Drift-induced Benjamin-Feir instabilities

Abstract

A modified version of the Ginzburg-Landau equation is introduced which accounts for asymmetric couplings between neighbors sites on a one-dimensional lattice, with periodic boundary conditions. The drift term which reflects the imposed microscopic asymmetry seeds a generalized class of instabilities, reminiscent of the Benjamin-Feir type. The uniformly synchronized solution is spontaneously destabilized outside the region of parameters classically associated to the Benjamin-Feir instability, upon injection of a non homogeneous perturbation. The ensuing patterns can be of the traveling wave type or display a patchy, colorful mosaic for the modulus of the complex oscillators amplitude.PACS numbers: 05.45. Xt, 82.40.Ck, 89.75.Kd The spontaneous ability of spatially extended systems to self-organize in space and time is proverbial and has been raised to paradigm in modern science [1,2]. Collective behaviors are widespread in nature and mirror, at the macroscopic level, the microscopic interactions at play among elementary constituents. Convection instabilities in fluid dynamics, weak turbulences and defects are representative examples that emblematize the remarkable capacity of assorted physical systems to yield coherent dynamics [3]. Rhythms production and the brain functions are prototypical illustrations drawn from biology [4,5], insect swarms and fish schools refer instead to ecological applications [6]. The degree of instinctive and unsupervised coordination which instigates the bottom-up cascade towards self-regulated patterns is however universal and, as such, has been invoked in many other fields [7][8][9][10], ranging from chemistry [11,12] to economy, via technology [13]. Instabilities triggered by random fluctuations are often patterns precursors. The imposed perturbation shakes e.g. an homogeneous equilibrium, seeding a resonant amplification mechanism that eventually materializes in magnificent patchy motifs, characterized by a vast gallery of shapes and geometries. Exploring possible routes to pattern formation, and unraveling novel avenues to symmetry breaking instability, is hence a challenge of both fundamental and applied importance.In the so-called modulational instability deviations from a periodic waveform are reinforced by nonlinearity, leading to spectral-sidebands and the breakup of the waveform into a train of pulses [14,15]. The phenomenon was first conceptualized for periodic surface gravity waves (Stokes waves) on deep water by Benjamin and Feir, in 1967 [16], and for this reason is customarily referred to as the Benjamin-Feir (BF) instability. The BF instability has been later on discussed [17,18] in the context of the Complex Ginzburg Landau equation (CGLE), a quintessential model for non linear physics, whose applications range from superconductivity, superfluidity and Bose-Einstein condensation to liquid crystals and strings in field theory [19]. In this Letter, we will revisit the BF instability in the framework of the CGLE, modified with the inclusion of a drift term. This latte...

Benjamin–Feir instabilities on directed networks

“…This choice enables us to progress analytically in delineating the region of the relevant parameters space for which the generalized BF instability is predicted to hold. As in [17] the homogeneous limit cycle, perturbed by a tiny externally imposed disturbance, can give rise to traveling waves or patchy intermittent patterns, which share a striking similarity with those obtained in the region classically deputed to the BF instability. In the second part of the paper we consider instead an ensemble made of non linear Ginzburg-Landau oscillators, coupled via a generic direct graph.…”

“…Conversely, when λ (α,αc) Re > 0, the instability interests a hierarchy of independent modes, possibly yielding structured patterns, the byproduct of the aforementioned unpredictable mixing. To validate this interpretative scenario, that we put forward building on the analogy with the discussion in [17], we report in panel (b) of Fig. 2 the quantity λ (α,αc) Re , scanning the region of (c 1 , c 2 ) where the asymmetry driven instability takes place.…”

“…Depending on the specific choice of the model parameters c 1 and c 2 more complex scenarios are however possible. As it happens in the continuum limit [17], the CGLE defined on an asymmetric discrete lattice can generate a rich gallery of patterns, reminiscent of the BF type. These patterns are however established outside the region deputed to the BF instability on a symmetric support, or, to state it differently, they occur for a choice of the parameters for which the synchronized configuration is classically predicted to be stable.…”

“…Starting from this setting, the aim of this paper is to revisit the analysis in [17] for a family of non linear coupled oscillators, hosted on a directed network. To this end, we initially specialize on a peculiar class of networks, characterized by a circulant adjacency matrix.…”

Desynchronization and pattern formation in a noisy feed-forward oscillator network

“…This is the so called Benjamin-Feir (BF) instability, named after the researchers who first identified the phenomenon working with periodic surface gravity waves (Stokes waves) on deep water [13]. Typically the condition for the onset of the deterministic instability can be straightforwardly worked out by means of a traditional linear stability analysis, which constraints the reaction parameters involved in the formulation of the problem [1,14,15,16].…”