Soliton mobility in disordered lattices

Defect-induced discrete breather in dissipative optical lattices with weak nonlinearity

It is widely believed that the discrete breather (DB) can only be created when the nonlinearity is strong in nonlinear systems. However, we here establish that this belief is incorrect. In this work, we systemically investigate the generation of DBs induced by coupling of the defects and nonlinearity for Bose-Einstein condensates in dissipative optical lattices. The results show that, only in a clean lattice is strong nonlinearity a necessary condition for generating of DB; whereas, if the lattice has a defect, the DBs can also be discovered even in weak nonlinearity, and its generation turns out to be controllable. In addition, we further reveal a critical interval of the defect in weak nonlinearity, within which DBs can be found, while outside DBs do not exist. Furthermore, we also explore the impact of multiple defects on the generation of DBs, and analyze the underlying physical mechanisms of these interesting phenomena. The results not only have the potential to be used for more precise engineering in the DB experiments, but also suggest that the DB may be ubiquitous since the defects and dissipation are unavoidable in real physics.

Nonlinear Schrödinger waves in a disordered potential: Branched flow, spectrum diffusion, and rogue waves

The problem of nonlinear Schrödinger (NLS) waves in a disordered potential arises in many physical occasions, such as hydrodynamics, optics, and cold atoms. It provides a paradigm for studying the interaction between nonlinearity and random effect, but the current results are far from perfect. In this paper, we systematically simulate the turbulent waves for the focusing NLS equation with dynamical (time-dependent) random potentials, where the enhanced branching structures evolve into branched soliton flows as the nonlinearity increases. In this process, the occurrence of rogue waves for short times results from the interplay of linear random focusing and modulation instability. While the nonlinear spectral analysis reveals that for longer times, it is due to a self-organization of larger solitons competing with breakup of intermediate solitons. On the other hand, we found that the strong nonlinearity can significantly increase the width of the linear (Fourier) spectrum for several time scales, but its spreading rate becomes suppressed, which has a dependence on the correlation length of the potential. We hope that our findings will facilitate a deeper understanding of the nonlinear waves interacting with disordered media.

Nearly integrable turbulence and rogue waves in disordered nonlinear Schrödinger systems

Integrable nonlinear Schrödinger (NLS) systems provide a platform for exploring the propagation and interaction of nonlinear waves. Extreme events such as rogue waves (RWs) are currently of particular interest. However, the presence of disorder in these systems is sometimes unavoidable, for example, in the forms of turbulent current in the ocean and random fluctuation in optical media, and its influence remains less understood. Here, we report numerical experiments of two nearly-integrable NLS equations with the effect of disorder showing that the probability of RW occurrence can be significantly increased by adding weak system noise. Linear and nonlinear spectral analyses are proposed to qualitatively explain those findings. Our results are expected to shed light on the understanding of the interplay between disorder and nonlinearity, and may motivate new experimental works in hydrodynamics, nonlinear optics, and Bose-Einstein condensates.

Anomalous diffusion of discrete solitons driven by evolving disorder

Anomalous diffusion is simulated in this paper by studying the transport of discrete solitons in a lattice with evolving disorder. We find a Richardson-type diffusion for the small solitons and a regime of transient diffusion for larger solitons within the ensemble-averaged description. As a comparison, the time-averaged observables present a ballistic scaling for both cases. However, distribution of these observables changes remarkably with the soliton size. Our results suggest violation of ergodicity for the solitons' diffusive processes, which are expected to shed light on further understanding of the discreteness-disorder-nonlinearity interaction.

Transient diffusion and two-regime localization of discrete breatherlike excitations in nonlinear Schrödinger lattice with disorder

We systematically simulate and analyze the motion of discrete breatherlike excitations (DBEs) in the nonlinear Schrödinger lattice with random potentials. A universal transient diffusion of the DBEs is observed for short timescales (t≲10^{3}). For longer timescales (t up to 10^{5}), the DBEs become localized. Such localization, depending on the DBE powers, has two different regimes: Regime I is the Anderson-like localization induced by the disorder, while Regime II is an enhanced localization attributed to both the disorder and discreteness. Our study is expected to shed light on understanding the interplay between disorder and strong nonlinearity, from the diffusive transport and localization properties of nonlinear localized excitations in random media.

Unidirectional transport of wave packets through tilted discrete breathers in nonlinear lattices with asymmetric defects

We consider the transfer of lattice wave packets through a tilted discrete breather (TDB) in opposite directions in the discrete nonlinear Schrödinger model with asymmetric defects, which may be realized as a Bose-Einstein condensate trapped in a deep optical lattice, or as optical beams in a waveguide array. A unidirectional transport mode is found, in which the incident wave packets, whose energy belongs to a certain interval between full reflection and full passage regions, pass the TDB only in one direction, while in the absence of the TDB, the same lattice admits bidirectional propagation. The operation of this mode is accurately explained by an analytical consideration of the respective energy barriers. The results suggest that the TDB may emulate the unidirectional propagation of atomic and optical beams in various settings. In the case of the passage of the incident wave packet, the scattering TDB typically shifts by one lattice unit in the direction from which the wave packet arrives, which is an example of the tractor-beam effect, provided by the same system, in addition to the rectification of incident waves.

Superdiffusive transport and energy localization in disordered granular crystals

We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder-an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)-and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment m̃(2) of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m̃(2)∼t(γ) (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m̃(2) and the inverse participation ratio P(-1) satisfy scaling relations m̃(2)∼t(γ) and P(-1)∼t(-η), and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η<0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ≈1.7 and η≈1) for all three types of disorder and for both types of initial conditions.