Nonlinear Order Research Articles

Highly accurate analytical solution for free vibrations of strongly nonlinear Duffing oscillator

Based on the suggested parameter, a new analytical perturbation technique is presented to obtain highly ordered accurate analytical solutions for nonlinear Duffing oscillator with nonlinearity of high order. Comparing the obtained results with the numerical and other previously published results reveals the usefulness and correctness of the present technique. It is shown that the results are valid for small and large amplitudes. Indeed, it is found that our proposed technique produces more accurate and computationally results than the rival known methods. The obtained results show the efficiency and capability of the present perturbation technique to be applied to various strongly nonlinear differential equations.

Nonrelativistic limit of solitary waves for nonlinear Maxwell–Klein–Gordon equations

We study the nonrelativistic limit of solitary waves from Nonlinear Maxwell–Klein–Gordon equations (NMKG) to Nonlinear Schrödinger–Poisson equations (NSP). It is known that the existence or multiplicity of positive solutions depends on the choices of parameters the equations contain. In this paper, we prove that for a given positive solitary wave of NSP, which is found in Ruiz’s work (J Funct Anal 237(2):655–674, 2006), there corresponds a family of positive solitary waves of NMKG under the nonrelativistic limit. Notably, our results contain a new result of existence of positive solutions to (NMKG) with lower order nonlinearity.

Origins and minimization of intermodulation distortion in a pseudo-differential CMOS beamforming receiver

This paper studies how nonlinear distortion is generated in the combination of an inverter-based low-noise amplifier and a passive mixer. The dominant nonlinearity appears to be the quadratic V_{gs}V_{ds} mixing term in the passive mixer that first causes low-frequency IM2 and then upconverts it to IM3. Adding a common-mode feedback (CMFB) cancels the IM2 in a pseudo-differential structure, and hence also reduces the IM3 caused by the cascaded second order nonlinearities significantly. The effect of CMFB gain, bandwidth and linearity were analyzed, and it is concluded that from the linearity point of view, the feedback circuit does not have to be very wideband since the dominant distortion products originate from baseband. Finally, the paper takes a look at the spurious tones rising in the mixing, and how to extend the analysis to include the actual frequency translation effect.

Out-of-equilibrium optomechanical resonance self-excitation

The fundamental sensitivity limit of atomic force microscopy is strongly correlated to the thermal noise of cantilever oscillation. A method to suppress this unwanted noise is to reduce the bandwidth of the measurement, but this approach is limited by the speed of the measurement and the width of the cantilever resonance, commonly defined through the quality factor Q. However, it has been shown that optomechanical resonances in interferometers might affect cantilever oscillations resulting in an effective quality factor Qeff. When the laser power is sufficiently increased cantilever oscillations might even reach the regime of self-oscillation. In this self-oscillation state, the noise of the system is partially determined by the interaction with laser light far from equilibrium. Here, we show and discuss how tuning of laser power leads to nonlinear optomechanical effects that can dramatically increase the effective quality factor of the cantilever leading to out-of-equilibrium noise. We model the effects using a fourth order nonlinearity of the damping coefficient.

Investigation of third order nonlinearity of Ethidium bromide doped deoxyribonucleic acid) DNA)

The concentrations-dependent refractive index n 2 and the nonlinear absorption coefficient β of Ethidium bromide dye-doped deoxyribonucleic acid (biological polymer DNA) solutions in the SDL regime at 532 nm are reported. The Z-scan technique was performed in two ways and two different wavelengths, 532nm and 473nm, the open aperture technique and the closed aperture technique. From open aperture Z-scan measurements it is found that the Ethidium bromide doped deoxyribonucleic acid films exhibited reverse saturable absorption. The coefficient of nonlinear refraction and nonlinear absorption coefficient at 473nm wavelength is greater than at 532nm.

Investigation on fiber‐based terahertz source

AbstractA terahertz (THz) source based on hybrid Opt‐THz fiber, which is made of concentric central optical fiber and external Terahertz fiber, was proposed. With the enhanced second order nonlinearity in the central optical fiber, THz waves are distributed through difference frequency by high‐power optical pump laser. The external THz fiber will be designed with hollow fiber structure, and the THz waves are propagated and accumulated along the THz fiber. The simulation shows that the maximum THz power can reach ~4.8 mW after a 13.7 m optimal fiber length. The proposed THz source will be able to compete with existing THz sources in terms of power, compactness, frequency tunability and room‐temperature operation.

GLOBAL ATTRACTORS FOR A CLASS OF SEMILINEAR PARABOLIC EQUATIONS INVOLVING GRUSHIN OPERATOR AND NOLINEARITIES OF ARBITRARY ORDER

We study the existence and uniqueness of weak solutions and the existence of global attractors to a class of semilinear parabolic equations involving the Grushin operator and nonlinearities of arbitrary order. The main novelty of our result is that no restriction on the upper growth of the nonlinearities is imposed

Linear and nonlinear dynamics of a plate with acoustic black hole, geometric and contact nonlinearity for vibration mitigation

A rectangular plate with a wedge profile creating an Acoustic Black Hole (ABH) termination is studied numerically. A particular emphasis is put on combining two different types of nonlinearity in order to improve the passive damping capacity of the ABH by transferring energy to the high-frequency range where it is more efficient. First, the addition of contact points to create a vibro-impact black hole (VI-ABH) is taken into account, following a previous study on beams. The contact nonlinearity allows for a rapid and efficient transfer of energy. Second, the large-amplitude vibrations of the plate in the ABH region where small thickness is reached, is also considered. The geometric nonlinearity is incorporated using a von Kármán plate model, and the regime of wave turbulence is shown to be triggered thus creating an energy flux from the low to the high frequencies. The linear characteristics of the ABH plate are first analyzed. Numerical results show the appearance of overdamped modes gathered in solution branches with constant number of half-waves in the transverse direction of the ABH, seen as a waveguide. The structure of the branches is shown to be more and more prominent when increasing the width of the plate, showing a transition from beam-like to full plate structure, with a fixed value for the fundamental cut-on frequency. The combination of both contact and geometric nonlinearities to improve the ABH effect is then reported. It is shown that the coexistence of both nonlinearities provides better passive damping efficacy.

Hybrid modeling approach for mode-locked laser diodes with cavity dispersion and nonlinearity

Semiconductor-based mode-locked lasers, integrated sources enabling the generation of coherent ultra-short optical pulses, are important for a wide range of applications, including datacom, optical ranging and spectroscopy. As their performance remains largely unpredictable due to the lack of commercial design tools and the poorly understood mode-locking dynamics, significant research has focused on their modeling. In recent years, traveling-wave models have been favored because they can efficiently incorporate the rich semiconductor physics of the laser. However, thus far such models struggle to include nonlinear and dispersive effects of an extended passive laser cavity, which can play an important role for the temporal and spectral pulse evolution and stability. To overcome these challenges, we developed a hybrid modeling strategy by unifying the traveling-wave modeling technique for the semiconductor laser sections with a split-step Fourier method for the extended passive laser cavity. This paper presents the hybrid modeling concept and exemplifies for the first time the significance of the third order nonlinearity and dispersion of the extended cavity for a 2.6 GHz III–V-on-Silicon mode-locked laser. This modeling approach allows to include a wide range of physical phenomena with low computational complexity, enabling the exploration of novel operating regimes such as chip-scale soliton mode-locking.

Precision Interaction Force Control of an Underactuated Hydraulic Stance Leg Exoskeleton Considering the Constraint from the Wearer

Hydraulic lower limb exoskeletons are wearable robotic systems, which can help people carry heavy loads. Recently, underactuated exoskeletons with some passive joints have been developed in large numbers for the purpose of decreasing the weight and energy consumption of the system. There are many control algorithms for a multi-joint fully actuated exoskeleton, which cannot be applied for underactuated systems due to the reduction in the number of control inputs. Besides, since the hydraulic actuator is not a desired force output source, there exist high order nonlinearities in hydraulic exoskeletons, which makes the controller design more challenging than motor driven exoskeleton systems. This paper proposed a precision interaction force controller for a 3DOF underactuated hydraulic stance leg exoskeleton. First, the control effect of the wearer is considered and the posture of the exoskeleton back is assumed as a desired trajectory under the control of the wearer. Under this assumption, the system dynamics are changed from a 3DOF underactuated system in joint space to a 2DOF fully actuated system in Cartesian space. Then, a three-level interaction force controller is designed in which the high-level controller conducts human motion intent inference, the middle level controller tracks human motion and the low-level controller achieves output force tracking of hydraulic cylinders. The MIMO adaptive robust control algorithm is applied in the controller design to effectively address the high order nonlinearities of the hydraulic system, multi-joint couplings and various model uncertainties. A gain tuning method is also provided to facilitate the controller gains selection for engineers. Comparative simulations are conducted, which demonstrate that the principal human-machine interaction force components can be minimized and good robust performance to load change and modeling errors can be achieved.

Higher order Hamiltonian approach for solving doubly clamped beam type N/MEMS subjected to the van der Waals attraction

In this paper, Hamiltonian approach for solving vibration in the first approximation is extended into a high order Hamiltonian approach method for n order approximation. The method is applied for computation of the frequency of doubly clamped beam type N/MEMS subjected to the van der Waals attraction and the Duffing oscillator with seventh order nonlinearity. The obtained results are compared with numerically and other analytically obtained results. It is concluded that the solution in the third approximation gives the most appropriate solution in comparison to other analytical methods. The time history diagrams and the phase plane diagrams calculated analytically are on the top of the curves obtained numerically.

On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation

Spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation are frequently used to model rogue waves and are typically unstable. In this paper we study the effects of dissipation and higher order nonlinearities on the stabilization of N-mode SPBs, 1≤N≤3, in the framework of a damped higher order NLS (HONLS) equation. We observe the onset of novel instabilities associated with the development of critical states resulting from symmetry breaking in the damped HONLS system. We develop a broadened Floquet characterization of instabilities of solutions of the NLS equation by showing that instabilities are associated with degenerate complex elements of not only the discrete, but also the continuous Floquet spectrum. As a result, the Floquet criteria for the stabilization of a solution of the damped HONLS centers around the elimination of all complex degenerate elements of the spectrum. For a given initial N-mode SPB, a short-time perturbation analysis shows that the complex double points associated with resonant modes split under the damped HONLS while those associated with nonresonant modes remain closed. The corresponding /damped HONLS numerical experiments corroborate that instabilities associated with nonresonant modes persist on a longer time scale than the instabilities associated with resonant modes.

Application of the Bispectral Wavelet Analysis for Searching Three-Wave Interactions in the Spectrum of Internal Waves

Purpose. The aim of the work is to test the bispectral wavelet analysis being applied as a tool for studying resonance interactions between the frequency components in the spectrum of internal waves (based on the example both of the model signals, the shape of which is similar to that of the solitons and boras, and the field observations data on temperature fluctuations resulted from the internal waves in the Gorlo Strait of the White Sea). Methods and Results. The paper represents a technique for detecting three-wave interactions in the internal waves’ spectrum. The method is based on the bispectral wavelet analysis. It permits to identify the interharmonic correlation and the magnitude of the quadratic phase relationship arising as a result of nonlinear interactions between the signal frequency components. In the first part of the paper, efficiency of the applied method was evaluated using the example of various artificial signals with quadratic nonlinearity in order to demonstrate the method features and advantages. In its second part, the method was used to analyze the temperature profiles obtained by scanning thermohaline sounding, in which the oscillations related to passing of the internal wave groups were recorded. It is shown that the waves with the 40 min period are generated due to quadratic nonlinearity. The auto-bicoherence function values confirm the fact that the higher harmonics are formed in the 60–120 min range as a result of the three-wave interactions. They change synchronously in time, and their amplitudes are proportional, that is typical of the initial stage of the waves’ nonlinear transformation. Absence of a periodic change in the biphase sign in the considered range indicates insignificant influence of the dispersion effects upon the short-period internal waves’ structure. Conclusions. The example of observations in the Gorlo Strait of the White Sea shows that the recorded asymmetric structure of the isotherm oscillations was formed being influenced by the three-wave interaction. Possibility of further application of the method for studying the processes of the internal waves’ nonlinear transformation and breaking is discussed.

Vibrational spectra, linear and nonlinear optical investigations on 3‑chloro 4-fluoro aniline and 2-iodo aniline for optical limiting applications

Detailed investigations on the linear and nonlinear optical properties of 3‑chloro 4-fluoro aniline (CFA) and 2-iodo aniline (IA) have been carried out to understand their applicability. Natural Bond Orbital (NBO) analysis reveals the potential nonlinearity of CFA and IA by virtue of their π-conjugations in the carbon-ring systems and the n→σ* and n→π* hyperconjugations. Vibrational spectra of the molecules, investigated using the FT-IR and FT-Raman analyses, manifest the incidence of Fermi interaction and Evans hole in IA. The UV–vis spectral analysis brings out the fine photo response of CFA and IA in the visible region.The optical absorption technique has been employed to analyze the optical band gap of the title crystals, through the investigation of the absorption edge, to assess their effectiveness in optical devices.The z-scan measurement brings to light the remarkable third order nonlinearity in IA, with an extraordinary drop in normalized transmittance down to 39% at high laser fluences, which is a clear sign of its potential use as an optical limiter. A study of the molecular docking postures of CFA and IA ligands binding at the active sites of the target proteins reveal them suitable for use as therapeutic agents having anti-inflammation and anti-cancer activities against target proteins.

Primeneniye bispektral'nogo veyvlet-analiza dlya poiska trekhvolnovykh vzaimodeystviy v spektre vnutrennikh voln

Purpose. The aim of the work is to test the bispectral wavelet analysis being applied as a tool for studying resonance interactions between the frequency components in the spectrum of internal waves (based on the example both of the model signals, the shape of which is similar to that of the solitons and boras, and the field observations data on temperature fluctuations resulted from the internal waves in the Gorlo Strait of the White Sea). Methods and Results. The paper represents a technique for detecting three-wave interactions in the internal waves’ spectrum. The method is based on the bispectral wavelet analysis. It permits to identify the interharmonic correlation and the magnitude of the quadratic phase relationship arising as a result of nonlinear interactions between the signal frequency components. In the first part of the paper, efficiency of the applied method was evaluated using the example of various artificial signals with quadratic nonlinearity in order to demonstrate the method features and advantages. In the second part, the method was used to analyze the temperature profiles obtained by scanning thermohaline sounding, in which the oscillations related to passing of the internal wave groups were recorded. It is shown that the waves with the 40 min period are generated due to quadratic nonlinearity. The auto-bicoherence function values confirm the fact that the higher harmonics are formed in the 60–120 min range as a result of the three-wave interactions. They change synchronously in time, and their amplitudes are proportional, that is typical of the initial stage of the waves’ nonlinear transformation. Absence of a periodic change in the biphase sign in the considered range indicates insignificant influence of the dispersion effects upon the short-period internal waves’ structure. Conclusions. The example of observations in the Gorlo Strait of the White Sea shows that the recorded asymmetric structure of the isotherm oscillations was formed being influenced by the three-wave interaction. Possibility of further application of the method for studying the processes of the internal waves’ nonlinear transformation and breaking is discussed.

Switchability of multimodal optical phases in a leaky and nonlinear quantum cavity

In the present report, the question of U(1) symmetry breaking in a system of atoms and electromagnetic fields, interacting inside a leaky cavity, filled with a nonlinear medium, is addressed. In particular, the ${Z}_{2}$ discrete symmetry of the system and the emergence of optical phases are fully discussed. For the nonlinearity of a general order, with an electromagnetic field of any number of modes, conditions under which the resulting field-field interactions destroy the U(1) (and ${Z}_{2}$) symmetry are determined. We then apply the theory to the case of a collection of two-level atoms and three electromagnetic modes. Taking one of the fields as a classically adjustable pumping one, it is demonstrated that the quantized field-field coupling profoundly depends upon the pump field strength. The presence of two competing phenomena, namely, the cavity photonic dissipation and nonlinearity, is shown to lead the system towards steady behavior. The steady-state solutions to the atomic population and field quadratures exhibit the normal and superradiant phases, depending on the strengths of pumping field and atom-field couplings. The conditions for the stability of such steady-state solutions are also discussed in detail. A notable result of the present article is that by adjusting the parameters involved in the system one can switch from the normal to electric and/or magnetic superradiant phases.

Stumped nature hyperjerk system with fractional order and exponential nonlinearity: Analog simulation, bifurcation analysis and cryptographic applications

This article presents the construction and consumption of hyperjerk system having chaotic nature with the commensurate ordered fractional derivative on rapidly digitalized emerging technologies. The system analyzed analytically using Lipschitz condition and numerically with the help of predictor corrector approaches. Originality of constructed system is confirmed using log error plots which gives satisfactory small values on finite time scale. The dynamical performances of the complex system are termed in multiple phase planes by the mean of their significant nature. Stability and bifurcation analysis around the fractional derivative is also studies for the various parameter of system to check the visibility of chaotic solution. The system is also designed in analog circuit simulator with the aid of operational amplifier, anti-parallel semiconductor diodes for validation of the system. With the help of random number generators (RNGs), binary array generated from the hyperjerk system and plugged in to the NIST 800–22 test suite to measure the randomness. The array having high randomness is used as strong cypher key for the cryptographic execution straightforwardly in both direction encryption and decryption.

Dissipative soliton dynamics and switching in split ring resonator based metamaterial with multi-photon absorption and diffusion

Compound dissipative soliton (DS) is obtained theoretically in Split Ring Resonator based metamaterial having higher order nonlinearities, multi-photon absorption and diffusion. Effects of both the multi-photon absorption and diffusion on the DS are found to be detrimental however can be compensated by a suitable external gain to stabilize the DS. The interaction dynamics of two DSs is studied for different phase difference and initial separation. The phase controlled soliton switching is demonstrated. Notably, a weak solitonic beam is demonstrated to control a strong one. The DSs are robust against the randomness of the metamaterial. The outcome of this investigation may be useful to design experiments that employ DS in higher order nonlinear metamaterials.

Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n -- 1 with n $\ge$ 3 and d $\ge$ 2. If $\epsilon$ $\ll$ 1 is the size of the initial datum, we prove that the lifespan T$\epsilon$ of solutions is O($\epsilon$ --A(n--2) --) where A $\not\equiv$ A(d, n) = 1 + 3 d--1 when n is even and A = 1 + 3 d--1 + max(4--d d--1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain T$\epsilon$ = O($\epsilon$ --6 --), much better than O($\epsilon$ --1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of $\sqrt$ $\Delta$ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.

WITHDRAWN: The effect of doped films on the third Order nonlinearity of novel organic dye

The research includes preparation of organic dyes as thin films, pure dyes as thin film, pure dyes doped with Poly methyl methacrylate (PMMA) and pure dyes doped PMMA with titanium dioxide nanoparticles (TiO 2 ) to study the linear and nonlinear optical properties. Organic dyes (pure and doped) are interesting materials for nonlinear optical (NLO) and its applications because of their high response nonlinearity and their extended spectral transparency. In this research organic compounds dyes derived from Auraintricarboxylic acid (Schiff's Base), were carried out. Thin films of solutions of dye materials have been deposited on glass substrates. The samples prepared in for concentrations 10 -3 and 10 -6 M and thin films were deposited via drop casting.