein condensate in the optical lattice by a periodic sublattice of dissipative sites