Power Law Nonlinearity Research Articles

Non-topological, topological and rogue wave Soliton solutions for Sharma Tasso Olver equation

This paper proposes three types of envelope solitons for the Sharma-Tasso-Olver equation with power-law nonlinearity. The ansatz approach is used to investigate the presence of these solitons identified during the solution derivation. As a result, we acquired the bright, singular, and dark solitons, which have a wide range of applications in applied science and engineering. Furthermore, the physical structures of the obtained solitons are investigated using two-dimensional, three-dimensional, and three-dimensional revolution plots with varying parameter values.

Blow-Up of States in the Dynamics Given by the Schrödinger Equation with a Power-Law Nonlinearity in the Potential

Blow-Up of States in the Dynamics Given by the Schrödinger Equation with a Power-Law Nonlinearity in the Potential

New exact solutions to the coupled (n + 1)-dimensional Klein–Gordon–Zakharov equation with power law nonlinearities

In this paper, a special class of exact solutions of the coupled higher-dimensional Klein–Gordon–Zakharov (KGZ) equation with power law nonlinearities is constructed explicitly by separation transformation method. One merit of this work is that when the dimension is larger than one, there is an arbitrary function [Formula: see text], which is used to search for more general exact solutions of the coupled higher-dimensional KGZ equation. As a result, the Jacobi elliptic function solutions and soliton solutions with free parameters are derived, which lead to rich localized nonlinear structures of the coupled higher-dimensional KGZ equation with power law nonlinearities.

Linking Entrepreneurial Activities and Community Prosperity/Poverty in United States Counties: Use of the Enterprise Dependency Index

More than 3000 U.S. counties are used to examine a hypothesis that the enterprise dependency index (population numbers/enterprise numbers, EDI) can serve as a measure of community prosperity/poverty. The theoretical derivation of EDI is presented. Then, a slightly nonlinear relationship between the total and poor populations of the counties is recorded. Poverty is slightly more systematically concentrated in smaller counties. The foregoing indicates that poverty forms part of the demographic–socioeconomic–entrepreneurial nexus of human settlements. The EDIs and poverty rates of counties are statistically significantly and positively correlated. The nonlinear power law, however, explains only about 45% of the variation, suggesting that the two measures are not identical. Further analyses confirm the independence of the two measures. Poverty is only one part of the two-part prosperity/poverty continuum. Measurement of poverty rates seemingly ignores the economic impacts of prosperity in communities. The analyses suggest that EDI, based on the ability of communities to ‘carry’ enterprises, is a more sensitive measure of community prosperity/poverty than the poverty rate. The hypothesis that EDI is a useful measure of community prosperity/poverty is accepted. Further research is, however, needed to optimize the use of this measure.

Sequel to “cubic‐quartic optical soliton perturbation with complex Ginzburg–Landau equation by the enhanced Kudryashov's method”

This paper serves as a sequel to previously reported results on cubic-quartic solitons with complex Ginzburg–Landau equation. This work is with various forms of power law nonlinearity that structures the self-phase modulation. The enhanced Kudryashov's approach gives the soliton solution. The conservation laws are also identified for the models.

Manipulating two-dimensional solitons in inhomogeneous nonlinear Schrödinger equation with power-law nonlinearity under [formula omitted]-symmetric Rosen–Morse and hyperbolic Scarff-II potentials

We construct two-dimensional soliton solutions for the inhomogeneous nonlinear Schrödinger (NLS) equation with power-law nonlinearity in two different types of parity-time (PT)-symmetric potentials, namely Rosen–Morse and hyperbolic Scarff-II potentials, through a similarity transformation. In each case, following three different kinds of dispersion parameters are considered: (i) exponential, (ii) periodic, and (iii) hyperbolic. We investigate the impact on the dynamical characteristics of solitons by varying the strengths of the inhomogeneity parameter. We also analyse the intensity variations of solitons at different propagation distances for three distinct dispersion profiles. Further, we observe that the intensity distribution of solitons stretches in space and that the width of it increases as the value of the power-law nonlinearity parameter increases. Our findings reveal that the obtained soliton solutions can be managed with the help of the strengths of both PT-symmetric potentials and dispersion parameters.

Non-Newtonian Flow in Bifurcated Piping and Arterial Networks

Flows in multi-branch piping systems are modeled for linear Newtonian and nonlinear Power Law, Bingham Plastic and Herschel-Bulkley fluids. The nonlinear models are often used to describe multiphase fluids consisting of liquid and solid phases, including cement slurries and drilling muds in petroleum engineering. Rheological and heterogeneous multiphase effects are important to blood flows in single and bifurcated arteries, capillaries and veins – biological applications are also emphasized to highlight their increasing importance in medical research. Unfortunately, conventional engineering approaches, many oversimplified, employ idealized assumptions for total energy conservation. Others are unrealistic and based on empirical “head loss” coefficients related to fittings, roughness, bend and expansion effects. In our approach, mass conservation is assumed, leading to momentum and energy reductions and a credible predictive algorithm is devised. Bifurcation results are given for several fluid rheologies, either in closed analytical form or using numerical algorithms based on closed form solutions. When the entry flow rate and all outlet pressures are given, along with needed geometric details, solutions give pressure drop versus flow rate relations useful in pumping and power calculations. Also calculated are pressure levels that ensure safe, burst-free operations and wall viscous shear stress values that support cleaning and remediation operations. Analytical formulas are derived for circular flow cross-sections while numerical extensions are summarized for general clogged flows (applications in piping conduits of arbitrary cross-sectional geometry). Example calculations are given demonstrating the usefulness and versatility of the new methods.

Bifurcations and exact soliton solutions for generalized Dullin–Gottwald–Holm equation with cubic power law nonlinearity

Bifurcations and exact soliton solutions for generalized Dullin–Gottwald–Holm equation with cubic power law nonlinearity

Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

AbstractThis study presents numerical simulations of generalized two‐dimensional (2D) and three‐dimensional (3D) Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, which are coupled nonlinear partial differential equations. A meshless collocation method based on barycentric rational interpolation is developed for space variable of the KGZ equations. For time discretization, an explicit low storage fourth order Runge Kutta method is proposed after transforming KGZ equations to system of ordinary differential equations by introducing auxiliary variables.L∞andL2error norms for some test problems are computed. Obtained numerical results and comparisons with finite element methods indicate that barycentric rational interpolation method is an efficient method for solving multidimensional generalized KGZ system numerically.

Pseudo-spectral approach for extracting optical solitons of the complex Ginzburg Landau equation with six nonlinearity forms

In this paper, a compelling pseudo-spectral approach namely split-step Fourier transform (SSFT) is utilized for the first time to report the complex Ginzburg Landau (CGL) equation. In this regard, six nonlinear forms of this notable equation are associated with this model, which are the kerr law nonlinearity, power law nonlinearity, quadratic-cubic nonlinearity, cubic-quintic nonlinearity, dual power nonlinearity, and polynomial law nonlinearity. Since opting for an analytical solution is complicated and only provided for a limited set of soliton solutions, the numerical solution might render a more generalized aspect in addressing versatile solutions for a wide range of arbitrary input pulses. Therefore, the SSFT scheme is efficiently employed to extract a plethora of optical solitons solutions such as kink waves, bright, dark, bright-dark, singular, singular periodic solitons. The obtained results, via MATLAB, are in total agreement with the findings explored form the exact analytical solutions. As a result, this numerical technique may provide a creditable mathematical tool to solve a wide range of other nonlinear evolution partial differential equations in the mathematical physics field.

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods: the Lie symmetry analysis, the generalized Riccati equation mapping approach, and the modified Kudryashov approach, we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation (NCDR) with power-law nonlinearity and density-dependent diffusion. Lie symmetry analysis is one of the powerful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables. By the Lie group technique, we obtain one-parameter invariant transformations, determining equations and corresponding vectors for the considered convection-diffusion-reaction equation. By treating the parameters of the governing equation as constants, the applied methods yield a variety of new closed-form solutions, including inverse function solutions, periodic solutions, exponential function solutions, dark solitons, singular solitons, combo bright-singular solitons, and the combine of bright-dark solitons and dark-bright solitons. Moreover, using the Bäcklund transformation of the generalized Riccati equation and modified Kudryashov method, we can construct multiple solitons and other solutions of the considered equation. The obtained new solutions of this work demonstrate that the used approaches are powerful and effective in dealing with nonlinear equations, and that these solutions are required to explain many biological and physical phenomena. Comparing our obtained solutions of this paper with the ones obtained in the literature, we see that our solutions are new and not reported elsewhere. These newly formed soliton solutions will be more beneficial in the various disciplines of ocean engineering, plasma physics, and nonlinear sciences.

New Localized and Periodic Solutions to a Korteweg–de Vries Equation with Power Law Nonlinearity: Applications to Some Plasma Models

Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, i.e., solitary waves, to the evolution equation, two different methods are devoted for this purpose. In the first one, new hypotheses with Cole–Hopf transformation are employed to find general solitary wave solutions. In the second one, the ansatz method with hyperbolic sech algorithm are utilized to obtain a general solitary wave solution. The obtained solutions recover the solitary wave solutions to all one-dimensional KdV equations with a power law nonlinearity, such as the KdV equation with quadratic nonlinearity, the modified KdV (mKdV) equation with cubic nonlinearity, the super KdV equation with quartic nonlinearity, and so on. Furthermore, two different approaches with two different formulas for the Weierstrass elliptic functions (WSEFs) are adopted for deriving some general periodic wave solutions to the evolution equation. Additionally, in the form of Jacobi elliptic functions (JEFs), the cnoidal wave solutions to the KdV-, mKdV-, and SKdV equations are obtained. These results help many authors to understand the mystery of several nonlinear phenomena in different branches of sciences, such as plasma physics, fluid mechanics, nonlinear optics, Bose Einstein condensates, and so on.

Kink soliton behavior study for systems with power-law nonlinearity

In this study, we investigate the kink soliton dynamics for power-law nonlinear systems. Based on the F-expansion method, we first derive the novel kink soliton solution of the nonlinear Schrödinger equation (NLSE) with third-order dispersion term, power-law dependent nonlinearity term, linear attenuation term, and self-steepness term under appropriate parameter settings. With pictorial demonstration, we show that the obtained kink soliton solution not only has the soliton features of the classical NLSE, but also has power-law features. The theoretical results presented in our work can be used to guide the observation of soliton behavior in power-law dependent media.

Stationary optical solitons having Kudryashov’s quintuple power law nonlinearity by extended [formula omitted]–expansion

In this paper, stationary optical solitons to nonlinear Schrödinger’s equation having nonlinear chromatic dispersion and Kudryashov’s quintuple power law nonlinearity are researched by the help of the extended version of G′/G–expansion approach, which is one of the effective integration algorithms. Linear and generalized temporal evolutions are considered. As a consequence bright, dark and singular solitons, complexiton solutions and other solutions for the governing model are recovered.

Formation, stability, and adiabatic excitation of peakons and double-hump solitons in parity-time-symmetric Dirac-δ(x)-Scarf-II optical potentials.

We introduce a class of physically intriguing PT-symmetric Dirac-δ-Scarf-II optical potentials. We find the parameter region making the corresponding non-Hermitian Hamiltonian admit the fully real spectra, and present the stable parameter domains for these obtained peakons, smooth solitons, and double-hump solitons in the self-focusing nonlinear Kerr media with PT-symmetric δ-Scarf-II potentials. In particular, the stable wave propagations are exhibited for the peakon solutions and double-hump solitons from some given parameters even if the corresponding parameters belong to the linear PT-phase broken region. Moreover, we also find the stable wave propagations of exact and numerical peakons and double-hump solitons in the interplay between the power-law nonlinearity and PT-symmetric potentials. Finally, we examine the interactions of the nonlinear modes with exotic waves, and the stable adiabatic excitations of peakons and double-hump solitons in the PT-symmetric Kerr nonlinear media. These results provide the theoretical basis for the design of related physical experiments and applications in PT-symmetric nonlinear optics, Bose-Einstein condensates, and other relevant physical fields.

Lie Group Classification of Generalized Variable Coefficient Korteweg-de Vries Equation with Dual Power-Law Nonlinearities with Linear Damping and Dispersion in Quantum Field Theory

Many physical phenomena in fields of studies such as optical fibre, solid-state physics, quantum field theory and so on are represented using nonlinear evolution equations with variable coefficients due to the fact that the majority of nonlinear conditions involve variable coefficients. In consequence, this article presents a complete Lie group analysis of a generalized variable coefficient damped wave equation in quantum field theory with time-dependent coefficients having dual power-law nonlinearities. Lie group classification of two distinct cases of the equation was performed to obtain its kernel algebra. Thereafter, symmetry reductions and invariant solutions of the equation were obtained. We also investigate various soliton solutions and their dynamical wave behaviours. Further, each class of general solutions found is invoked to construct conserved quantities for the equation with damping term via direct technique and homotopy formula. In addition, Noether’s theorem is engaged to furnish more conserved currents of the equation under some classifications.

Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity

<abstract><p>In this paper, we consider the global existence, regularizing decay rate and asymptotic behavior of mild solutions to Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then, we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.</p></abstract>

On the exact solutions and conservation laws of a generalized (1+2)-dimensional Jaulent-Miodek equation with a power law nonlinearity

In this paper, a generalized (1+2)-dimensional Jaulent-Miodek equation with a power law nonlinearity is examined, which arises in numerous problems in nonlinear science. The computed conservation laws reside in enormously crucial areas both at the foundations of nonlinear science such as biology, physics and other related areas. Exact solutions are acquired using the Lie symmetry method. In addition to exact solutions, we also present conservation laws. The arbitrary functions in the multipliers lead to infinitely many conservation laws.

Study of power law non-linearity in solitonic solutions using extended hyperbolic function method

<abstract><p>This paper retrieves the optical solitons to the Biswas-Arshed equation (BAE), which is examined with the lack of self-phase modulation by applying the extended hyperbolic function (EHF) method. Novel constructed solutions have the shape of bright, singular, periodic singular, and dark solitons. The achieved solutions have key applications in engineering and physics. These solutions define the wave performance of the governing models. The outcomes show that our scheme is very active and reliable. The acquired results are illustrated by 3-D and 2-D graphs to understand the real phenomena for such sort of non-linear models.</p></abstract>

Langrangian formulation and solitary wave solutions of a generalized Zakharov–Kuznetsov equation with dual power-law nonlinearity in physical sciences and engineering

This paper presents analytical studies carried out explicitly on a higher-dimensional generalized Zakharov–Kuznetsov equation with dual power-law nonlinearity arising in engineering and nonlinear science. We obtain analytic solutions for the underlying equation via Lie group approach as well as direct integration method. Moreover, we engage the extended Jacobi elliptic cosine and sine amplitude functions expansion technique to seek more exact travelling wave solutions of the equation for some particular cases. Consequently, we secure, singular and nonsingular (periodic) soliton solutions, cnoidal, snoidal as well as dnoidal wave solutions. Besides, we depict the dynamics of the solutions using suitable graphs. The application of obtained results in various fields of sciences and engineering are presented. In conclusion, we construct conserved currents of the aforementioned equation via Noether’s theorem (with Helmholtz criteria) and standard multiplier technique through the homotopy formula.