Interplay between modulational instability and self-trapping of wavepackets in nonlinear discrete lattices

Abstract

We investigate the modulational instability of uniform wavepackets governed by the discrete nonlinear Schrodinger equation in finite linear chains and square lattices. We show that, while the critical nonlinear coupling χMI above which modulational instability occurs remains finite in square lattices, it decays as 1/L in linear chains. In square lattices, there is a direct transition between the regime of stable uniform wavefunctions and the regime of asymptotically localized solutions with stationary probability distributions. On the other hand, there is an intermediate regime in linear chains for which the wavefunction dynamics develops complex breathing patterns. We analytically compute the critical nonlinear strengths for modulational instability in both lattices, as well as the characteristic time τ governing the exponential increase of perturbations in the vicinity of the transition. We unveil that the interplay between modulational instability and self-trapping phenomena is responsible for the distinct wavefunction dynamics in linear and square lattices.

Thresholds between modulational stability, rogue waves and soliton regimes in saturable nonlinear media

“…1c. Such behavior is also in full accordance with previous studies [31,32], which report this interplay. Here, by considering saturable nonlinear media, we explore the influence of saturation (ζ) and observe significant changes on the dynamic behavior.…”

“…1b). This result shows the critical nonlinearity, above which the uniformly stable solution does not survive, as being 0.195 < χ BS < 0.200, in full agreement with previous reports [31,32]. A further increasing of χ can drive the wave packet into localized regime, as shown in Fig.…”

“…Now, the fully extended solution is not always visited after a breath and the wave packet dynamics becomes very sensitive to the initial conditions. Such regime, so-called chaotic-like [31], becomes evident when we observe participation minima spreading over a wide range of values, signaling incommensurate breathing modes along dynamics (see Fig. 4a).…”

“…Modulational instability refers to the stability, with respect to any perturbation, of a wave with constant amplitude that propagates through a nonlinear dispersive medium [23,24]. Such phenomenon has been widely studied in different frameworks and is usually seen as the starting point for regimes of breathing modes and localized solutions [25][26][27][28][29][30][31][32]. Significant changes in the crossover between modulational instability and breather solutions have been reported by exploring the influence of next-nearest-neighbor coupling in nonlinear discrete lattices [25].…”

“…Studies on 1D and 2D nonlinear lattices show the critical nonlinear coupling, above which the modulational instability occurs, remaining finite for square lattices and inversely proportional to the length for 1D lattices. A direct transition between stable uniform wave functions and asymptotically localized solutions are reported for square lattices, while linear chains exhibit an intermediate regime in which the wave function dynamics develops complex breathing patterns [31]. The interplay between modulational instability and localized solutions has been evaluated by considering the presence of long-ranged hopping amplitudes in one-dimensional lattices, a prospect that influences the effective dimensionality of system [32].…”