Enhanced mobility of strongly localized modes in waveguide arrays by inversion of stability

Abstract

A model equation governing the amplitude of the electric field in an array of coupled optical waveguides embedded in a material with Kerr nonlinearities is derived and explored. The equation is an extended discrete nonlinear Schrödinger equation with intersite nonlinearities. Attention is turned towards localized solutions and investigations are made from the viewpoint of the theory of discrete breathers (DBs). Stability analysis reveals an inversion of stability between stationary one-site and symmetric or antisymmetric two-site solutions connected to bifurcations with a pair of asymmetric intermediate DBs. The stability inversion leads to the existence of high-intensity narrow mobile solutions, which can propagate essentially radiationless. The direction and transverse velocity of the mobile solutions can be controlled by appropriate perturbations. Such solutions may have an important application for multiport switching, allowing unambiguous selection of output channel. The derived equation also supports compact DBs, which in some sense yield the best possible solutions for switching purposes.

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

“…An example is given in [26, Section 2.2 & 2.3, pg. [9][10][11][12][13][14][15][16][17][18], where the minimization of a linear energy functional under a nonlinear constraint verified conditions for the co-existence of breather profiles. For instance, this alternative approach for (2.1) will show the existence of a regime for the hopping parameter α where an upper bound for the power is valid (see Remark II.3).…”

“…In the quantum lattice introduced therein, the inclusion of the nonlinear hopping term allows a fast energy propagation as long as α is high enough with respect to β. Such terms (the additional ones to the classical DNLS with cubic onsite nonlinearity and linear coupling between sites) have appeared in physical considerations within the modeling of waveguide arrays [18,20], establishing that in the case of large penetration length or closely spaced waveguides these terms are not negligible; however, it should be noted that in this case additional terms of the same (cubic) order should be included in the relevant modeling [18,20]. Nonlinear hopping terms appear also from FPU and KG chains of anharmonic oscillators coupled with anharmonic inter-site potentials, or mixed FPU/KG chains.…”

“…The physical interest in this particular model stems from various contexts, as the modeling of quantum lattices and waveguide arrays and the approximation of the dynamics Klein-Gordon (KG) and Fermi-Pasta-Ulam (FPU) chains [16][17][18][19][20][21]. Eq.…”

Superfluid dynamics of a Bose–Einstein condensate in a periodic potential

“…An equation similar to the DNL (15) has been derived in [30] to describe the dynamics of an electric field in an array of coupled optical waveguides embedded in a material with Kerr nonlinearities.…”

Dynamics of nonlinear excitations of helically confined charges

“…Within the degeneracy regime, however, the initial excitation focuses in the course of propagation reaching finally a rather localized state, serving as a hallmark of the existing nonlinearity. In order to quantify this nonlinearity we construct an effective DNLS model with additional nonlocal nonlinear terms [24,25]. Such a model is found to capture qualitatively well the localization and dispersion features of the original dynamics, providing a deeper insight into the observed effect.…”