Power Law Nonlinearity Research Articles

Optical solitons of the complex Ginzburg-Landau equation having dual power nonlinear form using $\varphi^{6}$-model expansion approach

This paper employs a novel $\varphi ^{6}$-model expansion approach to get dark, bright, periodic, dark-bright, and singular soliton solutions to the complex Ginzburg-Landau equation with dual power-law non-linearity. The dual-power law found in photovoltaic materials is used to explain nonlinearity in the refractive index. The results of this paper may assist in comprehending some of the physical effects of various nonlinear physics models. For example, the hyperbolic sine arises in the calculation of the Roche limit and the gravitational potential of a cylinder, the hyperbolic tangent arises in the calculation of the magnetic moment and the rapidity of special relativity, and the hyperbolic cotangent arises in the Langevin function for magnetic polarization. Frequency values, one of the soliton's internal dynamics, are used to examine the behavior of the traveling wave. Finally, some of the obtained solitons' three-, two-dimensional, and contour graphs are plotted.

Lie Symmetry Classification, Optimal System, and Conservation Laws of Damped Klein–Gordon Equation with Power Law Non-Linearity

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity utt+α(u)ut=(uβux)x+f(u). We carried out a complete Lie symmetry classification by finding forms for α(u) and f(u). This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach.

Continuum limit related to dispersion managed nonlinear Schrödinger equations

We consider the dispersion managed nonlinear Schrödinger equation with power-law nonlinearity and its discrete version of equations with step size h∈(0,1]. We prove that the solutions of the discrete equations strongly converge in L2(R) to the solution of the dispersion managed NLS as h→0 after showing the global well-posedness of the discrete equations.

Optical solitons and conservation laws for the concatenation model: Power–law nonlinearity

By investigating the concatenation model that incorporates a power-law nonlinearity, this paper provides an in-depth analysis, and the soliton solutions are derived. By utilizing the extended tanh-function technique and the enhanced Kudryashov’s scheme, bright and singular 1-solitons are yielded, as demonstrated in this paper. The conservation laws can be obtained through the multipliers approach, which subsequently exposes the conserved quantities.

Ground states in spatially discrete non-linear Schrödinger models

In the seminal work (Weinstein 1999 Nonlinearity 12 673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on . More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed power P (i.e. l 2 mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedke et al 1994 Phys. Rev. Lett. 73 1055 and Laedke et al 1996 Phys. Rev. E 54 4299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g. l 2 supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantity . In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameter ω varies, in contrast with the corresponding continuous NLS model.

Retrieval of optical solitons for nonlinear models with Kudryashov’s quintuple power law and dual-form nonlocal nonlinearity

Retrieval of optical solitons for nonlinear models with Kudryashov’s quintuple power law and dual-form nonlocal nonlinearity

Traveling Wave Solutions and Conservation Laws of a Generalized Chaffee–Infante Equation in (1+3) Dimensions

This paper aims to analyze a generalized Chaffee–Infante equation with power-law nonlinearity in (1+3) dimensions. Ansatz methods are utilized to provide topological and non-topological soliton solutions. Soliton solutions to nonlinear evolution equations have several practical applications, including plasma physics and the diffusion process, which is why they are becoming important. Additionally, it is shown that for certain values of the parameters, the power-law nonlinearity Chaffee–Infante equation allows solitons solutions. The requirements and restrictions for soliton solutions are also mentioned. Conservation laws are derived for the aforementioned equation. In order to comprehend the dynamics of the underlying model, we graphically show the secured findings. Hirota’s perturbation method is included in the multiple exp-function technique that results in multiple wave solutions that contain new general wave frequencies and phase shifts.

Grey-Black Optical Solitons, Homoclinic Breather, Combined Solitons via Chupin Liu’s Theorem for Improved Perturbed NLSE with Dual-Power Law Nonlinearity

In this article, we consider the improved perturbed nonlinear Schrödinger Equation (IP-NLSE) with dual power law nonlinearity, which arises in optical fibers and photovoltaic-photo-refractive materials. We found grey and black optical solitons of the governing equation by means of a suitable complex envelope ansatz solution. By using the Chupin Liu’s theorem (CLT) for the grey and black solitons, we evaluated new categories of combined optical soliton (COS) solutions to the IP-NLSE. The propagation behaviors for homoclinic breathers (HB), multiwaves and M-shape solitons will be analytically examined. All new analytical solutions will be found by an ansatz function scheme and suitable transformations. Multiwave solitons have been reported by using a three-waves technique. Furthermore, two kinds of interactions for M-shape soliton through exponential functions will be examined.

A fourth-order nonlinear Schrödinger equation involving power law and weak nonlocality: Its solitary waves and modulational instability analysis

The present paper examines the dynamical features of solitary waves in a weakly nonlinear medium. More precisely, the propagation of solitary waves in a system is modeled by a fourth-order nonlinear Schrödinger equation involving diffraction, power law nonlinearity, and weak nonlocality. Several localized waves classified as bright and dark solitons to the governing model are derived using ansatz methods. It is shown how power and nonlocality coefficients affect the dynamics of bright and dark solitons. Furthermore, the modulational instability of continuous waves in the presence of such different effects is studied. The results of the current paper represent a significant advancement in exploring the propagation of solitary waves in a nonlinear medium.

Impacts of a general power law on soliton for a (2+1)-dimensional Zakharov–Kuznetsov equation in magnetized quantum plasmas

In this article, under investigation is a (2+1)-dimensional Zakharov–Kuznetsov equation with a general power law nonlinearity arising from magnetized quantum plasmas model. The analytical periodic wave, bright soliton and bright–dark soliton solutions are computed for the equation via the F-expansion method. Furthermore, the impacts of the general power law parameter on the soliton is examined in details. It is interesting that the general power law can be used to control the amplitude and propagation speed, which can determine partly the kinetic energy. Particularly, as the general power law parameter increases, the soliton’ energy will decrease.

Prediction of Seismic Bearing Capacity Considering Nonlinearity and Dilatancy by Sequential Quadratic Programming

Most of the published literature regarding bearing capacity are often focused on linear and associative soils. Concerning the intrinsic strength nonlinearity in dilatancy soils, this study investigates the problem of the seismic bearing capacity in the framework of the kinematic theorem of limit analysis. The conventional linear Mohr–Coulomb criterion is substituted with a nonlinear power law criterion to depict the nonlinearity of the soil strength. The non-associative feature of soil materials is considered by defining a nonlinear dilatancy coefficient. A generalized tangential technique is accordingly introduced to linearize the strength envelope for making the nonlinear criterion tractable in the analysis. A non-symmetrical translational failure mechanism that is comprised of several rigid wedges is used to characterize the failure of the foundation at the limit state. Moreover, the seismic action is considered by the classic pseudo-static method. Based upon the energy equilibrium theory of the upper-bound limit analysis, new analytical solutions are derived from the work-balanced equation with nonlinearity and dilatancy. This rigorous upper-bound solution is formulated as a multivariate optimization problem and is readily addressed by sequential quadratic programming (SQP). To verify the reliability of the new expressions, the present results are compared with already posted solutions and the original pseudo-dynamic solutions. The comparative results show a good agreement with previous works, and the correctness and rationality of the new analytical solutions are validated. The detailed parametric study reveals that, in the non-associative flow soils, the ultimate bearing capacity is significantly decreased with a reduction in the dilatancy coefficient. Particularly in the linear condition, namely m = 1, the larger the internal friction angle is, the more obvious the influence of the non-associative feature on the bearing capacity is.

Dispersive optical solitons and conservation laws of Radhakrishnan–Kundu–Lakshmanan equation with dual–power law nonlinearity

The current paper implements three elegant approaches to recover a complete spectrum of optical solitons to the Radhakrishnan–Kundu–Lakshmanan equation with dual–power law of nonlinear refractive index. The conservation laws are also recovered by the usage of multipliers approach. The parameter constraints for the existence of such solitons are also enumerated. The numerical simulations of the recovered soliton solutions are also presented.

Power Normalized Gammachirp Cepstral (PNGC) coefficients-based approach for robust speaker recognition

Speaker identification or recognition task aims to identify persons from their voices. This paper introduces a new feature extraction approach for robust speaker recognition named Power Normalized Gammachirp Cepstral (PNGC). Key aspect of our method is the use of a biologically motivated auditory perceptual model. For speaker modeling, we use the Gaussian Mixture Model-Universal Background Model (GMM-UBM). The proposed approach includes the following main processing steps: (1) an efficient auditory filter model based on the normalized Gammachirp auditory Filterbank to estimate the cochlea spectral behavior, (2) an environmental noise compensation bloc that employs: a medium-time power analysis, an asymmetric noise-suppression module with a temporal masking module, and a frequency smoothing module to compensate the environmental noise effects, and (3) a power-law nonlinearity. Conducted experimentations on TIMIT and Aurora datasets proved that our proposed PNGC approach achieves an improved recognition accuracy compared to the MFCC and recent PNCC methods for noisy speaker recognition.

Acoustic Radiation Forces at the Crossroads of Ultrasound Exposimetry, HIFU, and Elastography.

The applications of the acoustic radiation force (ARF) continue to multiply and extend from elastography into high-intensity focused ultrasound (HIFU), diagnostic imaging, lithotripsy, sonochemistry, levitation, and microsonics yet fundamental principles remain shrouded in mystery. A well-known and popular equation often used for calculating ARF in elastography is in conflict with the equation commonly employed for calculating ARF for determining acoustic power in radiation force balances (RFBs). Controversies have sparked debate for over a century concerning the physical mechanisms underlying ARFs. For over four decades, the science of ultrasound exposimetry has steadily improved and has provided clues in terms of accumulated data about the characteristics of transmitted ultrasound fields. Concurrently, the availability and capability of predicting these fields have improved significantly. The author draws on these sources to re-examine the physical principles behind ARFs. Conflicts are shown to stem from idealized configurations and simplistic assumptions. By more fully accounting for the pulse shape and spectrum, the effect of frequency power law attenuation, diffraction, and nonlinearity, more accurate equations are developed for ARFs for practical applications which are more consistent with exposimetry observations. Simulations compare well to corrected 1.5 MHz RFB data. While some questions await resolution, the approach presented here settles several longstanding conflicts and provides a new broadband framework for future ARF work.

Investigation of optical soliton solutions of higher-order nonlinear Schrödinger equation having Kudryashov nonlinear refractive index

Purpose:In this paper, optical solitons of higher-order nonlinear Schrödinger equation with Kudryashov’s sextic power-law of nonlinear refractive index are investigated via the direct mapping method. The considered model identifies the optical soliton pulse propagation in the optical fibers. Deriving the optical solutions of investigated model such as sextic power is critical but difficult work. The primary aim of this paper is to graphically examine the impact of power-law nonlinearity (pLawNL) and chromatic dispersion (CD) parameters clarifying self-phase modulation (SPM) in the equation on soliton behavior as well as obtaining optical soliton solutions. Methodology:To according to the used technique, we first used the complex wave transform to generate the nonlinear ordinary differential equation (NLODE) form of the nonlinear Schrödinger equation (NLSE) with Kudryashov’s sextic power-law the nonlinear refractive index (SPLawNRI). Then, we were able to produce a system of linear equations in polynomial form by using the approach. Different solution sets including the values of the parameters of the studied equation and the suggested approach were produced by solving the linear system of equations. Findings:We acquired the optical soliton solutions of the main equation after inserting the sets and wave transformation into the solution functions suggested by the approach. The constraint conditions for the related solutions were suggested. We proved that the gained solutions satisfied the NLSE with Kudryashov’s SPLawNRI under the suggested constraint conditions. Originality:We present contour, 3D and 2D depictions in various simulations in figures to comment the obtained solution functions. Besides, we investigate the effects of the power-law nonlinearity parameter that expresses SPM in the main equation and the parameter that are the group velocity dispersion (GVD) or chromatic dispersion (CD) on soliton behavior. The results suggest that the utilized approach is efficient, reliable, and powerful to be readily applied to various NLSEs with higher-order or higher pLawNLs that characterize real-life problems.

Optimal design and assessment of tuned mass damper inerter with nonlinear viscous damper in seismically excited multi-storey buildings

In recent years, the tuned mass damper inerter (TMDI) has been demonstrated in several theoretical studies to be an effective vibration absorber for the seismic protection of non-isolated buildings. Its effectiveness relies on careful tuning of the TMDI stiffness and damping properties, while its performance improves with the increase of the inertance property which is readily scalable. Nevertheless, in all previous studies, the energy dissipative TMDI element has been modelled by a linear viscous damper. Still, commercial viscous dampers display a nonlinear velocity-dependent power law behavior. In this regard, this paper investigates, for the first time in literature, the potential of the TMDI fitted with nonlinear viscous damper (NVD) for seismic response protection of multi-storey buildings. This is supported by an efficient optimal nonlinear TMDI (NTMDI) tuning approach which accounts for any absorber connectivity to the building structure and employs statistical linearization to treat the nonlinear damping term. For the special case of white-noise excited undamped buildings, optimal NTMDI tuning is derived analytically in closed-form which is shown to be sufficiently accurate for lightly damped structures. Comprehensive numerical data are presented to delineate trends of optimal NVD damping coefficient with the NVD power-law exponent and the inertance. Further, nonlinear response history analysis results pertaining to optimally tuned NTMDI application for a benchmark 9-storey steel structure demonstrate that reduced NTMDI stroke and inerter force can be achieved with negligible change in storey drifts and floor acceleration performance by adopting lower NVD exponent values, leading to practically beneficial NTMDI deployments.

Convergence from power-law to logarithm-law in nonlinear fractional Schrödinger equations

In this paper, we show a connection between fractional Schrödinger equations with power-law nonlinearity and fractional Schrödinger equations with logarithm-law nonlinearity. We prove that ground state solutions of power-law fractional equations, as p → 2+, converge to a ground state solution of logarithm-law fractional equations. In particular, we provide a new proof to the existence of a ground state of logarithm-law fractional Schrödinger equations.

Dynamical system of optical soliton parameters by variational principle (super-Gaussian and super-sech pulses)

The parameter dynamics of super-sech and super-Gaussian pulses for the perturbed nonlinear Schrödinger’s equation with power-law nonlinearity is obtained in this article. The variational principle successfully recovers this dynamical system. The details of the variational principle with the implementation of the Euler–Lagrange’s equation to the nonlinear Schrödinger’s equation with power-law of nonlinearity described in this paper have not been previously reported.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH NONLINEARITIES, THAT ARE COMPOSITIONS OF EXPONENTIAL AND REGULARLY VARYING FUNCTIOS

One of the most actual problems of the modern qualitative theory of ordinary differential equations is the study of nonlinear and, especially, significantly nonlinear non-autonomous differential equations. Among the works in this area related to establishing the asymptotic properties of solutions, the largest part consists of studies of equations with power-law nonlinearities and nonlinearities asymptotically close to power-law nonlinearities, as well as with exponential nonlinearities. The premise of these studies was the study of the Emden–Fowler equation, partial cases of which are used in nuclear physics, gas dynamics, fluid mechanics, relativistic mechanics, and other fields of natural science. The existence conditions and asymptotic representations of a sufficiently wide class of solutions of substantially nonlinear second-order differential equations are found in the paper. This class of solutions was introduced in the works of V. M. Evtukhov for equations of the Emden-Fowler type of the nth order and specified for the equation of the second order. The investigated differential equations contain nonlinearities, which are compositions of exponential and correctly variable when the argument is directed to a special point of the functions. An important difference of this class of equations is the impossibility of even asymptotically representing the nonlinearity in the form of a product of functions, each of which depended either only on the unknown function or only on the derivative of the unknown function. The class of studied solutions contains properly variable solutions of such equations. In the work, asymptotic images are obtained both for the solutions of the studied class and for their first-order derivatives.

Optical 1-soliton solutions of Lakshmanan-Porsezian-daniel equation with power-law nonlinearity by the trial equation method

Optical 1-soliton solutions of Lakshmanan-Porsezian-daniel equation with power-law nonlinearity by the trial equation method