Power Law Nonlinearity Research Articles

The analytical solutions of long waves over geometries with linear and nonlinear variations in the form of power-law nonlinearities with solid inclined wall

In this paper, we derive the exact analytical solutions for the long-wave equation in both linear and nonlinear power-law form depth and breadth geometries containing a solid inclined wall. Firstly, we give general information about the concept of partial reflection and its components, and formulate the solid inclined wall boundary condition. For these specific power-law forms of depth and breadth geometries, we show that in the presence of the solid inclined wall, the long-wave equation admits solutions in terms of Bessel-Z functions and the Cauchy–Euler series. Since the presence of solid vertical wall removes the singular point from the domain, the solution admits both the first and the second kind of the Bessel functions and Cauchy–Euler series terms. We derive results for the general case and also discuss their significance using six different geometries with solid inclined wall.

Optical solitons perturbation for the concatenation system with power law nonlinearity

Optical solitons perturbation for the concatenation system with power law nonlinearity

Bright soliton of the third-order nonlinear Schrödinger equation with power law of self-phase modulation in the absence of chromatic dispersion

In this article, we are interested in two principal topics. First, the bright optical soliton solutions of the third-order (1+1)-nonlinear Schrödinger equation including power law nonlinearity with inter-modal and spatio-temporal dispersions are perused by taking advantage of the new Kudryashov method. Second, the impacts of power law nonlinearity parameters on soliton attitude are investigated for acquired bright soliton form. With the proposed technique, the bright optical soliton solution is acquired, and 3D, contour, and 2D plots are depicted. Then, the impact of power law nonlinearity parameters on the soliton attitude has been successfully demonstrated. As is clear from this perusal power law parameters have an important impact on the soliton attitude, and this impact alters based on the soliton form. As regards our investigation, this form of the equation has not been studied with the power law nonlinearity in the absence of the chromatic dispersion for nonlinear models and the proposed method has not been applied the introduced equation before. It is expected that the consequences which are acquired in this study will shed light on the studies in this field.

The analytical solutions of long waves over geometries with linear and nonlinear variations in the form of power-law nonlinearities with solid vertical wall

In harbor engineering long waves cause some engineering challenges such as avoiding harbor resonance. Due to similar problems, it is necessary to comprehend and predict their dynamics. Whilst the literature has extensively researched their dynamics for a variety of geometries, the nonlinear geometries with power-law forms are not very well studied. The analytical solutions of the long-wave equation over nonlinear depth and breadth profiles with power-law forms are described by h(x)=c1xa and b(x)=c2xc, where the parameters c1, c2, a, and c are some real constants, are just recently derived by one of the authors of this paper Bayındır and Farazande, (2021). With this motivation, in this paper, we extend our work and obtain the exact analytical solutions of the long-wave equation over linear and nonlinear in the power-law form depth and breadth geometries where a solid vertical wall is present. We show that the long-wave equation admits solutions in terms of Bessel-Z functions and the Cauchy–Euler series for these particular power-law forms of depth and breadth profiles in the presence of the solid vertical wall. We present our results for the general type of the geometries considered with solid vertical walls. Additionally, six various cases for the aforementioned geometries where a solid vertical wall is present are discussed.

Exact Solutions of Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity by Modified Exp-Function Method

In this work, the Benjamin-Bona-Mahoney-Burgers equation has been examined, which includes the dual power-law nonlinearity and diffraction term. By using the modified exp-function method, the exact solutions of the governing equation have been obtained. The resulting traveling wave solutions have been found to exhibit various characteristics, such as being dark solitons, periodic, and singular, depending on the values of certain constants. To further illustrate these solutions, 3D, 2D, and contour graphs have been displayed. To the best of our knowledge, this is the first time in literature that the dark solitons, periodic soliton, and singular soliton solutions of considered equations have been obtained by utilizing the modified exp-function method.

Quiescent Bright Optical Solitons for Radhakrishnan�Kundu�Lakshmanan Equation with Nonlinear Chromatic Dispersion and Power�Law of Self�Phase Modulation by Lie Symmetry

Quiescent Bright Optical Solitons for Radhakrishnan�Kundu�Lakshmanan Equation with Nonlinear Chromatic Dispersion and Power�Law of Self�Phase Modulation by Lie Symmetry

Optical Solitons for the Dispersive Concatenation Model with Power-Law Nonlinearity by the Complete Discriminant Approach

The present paper is a sequel to the paper on the topic with Kerr law nonlinearity. For power-law nonlinearity, it was derived through balancing principle that the solitons would exist for two values of the power-law parameter,  namely n = 1 or n = 2. The current paper derives dark and singular solitons for the latter value of n since the first case was already covered in a previous report that was dedicated to address the model with Kerr law.

Superconvergence analysis of an [formula omitted]-Galerkin mixed FEM for Klein–Gordon–Zakharov equations with power law nonlinearity

This paper aims to study an H1-Galerkin mixed finite element method (FEM) for Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity. The bilinear element and zero-order Raviart–Thomas element (Q11/Q10×Q01) are taken as approximation spaces for the original variables p, ϕ and the auxiliary variables u=∇p and ψ=∇ϕ, respectively. The supercloseness and superconvergence estimates of order O(h2+τ2) are presented for p and ϕ in the H1 norm and u and ψ in the L2 norm, which improve the known results that the original variable ϕ can only reach optimal error estimate shown in numerical experiments without theoretical analysis. Here h is the subdivision parameter and τ is the time step. Finally, some numerical results are provided to support the theoretical analysis.

O tochnykh resheniyakh mnogomernoy sistemy ellipticheskikh uravneniy so stepennymi nelineynostyami

Equations and systems of elliptic type with power-law nonlinearities are considered.Such equations are found in modeling distributed robotic formations, as well as in chemical kinetics,biology, astrophysics, and many other fields. The problem of constructing multidimensionalexact solutions is studied. It is proposed to use a special type of ansatz that reduces the problemto solving systems of algebraic equations. A number of multiparameter families of new exactmultidimensional solutions (both radially symmetric and anisotropic) represented by explicitformulas are obtained. Examples are given to illustrate the exact solutions found.

Bright Optical Solitons for the Concatenation Model with Power-Law Nonlinearity: Laplace-Adomian Decomposition

This paper is a numerical study of the bright optical solitons that emerge from the concatenation model which is considered with power-law nonlinearity. The Laplace-Adomian decomposition scheme is the integration methodology adopted in the paper. The surface plots are exhibited for a range of values of the power-law parameter.

Study of optical properties of DM–DM planar interface by incorporating non-linearity and NID space

Study of the optical characteristics of materials, objects and geometries has wider role and applications in science and engineering. Reflection and transmission coefficients are used to study the optical properties of planar geometries. Primary aim of the presented work is to study the reflection and transmission of second harmonic generation in non-integer dimensional environment. Nonlinear medium in our considered geometry is of second harmonic generation type hence we only studied the second order reflection and transmission characteristics. Solution of the nonlinear wave equation is obtained by using slowly varying amplitude approximation and method of separation of variables in NID space is used to study how the presence of NID medium alter the electromagnetic waves. As a special case of the original geometry we also studied the changes in the behavior of the coefficients when an epsilon negative and a mu negative NID mediums are used as a reflected half space. In all the cases impact of the non-linearity and the noninteger dimensions on the unknown coefficients are studied separately. It is studied that higher amplitude of the reflection coefficient can be achieved by using a dielectric magnetic NID medium compared to epsilon negative NID medium. On the other hand it is also noted that for epsilon near zero medium, amplitude of the reflection coefficient reaches zero for specific values of the nonlinear permeability and incident angle whereas, for mu negative NID medium amplitude of both the coefficients increases (decreases) by increasing the values of ϵic(ϵrc). It is also noted that dimensions of the NID mediums can be used as a controlling factor for the amplitude of the coefficients by keeping the same shape. For the verification of the analytical results nonlinear power law is used.

Bifurcation Analysis and Chaotic Behavior of the Concatenation Model with Power-Law Nonlinearity

This paper presents a comprehensive analysis of the concatenation model with power-law nonlinearity. The research encompasses multiple key aspects, providing a detailed exploration of the model's behavior and implications within the context of nonlinear dynamics and optics. The study commences with an in-depth bifurcation analysis, aiming to unravel the intricate dynamics and transitions within the system. This analysis not only uncovers the system's behavior under varying conditions but also sheds light on its stability and the emergence of bifurcation phenomena. Our research delves into the retrieval of soliton solutions within the model. The exploration of solitons is of paramount significance, offering insights into localized, self-sustaining waveforms that often play a crucial role in nonlinear systems. These soliton solutions are identified, characterized, and their relevance to the model is established. The paper addresses the complex dynamics of the system in the presence of perturbation terms. By incorporating perturbations into the analysis, we elucidate how external influences impact the system's behavior and lead to chaotic phenomena. This analysis helps uncover the system's sensitivity to external factors and provides a deeper understanding of chaotic behavior.

Optical solitons and conservation laws for the dispersive concatenation model with power-law nonlinearity

Optical solitons and conservation laws for the dispersive concatenation model with power-law nonlinearity

Nonlinear stochastic wave equations in 1D with fractional Laplacian, power-law nonlinearity and additive [formula omitted]-regular noise

A qualitative study of nonlinear, 1D stochastic fractional wave equations with dissipative nonlinearities of power-law form is conducted on (t,x)∈[0,+∞)×Dutt+σ2(−uxx)α−a1u+a2‖u‖L2(D)ρu−κut=b0∂W0∂t on (t,x)∈[0,+∞)×D with D=[0,L], where positive fractional α-powers of Laplace operator are allowed, perturbed by additive space–time random noise W0 with fairly general covariance operator Q with finite trace(Q)<+∞ (i.e. Q-regularity). The Q-regular space–time noise W0, which is white in time and generally spatially correlated, is supposed to have a Fourier expansion along the eigenfunctions of the Laplace operator under nonrandom, Dirichlet- and Neumann-type homogeneous boundary conditions. We focus on studying the generalized energy functional V incorporating the fractional diffusive energy part and the energy part which is due to the viscous damping force both in continuous and discrete time. In continuous time, the technique of Fourier expansions and appropriate truncation by finite-dimensional systems leads to the control on the generalized energy functional, and hence we may verify existence, uniqueness and boundedness of moments of Fourier series solutions u. The total mean energy E[E(t)] cannot grow more than linearly in time t (with or without damping). Moreover, under the absence of damping (i.e. κ=0), E[E(t)] is governed by a kind of trace formula in case of additive (state-independent), Q-regular space–time noise W0. For numerical computations and more adequate discretization, we suggest to take nonstandard, partial-implicit midpoint-type methods for the Fourier coefficients. These semi-analytical numerical methods (approximate Fourier series) possess the property of conserving the expected, total energy with random initial data under the absence of nonlinearities. Eventually, we estimate probabilities of large fluctuations of some interesting functionals.

Optical solitons for the concatenation model: Power-law nonlinearity

The current paper retrieves optical soliton solutions to the concatenation model that is with power-law nonlinearity. The traveling wave hypothesis is the starting point. The detailed mathematical analysis leads to the cnoidal waves and snoidal waves, which eventually lead to the bright and singular soliton solutions when the modulus of ellipticity reaches the appropriate limiting value.

Revisiting the face stability of circular tunnels driven in strength nonlinearity soils

The face stability of shield-driven tunnels in soil materials with distinct strength nonlinearity is re-evaluated, based on the kinematic limit analysis theorem. The nonlinear Power-Law failure criterion is piecewise approximated by a multi-tangent technique to improve the conventional single straight-line substitution. A new multi-cone rotational mechanism formed by n curvilinear cones is proposed to accommodate the nonlinearity of the strength envelope considering the associative flow rule. The analytical solution of the critical support pressure on tunnel faces is derived from the energy equilibrium equation. Calculation results show that the multi-tangent method can improve the calculated values by up to 20% compared to the generalized tangent method. According to the back-calculated stress vectors, it is interestingly found that the stress level is not the critical factor influencing the tunnel face stability, while the shear strength parameters are the direct factor. Although the upper-bound analysis is a kinematic method without involving stress analysis, the parametric form of the strength criterion gives a chance to infer the possible stress distribution on the rupture surface and explain the variational trend of the result based on the distribution of back-calculated stress vectors.

Exact Solutions of Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity by Modified Exp-Function Method

In this work, the Benjamin-Bona-Mahoney-Burgers equation is examined which includes the dual power law nonlinearity and diffraction term. The exact solutions of governing equation are obtained by exploting the modified exp-function method. For some specific values of constants, the obtained travelling wave soluions are dark soliton, periodic and singular in nature. Also, 3-D, 2-D and contour graphical representations of obtained solutions are displayed.

Exact Solutions of Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity by Modified Exp-Function Method

In this work, the Benjamin-Bona-Mahoney-Burgers equation is examined which includes the dual power law nonlinearity and diffraction term. The exact solutions of governing equation are obtained by exploting the modified exp-function method. For some specific values of constants, the obtained travelling wave soluions are dark soliton, periodic and singular in nature. Also, 3-D, 2-D and contour graphical representations of obtained solutions are displayed.

Soliton dynamics for generalized Chafee–Infante equation with power-law nonlinearity

Soliton dynamics for generalized Chafee–Infante equation with power-law nonlinearity

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.