Power Law Nonlinearity Research Articles

Bright and dark soliton solutions for a new fifth-order nonlinear integrable equation with perturbation terms

In this paper, we use the solitary wave ansatz method to carry out the integration of a new fifth-order nonlinear equation having perturbation terms. The perturbation terms that are considered are the first-order dispersion term, the power law nonlinearity term, and the fifth- order dispersion term. Both bright and dark soliton solutions are obtained. The physical parameters in the soliton solutions: amplitude, inverse width, and velocity are obtained as functions of the dependent model coefficients. The conditions of the existence of the derived solitons are presented.

Exact and traveling-wave solutions for convection–diffusion–reaction equation with power-law nonlinearity

Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection–diffusion–reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection–diffusion–reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena.

Divergence symmetries of critical Kohn-Laplace equations on Heisenberg groups

We show that every Lie point symmetry of semilinear Kohn-Laplace equations with a power-law nonlinearity on the Heisenberg group H 1 is a divergence symmetry if and only if the corresponding exponent takes a critical value.

Transmit beamforming for optimal second-harmonic generation

A simulation study of transmit ultrasound beams from several transducer configurations is conducted to compare second-harmonic imaging at 3.5 MHz and 11 MHz. Second- harmonic generation and the ability to suppress near field echoes are compared. Each transducer configuration is defined by a chosen f-number and focal depth, and the transmit pressure is estimated to not exceed a mechanical index of 1.2. The medium resembles homogeneous muscle tissue with nonlinear elasticity and power-law attenuation. To improve computational efficiency, the KZK equation is utilized, and all transducers are circular-symmetric. Previous literature shows that second-harmonic generation is proportional to the square of the transmit pressure, and that transducer configurations with different transmit frequencies, but equal aperture and focal depth in terms of wavelengths, generate identical second-harmonic fields in terms of shape. Results verify this for a medium with attenuation f1. For attenuation f1.1, deviations are found, and the high frequency subsequently performs worse than the low frequency. The results suggest that high frequencies are less able to suppress near-field echoes in the presence of a heterogeneous body wall than low frequencies.

Flow at ice‐divide triple junctions: 1. Three‐dimensional full‐Stokes modeling

Ice domes are either axisymmetric, high points along ridges, or ridge triple junctions. We model time‐dependent isothermal flow near triple junctions, solving the full set of mechanical equations with a nonlinear power law rheology. Forcing is applied through the boundary conditions, which affect flow patterns at outlets. Where such forcing is purely axisymmetric, an axisymmetric dome is formed. If a threefold symmetry in the forcing is applied, the axisymmetric dome breaks up into three ridges subtending angles of 120°. Sets of experiments where the forcing was not exactly threefold symmetric by angle or by amplitude caused the triple junction to migrate to a new steady state. Here, in steady state, the ridges join the triple junction at nearly 120°, but one ridge curves to satisfy the boundary forcing. The slope pattern in the immediate dome vicinity depends only on a dimensionless parameter, which is a function of the ice consistency, the accumulation, and the rheological power law index. Attempts to replicate the topography around Summit, Greenland, obtained a good fit with n = 3. At a triple junction the dome is really distinct from the surrounding ridges, contrary to the highest point of a single ridge divide. As a consequence, the Raymond effect is at its strongest at the dome and weakens considerably over one ice thickness as one moves away from the flow center. Along the ridges leaving the dome, the Raymond effect is still present and decreases with the ratio of the flow across and along the ridge. In the vicinity of the dome, horizontal strain rates vary strongly from uniaxial to biaxial. Large‐scale effects, represented in our model as fluxes at boundaries, seem to be the primary controls on dome position and shape.

Solutions of Zakharov-Kuznetsov equation with power law nonlinearity in (1+3) dimensions

This paper studies the Zakharov-Kuznetsov equation in (1+3) dimensions with an arbitrary power law nonlinearity. The method of Lie symmetry analysis is used to carry out the integration of the Zakharov-Kuznetsov equation. The solutions obtained are cnoidal waves, periodic solutions, singular periodic solutions, and solitary wave solutions. Subsequently, the extended tanh-function method and the G′/G method are used to integrate the Zakharov-Kuznetsov equation. Finally, the nontopological soliton solution is obtained by the aid of ansatz method. There are numerical simulations throughout the paper to support the analytical development.

Optical quasi-solitons by Lie symmetry analysis

This paper studies optical quasi-solitons by the aid of Lie group analysis. Nine types of nonlinearities are considered here. They are Kerr law, power law, parabolic law, dual-power law, polynomial law, triple-power law, saturable law, exponential law and log law nonlinearity. A closed form solution is obtained in each case.

Solitons and other nonlinear waves for the perturbed Boussinesq equation with power law nonlinearity

This paper studies the Boussinesq equation in the presence of a couple of perturbation terms. The traveling wave hypothesis is used to extract the soliton solution. Subsequently, other nonlinear wave solutions are also obtained by the aid of exponential function and G′G methods. The constraint relations are also indicated for the existence of these wave solutions.

Investigating temporal asymmetry using masking period patterns and models of peripheral auditory processing

Two experiments were conducted in conjunction with modeling to evaluate the role of peripheral nonlinearity and neural adaptation in the perception of temporally asymmetric sounds. In both experiments, maskers were broadband noises amplitude modulated with ramped and damped exponential modulators that repeated at 40 Hz. Masking period patterns (MPPs) were constructed by measuring detection threshold of a 5-ms, 1000-Hz tone burst as function of the signal's onset delay. Experiment I showed that varying modulator half-life from 1 to 16 ms led to differences in the damped and the ramped MPPs that were largest at the short half-lives and diminished at the longer half-lives. When masker level was varied (experiment II), the largest difference between ramped and damped MPPs occurred at moderate stimulus levels. Two peripheral auditory models were evaluated, one a simple auditory filter followed by a power-law nonlinearity and another, a model of auditory nerve processing [J. Acoust. Soc. Am. 126, 2390-2412 (2009)] that includes neural adaptation. Neither models predicted differences between the ramped and damped MPPs, providing indirect support that the central auditory system has a role in perceptual temporal asymmetry.

1-Soliton solution of the D( m, n) equation with generalized evolution

This paper obtains the 1-soliton solution of the nonlinear dispersive Drinfel’d–Sokolov equation with power law nonlinearity. In the first case the soliton solution is without the generalized evolution. The solitary wave ansatz method is used to carry out the integration. Subsequently, the He’s semi-inverse variational principle is used to integrate the equation with power law nonlinearity. Parametric conditions for the existence of envelope solitons are given.

Solving a system of integral equations by an analytic method

In this paper, the recurrent relation, for solving a system of linear and nonlinear Volterra and Fredholm integral equations by an analytic method, namely homotopy analysis method (HAM), is introduced. For the power-law nonlinearities, our strategy is directly applied and for the other kind of nonlinearities, it is necessary to rewrite them in the form of power-law nonlinearities. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. The efficiency of the approach will be shown by applying the procedure on some examples.

Bright and dark solitons of the Rosenau-Kawahara equation with power law nonlinearity

In this paper, the topological (dark) as well as non-topological (bright) soliton solutions to the Rosenau-Kawahara equation with power law nonlinearity are obtained by the solitary wave ansatz method. A couple of conserved quantities are also calculated for the case of bright soliton solution.

Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

A theoretical analysis is presented of the response of a lightly and nonlinearly damped mass–spring system in which the spring constant contains a small randomly fluctuating component. Damping is represented by a combination of linear and nonlinear power-law damping. System response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time. Leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of Stratonovich. The equations of amplitude and phase are given in two versions: Fokker–Planck equations for transient probability and Langevin equations for response in the time-domain. Solutions in closed-form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions. They are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle-point method. It is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping. Depending on these parameters different types of system behavior are found to be possible: response which decays exponentially to zero, response which leads to a stationary state of random behavior, and response which can either grow unboundedly or which approaches zero in a finite time.

Exact 1-soliton solution of the Zakharov equation in plasmas withpower law nonlinearity

This paper studies the Zakharov equation with power law nonlinearity. An exact 1-soliton solution is obtained by the ansatz method. The parameter regimes are identified in the process. The numerical simulation is also given to complete the study.

A Series Solution of the Cauchy Problem for Turing Reaction-diffusion Model

In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM). Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges rapidly to the exact solution of the problem. The efficiency of the approach will be shown by applying the procedure on two problems. Furthermore, the so-called homotopy-Pade technique (HPT) is applied to enlarge the convergence region and rate of solution series given by the HAM.

Modeling turbulent-bounded flow using non-Newtonian viscometric functions

Turbulent flows of Newtonian fluids have already been compared with non-Newtonian laminar flows. In this paper the analogy between these classes of flows is explored, and a new approach to derive a turbulent model based on a nonlinear constitutive equation is shown. In order to reach this aim, direct numerical simulation databases of turbulent channel flows are used and analyzed in the light of the classical parameters of non-Newtonian constitutive equations. The Reynolds stress tensor is expressed in terms of a set of basis tensors based on a projection of a nonlinear framework. The coefficients of the model are given as functions of the intensity of the mean strain tensor. The apparent turbulent viscosity, the first and second normal stress difference, are presented in function of the shear rate. A turbulent Weissenberg number, based on a characteristic turbulent time ratio of the first normal stress difference to the apparent viscosity is also presented. These material functions, exhibiting a shear-thinning behavior, are fitted with the power law (the Carreau-type) model. The range of the Reynolds number investigated was 180⩽Re τ⩽2000. One of the advantages of the new algebraic nonlinear power law constitutive equation derived in the paper is that its dependence is only on the mean velocity gradient and can be integrated up to the wall.

Universality of non-Ohmic shunt leakage in thin-film solar cells

We compare the dark current-voltage (IV) characteristics of three different thin-film solar cell types: hydrogenated amorphous silicon (a-Si:H) p-i-n cells, organic bulk heterojunction (BHJ) cells, and Cu(In,Ga)Se2 (CIGS) cells. All three device types exhibit a significant shunt leakage current at low forward bias (V<∼0.4) and reverse bias, which cannot be explained by the classical solar cell diode model. This parasitic shunt current exhibits non-Ohmic behavior, as opposed to the traditional constant shunt resistance model for photovoltaics. We show here that this shunt leakage (Ish), across all three solar cell types considered, is characterized by the following common phenomenological features: (a) voltage symmetry about V=0, (b) nonlinear (power law) voltage dependence, and (c) extremely weak temperature dependence. Based on this analysis, we provide a simple method of subtracting this shunt current component from the measured data and discuss its implications on dark IV parameter extraction. We propose a space charge limited (SCL) current model for capturing all these features of the shunt leakage in a consistent framework and discuss possible physical origin of the parasitic paths responsible for this shunt current mechanism.

Solutions to the Zakharov-Kuznetsov equation with higher order nonlinearity by mapping and ansatz methods

Solutions to the Zakharov-Kuznetsov equation with higher order nonlinearity are obtained using the mapping method. Several solutions are determined inclusing the cnoidal waves, shock waves, solitary waves, periodic singular waves and others. Finally, the ansatz method is applied to solve the equation with power law nonlinearity. It has been proved that the shock waves or topological solitons exist only for specific values of the power law parameter.

1-Soliton Solution of the Coupled Nonlinear Klein-Gordon Equations

This paper studies the coupled Klein-Gordon equations in (1+1) and (1+2) dimensions. The cubic law of nonlinearity and arbitrary power law nonlinearity are considered in this paper. The 1-soliton solution of the coupled system, for both cases, is obtained. The solitary wave ansatz is used to carry out the integration. KeyWords: Solitons; Integrability; Coupled Klein-Gordon Equations

A compact split step Padé scheme for higher-order nonlinear Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion

In this paper we propose a compact split step Padé scheme (CSSPS) to solve the scalar higher-order nonlinear Schrödinger equation (HNLS) with higher-order linear and nonlinear effects such as the third and fourth order dispersion effects, Kerr dispersion, stimulated Raman scattering and power law nonlinearity. The stability of this method has been proved. It has been shown as well that the CSSPS method gives the same results as classical numerical methods like the split step Fourier method and Crank–Nicholson (CN) method but it presents many advantages over theme. It is more efficient. This proposed scheme is well suited to higher-order dispersion effects and readily generalized for nonlinear and dispersion managed fibers. We tested this scheme for the case of the quintic nonlinearity and confirmed that this effect has no significant role on the propagation of single solitons.