Power Law Nonlinearity Research Articles

Stationary Solutions for the Nonlinear Dispersive Schrödinger Equation with Generalized Evolution

This paper carries out the integration of the nonlinear dispersive Schrodinger equation with generalized evolution by the aid of Lie symmetry analysis. We study three types of nonlinearity in this paper. These are power law nonlinearity, dual-power law nonlinearity, and finally log law nonlinearity. From the first two types of nonlinearity the special cases of the Kerr law and parabolic law nonlinearity are easily revealed.

Solitons and other solutions to quantum Zakharov–Kuznetsov equation in quantum magneto-plasmas

In this paper, the soliton solutions to the quantum Zakharov–Kuznetsov equation in quantum dusty plasmas have been investigated. A few integration tools have been applied to extract the soliton solutions to the governing equation. The topological, non-topological, singular soliton and complexiton solutions are obtained. Additionally, the periodic and singular periodic solutions are listed. Finally, the ansatz method has been employed to retrieve a singular 1-soliton solution to the equation for power law nonlinearity. In this case, a constraint condition naturally falls out and it serves as a pre-conditioning criteria for singular soliton to exist.

Topological Soliton Solution and Bifurcation Analysis of the Klein-Gordon-Zakharov Equation in(1+1)-Dimensions with Power Law Nonlinearity

This paper addresses the Klein-Gordon-Zakharov equation with power law nonlinearity in (1+1)-dimensions. The integrability aspect as well as the bifurcation analysis is studied in this paper. The numerical simulations are also given where the finite difference approach was utilized. There are a few constraint conditions that naturally evolve during the course of derivation of the soliton solutions. These constraint conditions must remain valid in order for the soliton solution to exist. For the bifurcation analysis, the phase portraits are also given.

Mixed Variational Formulation and Numerical Analysis of Thermally Coupled Nonlinear Darcy Flows

A mixed variational formulation and its finite element approximate procedure combined with a fixed point algorithm is presented for solving a thermally coupled nonlinear Darcy flow problem. Existence and uniqueness of the solution to the nonlinear mixed variational formulation are established. A more general result on uniqueness of the continuous problem is presented. Numerical analysis of the finite element approximation with the corresponding error estimates is given. Numerical results are presented confirming the expected rates of convergence and illustrating the influence of the exponent of the nonlinear power law constitutive equation.

Visual motion integration is mediated by directional ambiguities in local motion signals

The output of primary visual cortex (V1) is a piecemeal representation of the visual scene and the response of any one cell cannot unambiguously guide sensorimotor behavior. It remains unsolved how subsequent stages of cortical processing combine (“pool”) these early visual signals into a coherent representation. We (Webb et al., 2007, 2011) have shown that responses of human observers on a pooling task employing broadband, random dot motion can be accurately predicted by decoding the maximum likelihood direction from a population of motion-sensitive neurons. Whereas Amano et al. (2009) found that the vector average velocity of arrays of narrowband, two-dimensional (2-d) plaids predicts perceived global motion. To reconcile these different results, we designed two experiments in which we used 2-d noise textures moving behind spatially distributed apertures and measured the point of subjective equality between pairs of global noise textures. Textures in the standard stimulus moved rigidly in the same direction, whereas their directions in the comparison stimulus were sampled from a set of probability distributions. Human observers judged which noise texture had a more clockwise (CW) global direction. In agreement with Amano and colleagues, observers' perceived global motion coincided with the vector average stimulus direction. To test if directional ambiguities in local motion signals governed perceived global direction, we manipulated the fidelity of the texture motion within each aperture. A proportion of the apertures contained texture that underwent rigid translation and the remainder contained dynamic (temporally uncorrelated) noise to create locally ambiguous motion. Perceived global motion matched the vector average when the majority of apertures contained rigid motion, but with increasing levels of dynamic noise shifted toward the maximum likelihood direction. A class of population decoders utilizing power-law non-linearities can accommodate this flexible pooling.

Soliton solution and bifurcation analysis of the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation with power law nonlinearity

This paper addresses the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation with power law nonlinearity. First the soliton solution is obtained by the aid of traveling wave hypothesis and along with it the constraint conditions fall out naturally, in order for the soliton solution to exist. Subsequently, the bifurcation analysis of this equation is carried out and the fixed points are obtained. The phase portraits are also analyzed for the existence of other solutions.

Painlevé property and exact solutions for a nonlinear wave equation with generalized power-law nonlinearities

By employing a variety of techniques, we investigate several classes of solutions of a family of nonlinear partial differential equations (NLPDEs) with generalized nonlinearities, special cases of which include the Klein–Gordon equation, the Landau–Ginzburg–Higgs equation, the φ4 and φ6 equations, the Rayleigh wave equation. The Painlevé property for our class of equations is studied first, showing that there are integrable families of such equations satisfying the strong Painlevé property (under a traveling wave assumption). From the truncated Laurent expansions, we introduce the auto-Bäcklund transformation for the two families shown to admit the strong Painlevé property. A multi-parameter family of exact solutions is then constructed from these auto-Bäcklund transformations for each of the cases, leading to travelling wave solutions. From here, assuming only travelling wave solutions, we then discuss more general methods of obtaining travelling wave solutions for those cases which do not satisfy the strong Painlevé property. Such solutions constitute rare exact solutions to a complicated nonlinear partial differential equation.

Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity

In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.

Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.

Helmholtz Bright Spatial Solitons and Surface Waves at Power-Law Optical Interfaces

We consider arbitrary angle interactions between spatial solitons and the planar boundary between two optical materials with a single power-law nonlinear refractive index. Extensive analysis has uncovered a wide range of new qualitative phenomena in non-Kerr regimes. A universal Helmholtz-Snell law describing soliton refraction is derived using exact solutions to the governing equation as a nonlinear basis. New predictions are tested through exhaustive computations, which have uncovered substantially enhanced Goos-Hänchen shifts at some non-Kerr interfaces. Helmholtz nonlinear surface waves are analyzed theoretically, and their stability properties are investigated numerically for the first time. Interactions between surface waves and obliquely incident solitons are also considered. Novel solution behaviours have been uncovered, which depend upon a complex interplay between incidence angle, medium mismatch parameters, and the power-law nonlinearity exponent.

Nonlinear Power Laws in Stretched Flame Velocities in Finite Thickness Flames: A Numerical Study Using Realistic Chemistry

Stretched laminar flame velocities in stoichiometric finite thickness H2/air and CH4/air flames at atmospheric pressures—investigated through an implicit direct simulation method are that features the coupling of the fully compressible flow to the realistic chemistry and multicomponent transport properties. The method resolves all the chemical and flow length and time scales, which is essential for investigating highly stiff reacting flow systems with realistic chemistry. Eight flame configurations are investigated: outwardly and inwardly propagating H2/air and CH4/air in cylindrical and spherical geometries. Nonlinear power law relationships are observed between the velocity deficit S L − S n and the flame curvature 1/r u , viz with 0 < p < 1, where p depends upon the flame type and the propagation mode. (S L is the unstretched laminar flame velocity, and S n is the stretch flame velocity, r u is the flame radius of curvature.) The Markstein linear hypothesis for asymptotically thin flames corresponds to p = 1; our results show that such a linear hypothesis cannot be extended to realistic finite thickness flames under the conditions of the study. The outwardly propagating H2/air flames are an exception, displaying an entirely different anomalous non–power law relationship, which is indicative of the complex coupling between the thermochemistry and flame geometry.

A Boubaker Polynomials Expansion Scheme for Solving the Flow of Nonlinear Power-Law Fluid in Semi-Solid State

The problem of an uncompressible power-law fluid has long been the challenge in semi-solid forming area. In this paper, the flow of a power-law fluid film on an unsteady stretching surface is analyzed by the means of Boubaker Polynomials Expansion Scheme (BPES). Analytic solutions were given and compared with the numerical results for some real power-law index and the unsteadiness parameter in wide ranges. The good agreement between them showed BPES could be used effectively to solve the flow of nonlinear power-law fluid in semi-solid state.

Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method

This paper studies two nonlinear coupled evolution equations. They are the Zakharov equation and the Davey–Stewartson equation. These equations are studied by the aid of Jacobi’s elliptic function expansion method and exact periodic solutions are extracted. In addition, the Zakharov equation with power law nonlinearity is solved by traveling wave hypothesis.

The mechanism of non-linear photochemical oscillations in the mesopause region

Abstract. The mechanism of generation of 2-day photochemical oscillations in the mesopause region (80–90 km) has been studied analytically. The initial system of equations of chemical kinetics describing the temporal evolution of O, O3, H, OH and HO2 concentrations with allowance for diurnal variations of solar radiation has been simplified successively to a system of two nonlinear first-order time equations with sinusoidal external forcing. The obtained system has a minimum number of terms needed for generation of 2-day oscillations. Linearization of this system near the period-doubling threshold permits separating explicitly a particular case of the Mathieu equation ẍ + α · sin ω t · x = 0, in which the first sub-harmonic (ω/2) of the exciting force starts to grow exponentially when the amplitude of external forcing (α) exceeds its threshold value. Finally, a system of two simplest differential equations with power-law nonlinearity has been derived that allows analytical investigation of the effect of arising of reaction-diffusion waves in the mesospheric photochemical system.

Analytic Solutions of the Kadomtsev-Petviashvili Equation with Power Law Nonlinearity Using the Sine-Cosine Method

In this paper, a sine-cosine method is used to construct many periodic and solitary wave solutions to Kadomtsev-Petviashvili equation with power law nonlinearity. Many new families of exact traveling wave solutions of the Kadomtsev-Petviashvili equation with power law nonlinearity are successfully obtained.

Integral Equations for Computing AC Losses of Radially and Polygonally Arranged HTS Thin Tapes

In this paper, we derive integral equations for radially and polygonally arranged high-temperature superconductor thin tapes, and we solve them by finite-element method. The superconductor is modeled with a nonlinear power law, which allows the possibility of considering the dependence of the parameters on the magnetic field or the position. The ac losses are computed for a variety of geometrical configurations and for various values of the transport current. Differences with respect to existing analytical models, which are developed in the framework of the critical state model and only for certain values of the transport current, are pointed out.

Solitons and cnoidal waves of the Klein–Gordon–Zakharov equation in plasmas

This paper studies the Klein–Gordon–Zakharov equation with power-law nonlinearity. This is a coupled nonlinear evolution equation. The solutions for this equation are obtained by the travelling wave hypothesis method, (G′/G) method and the mapping method.

Optical solitons and conservation laws for driven nonlinear Schrödinger's equation with linear attenuation and detuning

This paper studies the optical solitons and conservation laws for the driven nonlinear Schrödinger's equation in the presence of linear attenuation and detuning. The soliton solutions are obtained, conservation laws are retrieved and the Lie symmetry generators are computed. Both Kerr law as well as the power law nonlinearity are considered.

Soliton Solutions of the Long-Short Wave Equation with Power Law Nonlinearity

This paper studies the generalized long-short wave equation with power law nonlinearity. There are several approaches that are used to solve this coupled system nonlinear evolution equations. The series solution approach yields the topological 1-soliton solution or shock wave solution. The ansatz method and the semiinverse variational principle leads to the non-topological 1-soliton of the equation. Additionally, the variational iteration method is used to study the equation. Finally, numerical simulations are also given to this equation.

Exact solutions for the nonlinear Schr�dinger equation with power law nonlinearity

Exact solutions for the nonlinear Schr�dinger equation with power law nonlinearity