Weak Nonlinear Regime Research Articles

Unconventional photon blockade in four mode coupled optomechanical system

In this study we theoretically and numerically analyze the photon statistics of a weakly driven optomechanical system comprising two optical and two mechanical modes where each mechanical mode is cross-coupled with the two cavity modes by a three-mode optomechanical interaction. We analytically determine the optimal conditions of unconventional photon blockade due to destructive interference between different transition pathways in weak Kerr nonlinear regime. We also numerically evaluate the second order photon correlation function and plot it against different system parameters to find that the numerical results are in agreement with analytical optimal conditions.

Plasma solitons in gated two-dimensional electron systems: Exactly solvable analytical model for the regime beyond weak nonlinearity

We analytically study plasma solitary waves, or solitons, in a two-dimensional electron system placed in close proximity to and between two ideal metallic gates. As a rule, solitons are described using a perturbative approach applicable only in the weak nonlinearity regime. In contrast, we analyze solitons considering a nonperturbative model. This framework enables an exact analytical description of the soliton shape. Moreover, it can be achieved in the regime beyond weak nonlinearity---when the concentration deviation due to the soliton is of the order of the equilibrium concentration. We determine the conditions required for a soliton to exist and derive the relationship between its amplitude, width, and velocity. We believe that our results obtained for the given model can provide valuable insight into the physics of nonlinear waves.

Thermal instability in a ferrimagnetic resonator strongly coupled to a loop-gap microwave cavity

We study nonlinear response of a ferrimagnetic sphere resonator (FSR) strongly coupled to a microwave loop gap resonator (LGR). The measured response in the regime of weak nonlinearity allows the extraction of the FSR Kerr coefficient and its cubic damping rate. We find that there is a certain range of driving parameters in which the system exhibits instability. In that range, self-sustained modulation of the reflected power off the system is generated. The instability is attributed to absorption-induced heating of the FSR above its Curie temperature.

The role of the non-linearity in controlling the surface roughness in the one-dimensional Kardar–Parisi–Zhang growth process

We explore linear control of the one-dimensional non-linear Kardar–Parisi–Zhang (KPZ) equation with the goal to understand the effects the control process has on the dynamics and on the stationary state of the resulting stochastic growth kinetics. In linear control, the intrinsic non-linearity of the system is maintained at all times. In our protocol, the control is applied to only a small number n c of Fourier modes. The stationary-state roughness is obtained analytically in the small-n c regime with weak non-linear coupling wherein the controlled growth process is found to result in Edwards–Wilkinson dynamics. Furthermore, when the non-linear KPZ coupling is strong, we discern a regime where the controlled dynamics shows scaling in accordance to the KPZ universality class. We perform a detailed numerical analysis to investigate the controlled dynamics subject to weak as well as strong non-linearity. A first-order perturbation theory calculation supports the simulation results in the weak non-linear regime. For strong non-linearity, we find a temporal crossover between KPZ and dispersive growth regimes, with the crossover time scaling with the number n c of controlled Fourier modes. We observe that the height distribution is positively skewed, indicating that as a consequence of the linear control, the surface morphology displays fewer and smaller hills than in the uncontrolled growth process, and that the inherent size-dependent stationary-state roughness provides an upper limit for the roughness of the controlled system.

Seltrapping in flat band lattices with nonlinear disorder

We study the transport properties of an initially localized excitation in several flat band lattices, in the presence of nonlinear (Kerr) disorder. In the weak nonlinearity regime, the dynamics is controlled by the degeneracy of the bands leading to a linear form of selftrapping. In the strong nonlinearity regime, the dynamics of the excitations depends strongly on the local environment around the initial excitation site that leads to a highly fluctuating selfrapping profile. For a binary nonlinear disorder, it is shown that the spreading of the flat band fundamental mode, is completely inhibited for a finite fraction of all cases. This fraction corresponds to the fraction of times the same value of (random) nonlinearity is assigned to all sites of the fundamental mode.

Modeling linear and nonlinear fluid flow through sheared rough-walled joints taking into account boundary stiffness

Abstract This study modeled four hundred fluid flow cases through shear-induced aperture fields of a rough-walled joint with self-affine surface characteristics under different boundary stress and stiffness conditions by solving the Navier-Stokes equations. The results show that as the hydraulic gradient increases from 10−8 to 101, the fluid flow experiences changes characterized by a linear regime, a weak nonlinear regime and a strong nonlinear regime. The permeability is a constant in the linear regime and starts to decline when entering the weak nonlinear regime, and the declining rate gradually converges when reaching the strong nonlinear regime. Such transition is poorly indicated by the global Reynolds number alone; it is instead primarily governed by the magnitudes and distribution characters of local Reynolds numbers. The boundary stiffness has linear/nonlinear relations with the permeability, depending on the magnitude of the joint aperture, which is governed by surface roughness, joint compressive strength and mechanical boundary conditions. With increasing boundary stiffness from 0 to 2.0 GPa/m, the ratio of permeability in y-direction to that in x-direction increases from 2.49 to 26.49 by approximately a factor of 10, suggesting that anomalous flow may prevail in sheared zones with stiff surrounding confinements.

Stability analysis of an interface between immiscible liquids in Hele-Shaw flow in the presence of a magnetic field

• Magnetic fluid interface in porous media. • Field stabilizing fingers. • Surface tension inhibits small scales. • Nonlinear regime of fingers interface. Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium . These penetrations are common in enhanced oil recovery , dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.

Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity, while the nonlinearity strength is characterized by $\varepsilon^2$ with $0 <\varepsilon \leq 1$ a dimensionless parameter. When $0 < \varepsilon \ll 1$, it is in the weak nonlinearity regime and the problem is equivalent to the NKGE with small initial data, while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. Four different FDTD methods are adapted to discretize the problem and rigorous error bounds of the FDTD methods are established for the long time dynamics, i.e. error bounds are valid up to the time at $O(1/\varepsilon^{\beta})$ with $0 \le \beta \leq 2$, by using the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. In the error bounds, we pay particular attention to how error bounds depend explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon\in (0,1]$, especially in the weak nonlinearity regime when $0 < \varepsilon \ll 1$. Our error bounds indicate that, in order to get ``correct'' numerical solutions up to the time at $O(1/\varepsilon^{\beta})$, the $\varepsilon$-scalability (or meshing strategy) of the FDTD methods should be taken as: $h = O(\varepsilon^{\beta/2})$ and $\tau = O(\varepsilon^{\beta/2})$. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.

Kerr nonlinearity in a superconducting Josephson metamaterial

We present a detailed experimental and theoretical analysis of the dispersion and non-linear Kerr frequency shifts of plasma modes in a one-dimensional Josephson junction chain containing 500 SQUIDs in the regime of weak nonlinearity. The measured low-power dispersion curve agrees perfectly with the theoretical model if we take into account the Kerr renormalisation of the bare frequencies and the long-range nature of the island charge screening by a remote ground plane. We measured the self- and cross-Kerr shifts for the frequencies of the eight lowest modes in the chain. We compare the measured Kerr coefficients with theory and find good agreement.

Nonlinear Rheology and Fracture of Disclination Network in Cholesteric Blue Phase III

Nonlinear rheological properties of chiral crystal cholesteryl oleyl carbonate (COC) in blue phase III (BPIII) were investigated under different shear deformations: large amplitude oscillatory shear, step shear deformation, and continuous shear flow. Rheology of the liquid crystal is significantly affected by structural rearrangement of defects under shear flow. One of the examples on the defect-mediated rheology is the blue phase rheology. Blue phase is characterized by three dimensional network structure of the disclination lines. It has been numerically studied that the rheological behavior of the blue phase is dominated by destruction and creation of the disclination networks. In this study, we find that the nonlinear viscoelasticity of BPIII is characterized by the fracture of the disclination networks. Depending on the degree of the fracture, the nonlinear viscoelasticity is divided into two regimes; the weak nonlinear regime where the disclination network locally fractures but still shows elastic response, and the strong nonlinear regime where the shear deformation breaks up the networks, which results in a loss of the elasticity. Continuous shear deformation reveals that a series of the fracture process delays with shear rate. The shear rate dependence suggests that force balance between the elastic force acting on the disclination lines and the viscous force determines the fracture behavior.

Nonlinear periodic wavetrains in thin liquid films falling on a uniformly heated horizontal plate

A thin liquid film falling on a uniformly heated horizontal plate spreads into fingering ripples that can display a complex dynamics ranging from continuous waves, nonlinear spatially localized periodic wave patterns (i.e., rivulet structures) to modulated nonlinear wavetrain structures. Some of these structures have been observed experimentally; however, conditions under which they form are still not well understood. In this work, we examine profiles of nonlinear wave patterns formed by a thin liquid film falling on a uniformly heated horizontal plate. For this purpose, the Benney model is considered assuming a uniform temperature distribution along the film propagation on the horizontal surface. It is shown that for strong surface tension but a relatively small Biot number, spatially localized periodic-wave structures can be analytically obtained by solving the governing equation under appropriate conditions. In the regime of weak nonlinearity, a multiple-scale expansion combined with the reductive perturbation method leads to a complex Ginzburg-Landau equation: the solutions of which are modulated periodic pulse trains which amplitude and width and period are expressed in terms of characteristic parameters of the model.

Zero eigenvalues of a photon blockade induced by a non-Hermitian Hamiltonian with a gain cavity

Under the weak driving limitation, we find that the strong photon blockade can be triggered when all the eigenvalues of the non-Hermitian Hamiltonian are equal to zero, so we name this the zero eigenvalues of a photon blockade induced by a non-Hermitian Hamiltonian. The possibility of realizing single- and two-photon blockades is investigated in the cavity QED system. A negative effective decay rate of the cavity is required, so the gain medium is needed to compensate the intrinsic loss rate of the cavity. The photon antibunching can be realized in both strong and weak nonlinear regimes, which is different from the conventional photon blockade and unconventional photon blockade. Our scheme has paved an avenue towards the study of photon blockades.

Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical and planar geometries

The relationship between the weakly nonlinear (WN) solutions of the Rayleigh-Taylor instability in spherical geometry [Zhang et al., Phys. Plasmas 24, 062703 (2017)] and those in planar geometry [Wang et al., Phys. Plasmas 19, 112706 (2012)] is analyzed. In the high-mode perturbation limit (Pn(cos θ), n≫1), it is found that at the equator, the contributions of mode P2n along with its neighboring modes, mode P3n along with its neighboring modes, and mode Pn at the third order along with its neighboring modes are equal to those of the second harmonic, the third harmonic, and the third-order feedback to the fundamental mode, respectively, in the planar case with a perturbation of the same wave vector and amplitude as those at the equator. The trends of WN results in spherical geometry towards the corresponding planar counterparts are found, and the convergence behaviors of the neighboring modes of Pn, P2n, and P3n are analyzed. Moreover, the spectra generated from the high-mode perturbations in the WN regime are provided. For low-mode perturbations, it is found that the fundamental modes saturate at larger amplitudes than the planar result. The geometry effect makes the bubbles at or near the equator grow faster than the bubbles in planar geometry in the WN regime.

Scattering and inverse scattering for nonlinear quantum walks

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schr\"odinger equations and discrete nonlinear Schr\"odinger equations but it seems to be the first time to be applied to QW.

Linear and Weakly Nonlinear Models of Wind Generated Surface Waves in Finite Depth

This work regards the extension of the Miles’ and Jeffreys’ theories of growth of wind-waves in water of finite depth. It is divided in two major sections. The first one corresponds to the surface water waves in a linear regimes and the second one to the surface water waver considered in a weak nonlinear, dispersive and anti-dissipative regime. In the linear regime, we extend the Miles’ theory of wind wave amplification to finite depth. The dispersion relation provides a wave growth rate depending to depth. A dimensionless water depth parameter depending to depth and a characteristic wind speed, induces a family of curves representing the wave growth as a function of the wave phase velocity and the wind speed. We obtain a good agreement between our theoretical results and the data from the Australian Shallow Water Experiment as well as the data from the Lake George experiment. In a weakly nonlinear regime the evolution of wind waves in finite depth is reduced to an anti-dissipative Korteweg-de Vries-Burgers equation and its solitary wave solution is exhibited. Anti-dissipation phenomenon accelerates the solitary wave and increases its amplitude which leads to its blow-up and breaking. Blow-up is a nonlinear, dispersive and anti-dissipative phenomenon which occurs in finite time. A consequence of anti-dissipation is that any solitary waves’ adjacent planes of constants phases acquire different velocities and accelerations and ends to breaking which occurs in finite space and in a finite time prior to the blow-up. It worth remarking that the theoretical amplitude growth breaking time are both testable in the usual experimental facilities. At the end, in the context of wind forced waves in finite depth, the nonlinear Schr odinger equation is derived and for weak wind inputs, the Akhmediev, Peregrine and Kuznetsov-Ma breather solutions are obtained

Periodic squeezing in a polariton Josephson junction

The use of a Kerr nonlinearity to generate squeezed light is a well-known way to surpass the quantum noise limit along a given field quadrature. Nevertheless, in the most common regime of weak nonlinearity, a single Kerr resonator is unable to provide the proper interrelation between the field amplitude and squeezing required to induce a sizable deviation from Poissonian statistics. We demonstrate experimentally that weakly coupled bosonic modes allow exploration of the interplay between squeezing and displacement, which can give rise to strong deviations from the Poissonian statistics. In particular, we report on the periodic bunching in a Josephson junction formed by two coupled exciton-polariton modes. Quantum modeling traces the bunching back to the presence of quadrature squeezing. Our results, linking the light statistics to squeezing, are a precursor to the study of nonclassical features in semiconductor microcavities and other weakly nonlinear bosonic systems.

Phonon antibunching effect in coupled nonlinear micro/nanomechanical resonator at finite temperature

In this study, we investigate the phonon antibunching effect in a coupled nonlinear micro/nanoelectromechanical system (MEMS/NEMS) resonator at a finite temperature. In the weak driving limit, the optimal condition for phonon antibunching is given by solving the stationary Liouville-von Neumann master equation. We show that at low temperature, the phonon antibunching effect occurs in the regime of weak nonlinearity and mechanical coupling, which is confirmed by analytical and numerical solutions. We also find that thermal noise can degrade or even destroy the antibunching effect for different mechanical coupling strengths. Furthermore, a transition from strong antibunching to bunching for phonon correlation has been observed in the temperature domain. Finally, we find that a suitably strong driving in the finite-temperature case would help to preserve an optimal phonon correlation against thermal noise.

Study of plasmonic slot waveguides with a nonlinear metamaterial core: semi-analytical and numerical methods

Two distinct models are developed to investigate the transverse magnetic stationary solutions propagating in one-dimensional anisotropic nonlinear plasmonic structures made from a Kerr-type nonlinear metamaterial core embedded between two semi-infinite metal claddings. The first model is semi-analytical, in which we assume that the anisotropic nonlinearity depends only on the transverse component of the electric field and that the nonlinear refractive index modification is small compared to the linear one. This method allows us to derive analytically the field profiles and nonlinear dispersion relations in terms of the Jacobi elliptical functions. The second model is fully numerical and is based on the finite element method in which all the components of the electric field are considered in the Kerr-type nonlinearity, with no presumptions as to the nonlinear refractive index change. Our finite-element-based model is valid beyond the weak nonlinearity regime and generalizes the well-known single-component fixed power algorithm that is usually used. Examples of the main cases are investigated, including those with strong spatial nonlinear effects at low power. Loss issues are reduced through the use of a gain medium in the nonlinear metamaterial core. Using anisotropic nonlinear FDTD simulations, we provide some results for the properties of the main solution.

Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical geometry

In this research, a weakly nonlinear (WN) model for the incompressible Rayleigh-Taylor instability in cylindrical geometry [Wang et al., Phys. Plasmas 20, 042708 (2013)] is generalized to spherical geometry. The evolution of the interface with an initial small-amplitude single-mode perturbation in the form of Legendre mode (Pn) is analysed with the third-order WN solutions. The transition of the small-amplitude perturbed spherical interface to the bubble-and-spike structure can be observed by our model. For single-mode perturbation Pn, besides the generation of P2n and P3n, which are similar to the second and third harmonics in planar and cylindrical geometries, many other modes in the range of P0–P3n are generated by mode-coupling effects up to the third order. With the same initial amplitude, the bubbles at the pole grow faster than those at the equator in the WN regime. Furthermore, it is found that the behavior of the bubbles at the pole is similar to that of three-dimensional axisymmetric bubbles, while the behavior of the bubbles at the equator is similar to that of two-dimensional bubbles.

The mise en scéne of memristive networks: effective memory, dynamics and learning

We discuss the properties of the dynamics of purely memristive circuits using a recently derived consistent equation for the internal memory variables of the involved memristors. In particular, we show that the number of independent memory states in a memristive circuit is constrained by the circuit conservation laws, and that the dynamics preserves these symmetry by means of a projection on the physical subspace. Moreover, we discuss other symmetries of the dynamics under various transformations of the involved variables, and study the weak and strong non-linear regimes of the dynamics. In the strong regime, we derive a conservation law for the internal memory variable. We also provide a condition on the reality of the eigenvalues of Lyapunov matrices. The Lyapunov matrix describes the dynamics close to a fixed point, for which show that the eigenvalues can be imaginary only for mixtures of passive and active components. Our last result concerns the weak non-linear regime, showing that the internal memory dynamics can be interpreted as a constrained gradient descent, and provide the functional being minimized. This latter result provides another direct connection between memristors and learning.