Explore the vast range of titles available.

In this paper, we use the modified exp−ψθ-function method to observe some of the solitary wave solutions for the microtubules (MTs). By treating the issues as nonlinear model partial differential equations describing microtubules, we were able to solve the problem. We then found specific solutions to the nonlinear evolution equation (NLEE) covering various parameters that are particularly significant in biophysics and nanobiosciences. In addition to the soliton-like pulse solutions, we also find the rational, trigonometric, hyperbolic, and exponential function characteristic solutions for this equation. The validity of the method we developed and the fact that it provides more solutions are demonstrated by comparison to other methods. We next use the software Mathematica 10 to generate 2D, 3D, and contour plots of the precise findings we observed using the suggested technique and the proper parameter values.

This paper aims at computing M-lump solutions for the ( 3 + 1 ) -dimensional nonlinear evolution equation. These solutions in all directions decline to an identical state obtained by employing the “long wave” limit with respect to the N-soliton solutions which are got by using the direct methods. Subsequently, we discuss the dynamic properties of the M-lump solutions which describe the multiple collisions of lumps. Based on the obtained lump solutions, the lump–kink solutions are also obtained. In addition, the periodic interactive solutions are given.

We investigate a (3 + 1)-dimensional nonlinear evolution equation which is a higher-dimensional generalization of the Korteweg–de Vries equation. On the basis of the decomposition approach, the N -antidark soliton solution on a finite background is constructed by using the Darboux transformation together with the limit technique. The asymptotic analysis for the N -antidark soliton solution is performed, and the collision between multiple antidark solitons is proved to be elastic. Under the velocity resonant mechanism, the antidark soliton molecules on the ( x ,  t ), ( y ,  t ), ( y ,  z ) and ( t ,  z ) planes are found instead of the ( x ,  y ) and ( x ,  z ) planes. Based on the three- and the four-antidark soliton solutions, the elastic collision between a soliton molecule and a common soliton and the elastic collision between two soliton molecules are analytically demonstrated, respectively. These results may be useful for the study of soliton molecules in hydrodynamics and nonlinear optics.

The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces is established. The strong convergence of this technique is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to variational inequalities, hierarchical minimization problems, and nonlinear evolution equations are included.

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models.

A number of solitary wave solutions for microtubules (MTs) are observed in this article by using the modified exp-function approach. We tackle the problem by treating the results as nonlinear RLC transmission lines, and then finding exact solutions to Nonlinear Evolution Equation (NLEE) containing parameters of particular importance in biophysics and nanobiosciences. For this equation, we find trigonometric, hyperbolic, rational, and exponential function solutions, as well as soliton-like pulse solutions. A comparison with other approach indicates the legitimacy of the approach we devised as well as the fact that our method offers extra solutions. Finally, we plot 2D, 3D and contour visualizations of the exact results that we observed using our approach using appropriate parameter values with the help of software Mathematica 10.

The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.

Two nonlinear evolution equations (NLEEs) viz Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations are derived from the dust hydrodynamic model of collisionless, unmagnetized dusty plasma with electrons following non-extensive velocity distribution, i.e. q distribution. Regular as well as singular analytic solutions of KdV and MKdV equations are obtained separately. Two singular soliton solutions of both KdV and MKdV equations are obtained separately by the Hirota Bilinear method. The efficiency and interactive approach of the Hirota Bilinear method is used to obtain the multiple singular soliton solutions for its beautiful algebraic technique. The significance of multi-singular soliton interaction has been studied for the derived NLEEs for regular as well as critical parameter KdV singular solitons and MKdV singular solitons in the critical parameter set.

In this article, we present a hyperbolic secant-squared distribution via the nonlinear evolution equation. Namely, for this equation, the probability density function of the hyperbolic secant-squared (HSS) distribution has been determined. The density of our model has a variety of shapes, including symmetric, left-skewed, and right-skewed. Eight distinct frequent list estimation methods have been proposed for estimating the parameters of our models. Additionally, these estimation techniques have been used to examine the behavior of the HSS model parameters using data sets that were generated randomly. To demonstrate how the findings may be used to model real data using the HSS distribution, we also use real data. Finally, the proposed justification can be applied to a variety of other complex physical models.