Power Law Nonlinearity Research Articles

The Relationship between Nonlinearity of Asphalt Binders and Asphalt Mixture Permanent Deformation

ABSTRACT The present work deals with the influence of the nonlinear creep response of asphalt binders on asphalt mixture permanent deformation. Asphalt binders were tested using a Dynamic Shear Rheometer with cone and plate geometry. The mixtures were tested in unconfined compression using cylindrical samples. The resulting responses of binders and mixtures were fitted successfully using nonlinear power law models. Comparison of the fitted responses of the binders and mixtures indicated there is a strong correlation between the binder nonlinearity and the mixture nonlinearity. The degree of mixture nonlinearity is found to be highly defined by the binder nonlinearity, but it was also influenced by the aggregate gradation. The results highlight the importance of non-linear characterization of binders in studying permanent deformation of mixtures.

A study of optical solitons with Kerr and power law nonlinearities by He's variational principle

This paper studies optical solitons, in presence of perturbation terms, by the aid of He's variational principle. The inter-modal dispersion, self-steepening, nonlinear dispersion and Raman scattering are all treated as perturbation terms. Both Kerr law as well as power law nonlinearities are considered in this paper.

Topological and non-topological exact soliton solution of the power law KdV equation

This paper obtains the exact 1-soliton solution of the perturbed Korteweg–de Vries equation with power law nonlinearity. Both topological as well as non-topological soliton solutions are obtained. The solitary wave ansatz is used to carry out this integration. The domain restrictions are identified in the process and the parameter constraints are also obtained. Finally, the numerical simulations are implemented in the paper.

A series solution of the Cauchy problem for the generalized d-dimensional Schrödinger equation with a power-law nonlinearity

In this paper, the Cauchy problem for generalized d -dimensional Schrödinger equation with a power-law nonlinearity is studied. Three methods, homotopy analysis method (HAM), homotopy perturbation method (HPM) and Adomian decomposition method (ADM), are applied to obtain series pattern solutions of the mentioned Cauchy problem. The recurrent relations, for solving the mentioned Cauchy problem, is introduced. For some cases of given initial conditions, we obtain the closed form of the exact solutions.

Topological exact soliton solution of the power law KdV equation

This paper obtains the exact 1-soliton solution of the perturbed Korteweg–de Vries equation with power law nonlinearity. The topological soliton solutions are obtained. The solitary wave ansatz is used to carry out this integration. The domain restrictions are identified in the process and the parameter constraints are also obtained. It has been proved that topological solitons exist only when the KdV equation with power law nonlinearity reduces to simply KdV equation.

Momentum and thermal boundary layers over a two-dimensional or axisymmetric non-linear stretching surface in a stationary fluid

The laminar boundary layer flow over a non-linearly stretching two-dimensional sheet, or axisymmetric plane or the body of revolution arising from non-linear power law stretching velocity has been presented. The analysis leads to non-linear self-similar equations of Afzal and Varshney [N. Afzal, I. S. Varshney, The cooling of a low heat resistance stretching sheet moving through a fluid, Heat Mass Transfer 14 (1980) 289–293], irrespective of the fact whether the stretching generates two-dimensional or axisymmetric flow. In fact, the surface stretching parameter β contains the property of two-dimensional or axisymmetric geometry of the flow. The non-similar solutions represented by a power series are considered, where and the higher order terms lead to linear set of ordinary differential equations. The linearity of these equations have been exploited by splitting them into universal functions which can tabulated once and for all, and numerical solution of these universal functions have been tabulated (but not presented in this paper). The solution to the universal functions have been applied for the two-dimensional stretching sheet velocity U w ( x ) = U wo ( 1 + a w x ) for a w > 0 and a w < 0 have been encouraging. For self-similar equations, a series solution for a non-linear stretching of sheet with suction and blowing over a permeable surface have been also presented here. The thermal boundary layer closed form solution, series solution and the asymptotic solutions for very large and very small values of Prandtl numbers are also presented.

Topological 1-soliton solution of the Biswas–Milovic equation with power law nonlinearity

This paper studies the topological or dark solitons due to the Biswas–Milovic equation with power law nonlinearity. The coefficients of the dispersion and nonlinearity terms are time dependent. The damping (gain) term with time-dependent coefficient is also taken into consideration. The solitary wave ansatz is used to carry out the integration. The only requirement is that the coefficient of the damping term is Riemann integrable.

New exact solutions for some power law nonlinear diffusion equation

An extended auxiliary equation method for exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. This, together with a convenient characterization, affords new exact traveling wave solutions of some classes of nonlinear power law diffusion equations to be obtained.

Dark optical solitons in power law media with time-dependent coefficients

This Letter talks about the dynamics of dark optical solitons that are governed by the nonlinear Schrödinger's equation with power law nonlinearity. The solitons are considered in presence of linear attenuation, third order dispersion and self-steepening terms, all with time-dependent coefficients. The solitary wave ansatz is used to carry out the integration and an exact soliton solution is obtained. It is only necessary that these time-dependent coefficients are Riemann integrable.

Linear stability of selfsimilar solutions of unstable thin-film equations

We study the linear stability of selfsimilar solutions of long-wave unstable thin-film equations with power-law nonlinearities u_t = −(u^n u_{xxx} + u^m u_x)_x\qquad \text{for }0 &lt; n &lt; 3, \ n≤ m. Steady states, which exist for all values of m and n above, are shown to be stable if m ≤ n + 2 when 0 &lt; n ≤ 2 , marginally stable if m ≤ n + 2 when 2 &lt; n &lt; 3 , and unstable otherwise. Dynamical selfsimilar solutions are known to exist for a range of values of n when m = n + 2 . We carry out the analysis of the stability of these solutions when n= 1 and m = 3 . Spreading selfsimilar solutions are proven to be stable. Selfsimilar blowup solutions with a single local maximum are proven to be stable, while selfsimilar blowup solutions with more than one local maximum are shown to be unstable. The equations above are gradient flows of a nonconvex energy on formal infinite-dimensional manifolds. In the special case n = 1 the equations are gradient flows with respect to the Wasserstein metric. The geometric structure of the equations plays an important role in the analysis and provides a natural way to approach a family of linear stability problems.

Topological 1-soliton solution of Kadomtsev–Petviashvili equation with power law nonlinearity

This paper studies the Kadomtsev–Petviasvili equation with power law nonlinearity. Topological 1-soliton solution is obtained and the parameter domain is identified for these solitons to exist. The solitary wave ansatz is used to obtain this solution.

A study of Langmuir waves in plasmas

This paper talks about Langmuir waves in plasmas. The integration of the governing nonlinear Schrödinger’s equation with perturbation terms is carried out. Both Kerr law as well as the power law nonlinearity are considered.

Solitary wave solution of the Zakharov–Kuznetsov equation in plasmas with power law nonlinearity

In this paper, we obtain an exact 1-soliton solution of the Zakharov–Kuznetsov equation, with power law nonlinearity, by the solitary wave ansatz method. A couple of conserved quantities of this equation are also calculated by using this 1-soliton solution.

Solitons in Plasmas: A Lie Symmetry Approach

This paper talks about the stationary solitons for Langmuir waves in plasmas that are described by the Nonlinear Schrodinger’s equation with power law nonlinearity. The integration is carried out by the usage of Lie symmetry in presence of perturbation terms.

1-soliton solution of the generalized Zakharov–Kuznetsov modified equal width equation

This paper obtains the solitary wave solution of the generalized Zakharov–Kuznetsov modified equal width equation. The solitary wave ansatz method is used to carry out the integration of this equation. A couple of conserved quantities are calculated. The domain restriction is identified for the power law nonlinearity parameter.

Solitary waves for power-law regularized long-wave equation and R(m,n) equation

This paper integrates the regularized long-wave equation with power-law nonlinearity using the solitary-wave ansatz. A few of the conserved quantities are calculated by using the 1-soliton solution. This technique is then extended to obtain the solitary-wave solution of the R(m,n) equation and a conserved quantity is also calculated for this generalized equation.

1-Soliton solution of the generalized Zakharov–Kuznetsov equation with nonlinear dispersion and time-dependent coefficients

In this Letter, the 1-soliton solution of the Zakharov–Kuznetsov equation with power law nonlinearity and nonlinear dispersion along with time-dependent coefficients is obtained. There are two models for this kind of an equation that are studied. The constraint relation between these time-dependent coefficients is established for the solitons to exist. Subsequently, this equation is again analysed with generalized evolution. The solitary wave ansatz is used to carry out this investigation.

Dynamics of solitons in plasmas for the complex KdV equation with power law nonlinearity

This paper obtains the 1-soliton solution of the complex KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The soliton perturbation theory for this equation is developed and the soliton cooling is observed for bright solitons. Finally, the dark soliton solution is also obtained for this equation.

Topological and Non-topological Solitons for the Generalized Zakharov-Kuznetsov Modified Equal Width Equation

This paper obtains the topological and non-topological solitary wave solution of the generalized Zakharov-Kuznetsov modified equal width equation. The solitary wave ansatz method is used to carry out the integration of this equation. A couple of conserved quantities are calculated for the non-topological solitons. The domain restriction is identified for the power law nonlinearity parameter.

Optical solitons in 1+2 dimensions with non-Kerr law nonlinearity

This paper integrates the nonlinear Schrodinger’s equation in 1+2 dimensions with power and dual-power law nonlinearity. An exact 1-soliton solution is obtained in closed form using the solitary wave ansatze for these two laws. The special cases of Kerr law and parabolic law naturally fall out of these two laws.