The zone boundary mode in periodic nonlinear electrical lattices

We study the existence and linear stability of the zone boundary mode in a nonlinear electrical lattice consisting of N inductors and N voltage-dependent capacitors with periodic boundary conditions. The inductances are allowed to alternate, while the capacitors are identical and each have a quadratic dependence on voltage. By block-diagonalizing a 2N×2N Floquet problem, we reduce the question of the stability of the mode to a single Hill’s equation that is analyzed using methods of perturbation theory and averaging. We show that periodicity of the lattice inductances degrades stability, and also show that the instability threshold is proportional to N⁻². Numerical computations validate the perturbative results.     

Pattern formation by spatially incoherent light in a nonlinear ring cavity

Modulation instability and pattern formation by spatially incoherent light is investigated experimentally in a nonlinear ring cavity using a photorefractive strontium barium niobate crystal as the nonlinear medium. A step-like threshold for the onset of pattern formation is observed experimentally for the case of high optical feedback. When compared to the case without feedback, this threshold is shifted towards smaller nonlinearities and a significant increase of the modulation degree of the obtained patterns is obtained. Our measurements also show that, above threshold, the dominating spatial frequency of the patterns decreases monotonically with both increasing nonlinearity and increasing feedback. PACS 42.65.Tg; 42.65.Sf; 89.75.Kd     

Exact transverse solitary and periodic wave solutions in a coupled nonlinear inductor-capacitor network

Through two methods, we investigate the solitary and periodic wave solutions of the differential equation describing a nonlinear coupled two-dimensional discrete electrical lattice. The fixed points of our model equation are examined and the bifurcations of phase portraits of this equation for various values of the front wave velocity are presented. Using the sineGordon expansion method and classic integration, we obtain exact transverse solutions including breathers, bright solitons,and periodic solutions.     

Pattern recognition minimizes entropy production in a neural network of electrical oscillators

We investigate the physical principle driving pattern recognition in a previously introduced Hopfield-like neural network circuit (Hölzel and Krischer, 2011 [13]). Effectively, this system is a network of Kuramoto oscillators with a coupling matrix defined by the Hebbian rule. We calculate the average entropy production 〈dS/dt〉$〈\mathrm{d}S/\mathrm{d}t〉$ of all neurons in the network for an arbitrary network state and show that the obtained expression for 〈dS/dt〉$〈\mathrm{d}S/\mathrm{d}t〉$ is a potential function for the dynamics of the network. Therefore, pattern recognition in a Hebbian network of Kuramoto oscillators is equivalent to the minimization of entropy production for the implementation at hand. Moreover, it is likely that all Hopfield-like networks implemented as open systems follow this mechanism.     

New mechanism of solitons formation at diffraction on a periodic inhomogeneity induced in a cubic nonlinear medium

Formation of solitons during the propagation of the initial Gaussian pulse in a Kerr medium and its diffraction on the induced periodic layer structure are described using the computer simulation. Dependence of the characteristics and the number of the resulting solitons on the parameters of the structure is demonstrated.     

Pattern formation and encoding rhythms analysis on a spiking/bursting neuronal network

In this work we study the formation of patterns of neuronal activity when some input are presented to the network. For this task a recently developed model of neuron is utilized. This model requires a very low computational effort but presents many characteristics of more complex models such as, spiking, bursting and sub-threshold oscillations, and therefore the realistic study of the behavior of big ensembles of neurons can be aborded, even under real time conditions. New results of the application of the wavelet transform technique to the analysis of pattern formation and the possible encoding of rhythms are presented; they show that this simple, low-computational, neuron model behaves much like more complex ones.     

Pattern formation in the Schlögl model of nonlinear kinetics

To study the effect of a local, spatial inhomogeneity on the progress of a chemical reaction which may exhibit a nonequilibrium phase transition, we have studied a (slightly) generalized version of a model proposed originally by Schlögl. Focusing on that regime of parameter space where the reaction sequence A + 2X[lrarr2]3X and B + X[lrarr2]C allows a first-order transition, we consider the consequences of introducing a spatially-varying diffusion coefficient characterized by a correlation length which calibrates the region over which the spatial inhomogeneity persists. We find that if the inhomogeneity is localized, only small quantitative differences are found between the exact solution reported earlier by Schlögl and the solutions generated here. However, as the correlation length becomes larger, abruptly at a certain critical value, a qualitative change in the nature of the solutions is found, with apparent oscillations produced in the concentration variable of the problem. We interpret this behavior as indicating the onset of pattern formation, and suggest that this behavior may be of importance in those radiation-induced phenomena where high-energy intermediates are produced.     

Avalanches in a nonlinear oscillator chain in a periodic potential

We consider the dynamics of a chain of coupled units evolving in a periodic substrate potential. The chain is initially in a flat state and situated in a potential well. A bias force, acting as a weak driving mechanism, is applied at a single unit of the chain. We study the instigation of directed transport in two types of system: (i) a microcanonical situation associated with deterministic and conservative dynamics and (ii) the Langevin dynamics when the system is in contact with a heat bath. Interestingly, for the deterministic and conservative dynamics the directed transport is drastically enhanced compared with its Langevin counterpart. In particular, in the deterministic and conservative regime a self-organised redistribution of energy triggers huge-sized avalanches yielding ultimately accelerated transport of the chain. In contrast, in the thermally-assisted process between avalanches the chain settles always into a pinned metastable state impeding continual accelerated chain motion.     

Pattern formation in a passive incoherent ring resonator system based on nonlinear material with the self-focusing non-instantaneous Kerr response

We have analyzed the pattern formation in a nonlinear medium with self-focusing non-instantaneous Kerr response by employing the passive incoherent ring resonator system. In such a system, coherence time of the light is much shorter than the time of one round trip in the resonator. This delayed response of the nonlinearity can amplify the noise of certain spatial frequencies of the perturbed wave field and thus patterns can form when nonlinear gain (i.e., amplification of the noises) overcomes the loss (i.e., a well defined cavity threshold set by the coherence properties) in a single pass. The expression for the spatial spectral density of the perturbed wave field, which is the characteristic parameters of the pattern formation, have been derived in the case of lowest order approximation. It is found that for a specific value of the spatial frequency of the perturbed wave field, the intensity feedback of the cavity is much effective factor rather than the crystal thickness of the nonlinear media and amplitude of the incoming beam in the cavity for the enhancement of the spatial spectral density of the intensity pattern, which greatly improved the performance and applications of the pattern formation such as information processing, symmetry-breaking, and dynamics in non-equilibrium systems.     

Generalization of a nonlinear friction relation for a dimer sliding on a periodic substrate

An atomic cluster moving along a solid surface can undergo dissipation of its translational energy through a direct mode, involving the coupling of the center-of-mass motion to thermal excitations of the substrate, and an indirect mode, due to damping of the internal motion of the cluster, to which the center-of-mass motion can be coupled as a result of surface potential. Focussing only on the less well understood indirect mode, on the basis of numerical solutions, we present, departures from a recently reported simple relationship between the force and velocity of nonlinear friction. A generalization of the analytic considerations that earlier led to that relationship is carried out and shown to explain the departures satisfactorily. Our generalization treats for the system considered (dimer sliding over a periodic substrate) the complete dependence on several of the key parameters, specifically internal dissipation, natural frequency, substrate corrugation, and length ratio. Further predictions from our generalizations are found to agree with new simulations. The system analyzed is relevant to nanostructures moving over crystal surfaces.     

Focusing-to-defocusing crossover in nonlinear periodic structures

We demonstrate experimentally a transition from nonlinear beam trapping to defocusing in a two-dimensional periodic photonic structures by varying the modulation depth of the lattice. The observed effect illustrates the fundamental crossover from discrete to cw transport mechanisms. At the threshold modulation, the output beam is highly sensitive to refractive index and power variations, which can be potentially applied for high-sensitivity refractive index or temperature sensing.     

一类非线性方程的新周期解

把Jacobi椭圆函数展开法扩展到Jacobi椭圆余弦函数和第三类Jacobi椭圆函数的有限展开法,并给出了一类非线性波动方程的新周期解,并且应用这种方法得到的周期解也可以退化为冲击波解或孤波解. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤波解     

Exact solution of a Langevin equation with nonlinear periodic noise

An exact stochastic average of a Langevin equation with a multiplicative nonlinear periodic noise is performed. The noise is described by an arbitrary periodic function of the diffusion Wiener-Lévy stochastic process. The solution of this stochastic equation is given by periodic solutions of the Hill equation.     

Random-phase solitons in nonlinear periodic lattices

We predict the existence of random phase solitons in nonlinear periodic lattices. These solitons exist when the nonlinear response time is much longer than the characteristic time of random phase fluctuations. The intensity profiles, power spectra, and statistical (coherence) properties of these stationary waves conform to the periodicity of the lattice. The general phenomenon of such solitons is analyzed in the context of nonlinear photonic lattices.     

Effects of periodic modulation on the nonlinear Landau--Zener tunneling

We study the Landau-Zener tunneling of a nonlinear two-level system by applying a periodic modulation on its energy bias. We find that the two levels are splitting at the zero points of the zero order Bessel function for high-frequency modulation. Moreover, we obtain the effective coupling constant between two levels at the zero points of the zero order Bessel function by calculating the final tunneling probability at these points. It seems that the effective coupling constant can be regarded as the approximation of the higher order Bessel function at these points. For the low-frequency modulation, we find that the final tunneling probability is a function of the interaction strength. For the weak inter-level coupling case, we find that the final tunneling probability is more disordered as the interaction strength becomes larger.     

Longitudinal structure formation in a nonlinear resonator

A fiber ring resonator, synchronously driven by picosecond pulses, exhibits a spatiotemporal instability. A rapid modulation in the pulse envelope is spontaneously formed and evolves dynamically. This structure may be called one-dimensional optical turbulence. Single-shot detection of these structures is not viable. We report how to obtain experimental information which, when combined with numerical simulation, provides an adequate picture of the processes at work. Intrapulse correlations are studied; they reveal clear differences between dispersion of either sign. Also, they reveal the formation of soliton-like structures in the turbulent regime. A criterion is given that allows the prediction of the degree of complexity.