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 onedimensional 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 onedimensional Schrödinger equation in the array of identical Gaussianshaped dissipations with three parameters, the integral dissipation strength Γ_{0}, the width σ and the array period d. In the linear case, setting σ → 0, while keeping Γ_{0} fixed, we get an array of zerowidth dissipations given by the Dirac deltafunctions, i.e. the complex KronigPenney 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 = π/d, of an analogue of the Brillouin zone. By using numerical simulations we confirm that the weakly decaying modes persist for σ such that σ/d ≪ 1 and have the same Bloch index. The nondecaying modes persist also if a realvalued periodic potential is added to the spatially periodic array of dissipations, with the period of the dissipative array being a multiple of that of the periodic potential. We also consider evolution of the solitonshaped 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 = 2π/d, or at the boundary, k = π/d, a significant fraction of the pulse escapes the dissipation forming a stationary nonlinear mode with the solitonshaped envelope and the Fourier spectrum consisting of two peaks centered at k and k.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 February 2014
 DOI:
 10.1088/17518113/47/8/085202
 arXiv:
 arXiv:1401.1687
 Bibcode:
 2014JPhA...47h5202F
 Keywords:

 Mathematical Physics;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Physics  Optics;
 Quantum Physics
 EPrint:
 20 pages, 11 figures (best viewed in color). Accepted to Journal of Physics A: Mathematical and Theoretical