Symmetric Nonlinearity Research Articles

Rational soliton solutions in the parity-time-symmetric nonlocal coupled nonlinear Schrödinger equations

In this paper, via the Darboux transformation (DT) method, we study the nonlocal coupled nonlinear Schrödinger (NLS) equations with parity-time (PT)-symmetric nonlinearities. We construct the N-fold iterative generalized DT and derive rational vector soliton solutions from a non-vanishing background. The first-order rational solution in each component can exhibit elastic interactions of antidark-antidark, dark-antidark, and antidark-dark rational solitons whose dynamics are the same as ones in nonlocal scalar NLS equation. The second-order rational solution possesses abundant interaction patterns. They not only can display elastic interaction between two rational solitons with combined-peak-valley profiles, but also can exhibit the dynamics of localized nonlinear excitation for rational antidark solitons generating high and steep hump wave patterns in the region of the interaction. We expect that the results in this work will be useful to study rational solitons in coupled PT-symmetric optical waveguides.

A string of Peregrine rogue waves in the nonlocal nonlinear Schrödinger equation with parity-time symmetric self-induced potential

Dynamic wave localization phenomena draw fundamental and technological interests in optics and photonics. Based on the recently proposed (Ablowitz and Musslimani, 2013) continuous nonlocal nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity (PTNLSE), a numerical investigation has been carried out for two first order Peregrine solitons as the initial ansatz. Peregrine soliton, as an exact solution to the PTNLSE, evokes a very potent question: what effects does the interaction of two first order Peregrine solitons have on the overall optical field dynamics. Upon numerical computation, we observe the appearance of Kuznetsov–Ma (KM) soliton trains in the unbroken PT-phase when the initial Peregrine solitons are in phase. In the out of phase condition, it shows repulsive nonlinear waves. Quite interestingly, our study shows that within a specific range of the interval factor in the transverse co-ordinate there exists a string of high intensity well-localized Peregrine rogue waves in the PT unbroken phase. We note that the interval factor as well as the transverse shift parameter play important roles in the nonlinear interaction and evolution dynamics of the optical fields. This could be important in developing fundamental understanding of nonlocal non-Hermitian NLSE systems and dynamic wave localization behaviors.

Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant Coefficients

We consider the nonlinear curl-curl problem \nabla\times\nabla\times U + V(x) U=f(x, |U|^2)U in \mathbb R^3 related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient V and subcritical cylindrically symmetric nonlinearity f . The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions [J. Funct. Anal. 41 (1981)(2), 236–275], new rearrangement type inequalities by Brock [Proc. Indian Acad. Sci. Math. Sci. 110 (2000), 157–204] and the recent extension of the Nehari-manifold technique from Szulkin and Weth [Handbook of Nonconvex Analysis and Applications (2010), pp. 597–632].

Oscillators with symmetric and asymmetric quadratic nonlinearity

In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both oscillators are modeled as ordinary second-order differential equations with strong quadratic nonlinearities: one with positive quadratic term and the second with a quadratic term which changes the sign. Solutions for both equations are obtained in the form of Jacobi elliptic functions. For the asymmetric oscillator, conditions for the periodic motion are determined, while for the symmetric oscillator a new approximate solution procedure based on averaging is developed. Obtained results are tested on an optomechanical system where the motion of a cantilever in the intracavity field is oscillatory. Two types of quadratic nonlinearities in the system are investigated: symmetric and asymmetric. The advantage and disadvantage of both models is discussed. The analytical procedure suggested in the paper is applied. The obtained solution agrees well with a numerical one.

Peregrine rogue wave dynamics in the continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity

In this work, we have studied the peregrine rogue wave dynamics, with a solitons on finite background (SFB) ansatz, in the recently proposed (Ablowitz and Musslimani, (2013) [31]) continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity. We have found that the continuous nonlinear Schrödinger system with PT-symmetric nonlinearity also admits Peregrine soliton solution. Motivated by the fact that Peregrine solitons are regarded as prototypical solutions of rogue waves, we have studied Peregrine rogue wave dynamics in the c-PTNLSE model. Upon numerical computation, we observe the appearance of low-intense Kuznetsov–Ma (KM) soliton trains in the absence of transverse shift (unbroken PT-symmetry) and well-localized high-intense Peregrine rogue waves in the presence of transverse shift (broken PT-symmetry) in a definite parametric regime.

Two-component solitons in a chain of dimers with Parity-Time symmetric nonlinearity

We introduced a coupled waveguide arrays with intrinsic Parity-Time (PT)-symmetric nonlinearity. The system is described by a set of coupled Schrödinger equations with linear couplings and a complex nonlinearity. Two-component soliton modes are found to exist in this system. The stability of the solitons are investigated and the stable map is plotted numerically in the parameter plane of linear coupling and the strength of PT nonlinearity. The effects of the PT nonlinearity on the amplitude and the effective area of the soliton modes are studied. It is found that the PT nonlinearity can manipulate the amplitude and the propagation constant of the soliton modes. The mobility of these soliton modes are studied numerically. The soliton modes can be kicked to move and increase the PT nonlinearity will enhance the mobility. Collision of two soliton modes are investigated and the results indicates that we can control the types and properties of the collision by adjusting the linear coupling and the PT nonlinearity.

Defect solitons in parity-time symmetric superlattices with focusing saturable nonlinearity

In this paper, we find that defect superlattice solitons (DSSs) exist at the defect site in one-dimensional optical superlattices with a parity-time (PT) symmetric potential and focusing saturable nonlinearity. We discuss the existence and stability of DSSs numerically and find that both the saturation parameter s and the defect strength ε can affect the power, the existence and stable regions of solitons significantly. For given saturation parameter s and defect strength ε (propagation constant μ), the power of DSSs increases with the increase (decrease) of μ (ε). For positive and zero defects, DSSs only exist in the semi-infinite band gap, whereas for negative defects, DSSs exist not only in the semi-infinite band gap but also in the first finite band gap. Besides, our numerical calculations show that when the saturation nonlinearity increases up to some level, the increasing rate of soliton power with μ is very fast and the stable regions of DSSs will become obviously narrow and eventually disappear with a decrease of ε for negative defects.

Adaptive control of stochastic Hammerstein systems with dead-zone input non-linearity

This article presents a new control scheme for H systems with symmetric dead-zone input non-linearity and coloured noise. Based on the parameterized H model, we derive a dead-zone compensating control law and a recursive estimator. Theoretical analysis shows that global convergence and closed-loop stability can be guaranteed using the proposed algorithms. What is more, theoretical results show that the output tracking error is convergent with a specific residue, which is related to the variance of the noise and the parameter estimate. Simulation results confirm that the proposed method leads to satisfactory output tracking performance.

Stochastic linearisation approach to performance analysis of feedback systems with asymmetric nonlinear actuators and sensors

This paper considers feedback systems with asymmetric (i.e., non-odd functions) nonlinear actuators and sensors. While the stability of such systems can be investigated using the theory of absolute stability and its extensions, the current paper provides a method for their performance analysis, i.e., reference tracking and disturbance rejection. Similar to the case of symmetric nonlinearities considered in earlier work, the development is based on the method of stochastic linearisation (which is akin to the describing functions, but intended to study general properties of dynamics, rather than periodic regimes). Unlike the symmetric case, however, the nonlinearities considered here must be approximated not only by a quasilinear gain, but a quasilinear bias as well. This paper derives transcendental equations for the quasilinear gain and bias, provides necessary and sufficient conditions for existence of their solutions, and, using simulations, investigates the accuracy of these solutions as a tool for predicting the quality of reference tracking and disturbance rejection. The method developed is then applied to performance analysis of specific systems, and the effect of asymmetry on their behaviour is investigated. In addition, this method is used to justify the recently discovered phenomenon of noise-induced loss of tracking in feedback systems with PI controllers, anti-windup, and sensor noise.

Modified gravitational collapse, or the wonders of the MOND

There are many hot discussions in the literature about two competing paradigms in galactic and extra-galactic astronomy and cosmology, namely the Dark Matter and the Modified Newtonian Dynamics (MOND). It is very difficult to challenge MOND from the cosmological side because a full relativistic realisation is needed in the first place, and any failure can then be attributed to a particular model, and not to the MOND itself. We propose to study non-relativistic stages of gravitational collapse in MOND which, we argue, is a relevant task for this competition. Spherically symmetric dust cloud collapse and intrinsic unavoidable non-linearities of the deep MOND regime are discussed. We conclude that complicated, both numerical and analytic, studies of modified gravitational dynamics are needed in order to assess the viability of MOND.

Reexamining the PPP hypothesis: A nonlinear asymmetric heterogeneous panel unit root test

In this study, we re-examine the PPP hypothesis in the light of the new developments in the unit root testing literature. The recent theoretical findings have pointed out that the real exchange rate series exhibit asymmetric nonlinear behavior. A unit root test applied to analyze the PPP hypothesis therefore, should also take into account this asymmetry inherent in the real exchange rate. Different unit root tests that consider the presence of these data features have been developed in the time series literature. However, a true attempt to test the PPP hypothesis should take a panel data approach. To this end, we propose a nonlinear heterogeneous panel unit root test where the alternative hypothesis allows for symmetric or asymmetric exponential smooth transition autoregressive nonlinearity and provide its finite sample properties. We apply our test to the real exchange rates of the 15 European Union countries against the US dollar. While the results of the linear and symmetric nonlinear heterogeneous panel unit root tests are against the PPP hypothesis, the asymmetric nonlinear heterogeneous panel test that we propose gives support for the PPP hypothesis as expected. Therefore, the conclusions drawn from the linear panel unit root tests or the nonlinear panel unit root tests that do not take asymmetry into account might be misleading.

Stationary modes and integrals of motion in nonlinear lattices with a -symmetric linear part

We consider finite-dimensional nonlinear systems with a linear part described by a parity-time (-)symmetric operator. We investigate bifurcations of stationary nonlinear modes from the eigenstates of the linear operator and consider a class of -symmetric nonlinearities allowing the existence of families of nonlinear modes. We pay particular attention to situations when the underlying linear -symmetric operator is characterized by the presence of degenerate eigenvalues or an exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing the system to admit one or several integrals of motion, which turn out to be determined by the pseudo-hermiticity of the nonlinear operator.

Finding exact periodic solutions in Lure systems through periodic signal mapping

A solution of the periodic problem in a nonlinear system comprising a single-valued symmetric nonlinearity (Lure system) and linear dynamics is presented. The solution is designed as an iterative application of Periodic Signal Mapping to refine the approximate solution obtained through the describing function method. The algorithm is based upon the transformation of the original nonlinear system into an equivalent nonlinear system for which the filtering hypothesis is satisfied exactly, and for that reason the latter being suitable for application of the developed algorithm. The solution sought for is a fixed point of the periodic signal mapping. Conditions of asymptotic convergence of the proposed algorithm are given. The proposed approach is illustrated by two examples of analysis of periodic motions in a relay feedback system and chattering in a terminal sliding mode.

Experimental Observation of Self-Accelerating Beams in Quadratic Nonlinear Media

We present the experimental observation of 1D and 2D self-accelerating nonlinear beams in quadratic media, which are also the first nonlinear self-accelerating beams in any symmetric nonlinearity. Notably, we show that the intensity peaks of the first and second harmonics are asynchronous with respect to one another, but the coupled harmonics exhibit joint acceleration within the nonlinear medium. Finally, we demonstrate the impact of self-healing effects on the jointly accelerating first and second harmonics.

Memristor based-elements for chaotic circuits

The peculiar features of the memristor, a fundamental passive two-terminal element characterized by nonlinear relationship between charge and flux, promise to revolutionize integrated circuit design in the next few decades. Besides its most popular potential applications, ultra-dense non-volatile memories and brain-simulating systems, much research has been lately devoted to their use in chaotic circuits. Although the physical memristor is inherently-asymmetric, complementary behaviors arise in devices with opposite polarity. In this work we demonstrate that this makes it feasible to devise a number of practical realizations of a monotone-increasing odd-symmetric charge-flux nonlinearity suitable for chaos-based applications through the sole use of physical memristors of the kind recently fabricated at Hewlett-Packard Labs. Confirmation for such claim is obtained through comparison of chaotic behavior of two modified Chua's oscillators, in which the nonlinear diode is replaced in one case with an artificial memristor with symmetric nonlinearity and in the other case with one of the proposed symmetric memristor combinations

Non-Centered Spike-Triggered Covariance Analysis Reveals Neurotrophin-3 as a Developmental Regulator of Receptive Field Properties of ON-OFF Retinal Ganglion Cells

The functional separation of ON and OFF pathways, one of the fundamental features of the visual system, starts in the retina. During postnatal development, some retinal ganglion cells (RGCs) whose dendrites arborize in both ON and OFF sublaminae of the inner plexiform layer transform into RGCs with dendrites that monostratify in either the ON or OFF sublamina, acquiring final dendritic morphology in a subtype-dependent manner. Little is known about how the receptive field (RF) properties of ON, OFF, and ON-OFF RGCs mature during this time because of the lack of a reliable and efficient method to classify RGCs into these subtypes. To address this deficiency, we developed an innovative variant of Spike Triggered Covariance (STC) analysis, which we term Spike Triggered Covariance – Non-Centered (STC-NC) analysis. Using a multi-electrode array (MEA), we recorded the responses of a large population of mouse RGCs to a Gaussian white noise stimulus. As expected, the Spike-Triggered Average (STA) fails to identify responses driven by symmetric static nonlinearities such as those that underlie ON-OFF center RGC behavior. The STC-NC technique, in contrast, provides an efficient means to identify ON-OFF responses and quantify their RF center sizes accurately. Using this new tool, we find that RGCs gradually develop sensitivity to focal stimulation after eye opening, that the percentage of ON-OFF center cells decreases with age, and that RF centers of ON and ON-OFF cells become smaller. Importantly, we demonstrate for the first time that neurotrophin-3 (NT-3) regulates the development of physiological properties of ON-OFF center RGCs. Overexpression of NT-3 leads to the precocious maturation of RGC responsiveness and accelerates the developmental decrease of RF center size in ON-OFF cells. In summary, our study introduces STC-NC analysis which successfully identifies subtype RGCs and demonstrates how RF development relates to a neurotrophic driver in the retina.

Front dynamics in heterogeneous diffusive media

We herein consider two-component reaction–diffusion systems with a specific bistable and odd symmetric nonlinearity, which have the bifurcation structure of pitchfork type traveling front solutions with opposite velocities. We introduce a spatial heterogeneity, for example, a Heaviside-like abrupt change at the origin in the space, into diffusion coefficients. Numerically, the responses of traveling fronts via the heterogeneity can be classified into four types of behavior depending on the strength of the heterogeneity, which, in the present paper, is represented by the height of the jump: passage, stoppage, and two types of reflection. The goal of the present paper is to reduce the PDE dynamics to finite-dimensional ODE systems on a center manifold and show the mathematical mechanism for producing the four types of response in the PDE systems using finite-dimensional ODE systems. The reduced ODE systems include the terms (referred to as heterogeneous perturbations) originating from the interaction between traveling front solutions and the heterogeneity, which is very important for determining the dynamics of the ODE systems. In the present paper, we succeed in calculating these heterogeneous perturbations exactly and explicitly.

Quantum properties of the three-mode squeezed operator: Triply concurrent parametric amplifiers

In this paper, we study the quantum properties of the three-mode squeezed operator. This operator is constructed from the optical parametric oscillator based on the three concurrent χ(2) nonlinearities. We give a complete treatment for this operator including the symmetric and asymmetric nonlinearity cases. The action of the operator on the number and coherent states is studied in the framework of squeezing, second-order correlation function, Cauchy–Schwartz inequality and single-mode quasiprobability function. The nonclassical effects are remarkable in all these quantities. We show that the nonclassical effects generated by the asymmetric case–for certain values of the system parameters–are greater than those of the symmetric one. This reflects the important role for the asymmetry in the system. Moreover, the system can generate particular types of the superposition states.

Modeling and detecting localized nonlinearity in continuum systems with a multistage transform

A general method is presented for modeling spatially extended systems that may contain a localized source of nonlinearity. It has direct applications to structural health monitoring (SHM) where physical damage may cause such nonlinearity and also communications channels which may exhibit localized nonlinearity due to bad electrical contacts or component nonlinearity. The method uses a multistage nonlinear transform in order to model the system dynamics. We discuss the application to SHM and provide a preliminary test of the method with experimental data from a randomly shaken beam with loose bolts. We discuss the application to telecommunications, provide an experimental observation of symmetric nonlinearity in a "bad" electrical contact, and provide a preliminary test of using this method to remove nonlinear echo (and thereby improve data rate) on a telephone line used for data transmission.

A simple unit root test against asymmetric STAR nonlinearity with an application to real exchange rates in Nordic countries

Existing tests of the unit root hypothesis against the alternative hypothesis of exponential smooth transition autoregressive (ESTAR) nonlinearity implicitly assume symmetry under the alternative. This paper proposes a simple unit root test against the alternative of symmetric or asymmetric ESTAR nonlinearity. In the event that the unit root hypothesis is rejected, a simple test of symmetric versus asymmetric ESTAR nonlinearity is also proposed. The asymptotic distributions of the test statistics are straightforward to establish and finite-sample performance is studied with Monte Carlo simulations. An empirical application involving the real exchange rates of four Nordic countries against the U.S. dollar illustrates the usefulness of the new tests.