Lie symmetry analysis and invariant solutions of (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, the (3+1)-dimensional constant coefficient of Date–Jimbo–Kashiwara–Miwa (CCDJKM) equation is studied. All of the vector fields, infinitesimal generators, Lie symmetry reductions and different similarity reduction solutions are constructed. Due to the arbitrary functions in the infinitesimal generators, the (3+1)-dimensional CCDJKM equation can further be reduced to many (2+1)-dimensional partial differential equations. The explicit solutions of the similarity reduction equations, which include the quasi-periodic wave solution, the interaction solution between the periodic wave and a kink soliton and the trigonometric function solutions, are constructed with proper arbitrary function selection, and these new exact solutions are given out analytically and graphically.

Periodic wave solution of the Kundu-Mukherjee-Naskar equation in birefringent fibers via the Hamiltonian-based algorithm

The Kundu-Mukherjee-Naskar equation can be used to address certain optical soliton dynamics in the (2+1) dimensions. In this paper, we aim to find its periodic wave solution by the Hamiltonian-based algorithm. Compared with the existing results, they have a good agreement, which strongly proves the correctness of the proposed method. Finally, the numerical results are presented in the form of 3-D and 2-D plots. The results in this work are expected to shed a bright light on the study of the periodic wave solution in physics.

Study on the (3+1)-dimensional KP equation with lump solution and its collision phenomena

The (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied in this paper by constructing the Hirota bilinear form. The lump solution of the equation is obtained by bilinear form, and the conditions for the existence of the solution are obtained. The picture description of lump solution is further given. On the other hand, we also give the collision phenomena of lump solution, periodic wave solution and a single-kink soliton solution when the (3+1)-dimensional KP equation reduces to [Formula: see text] and [Formula: see text] by means of the Hirota method. The collision phenomenon is shown in the 3D plot description, the dynamic characteristics of the collision are also analyzed.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

Ion-acoustic stable oscillations, solitary, periodic and shock waves in a quantum magnetized electron–positron–ion plasma

Nonlinear stable oscillations, solitary, periodic and shock waves in electron–positron–ion (EPI) quantum plasma in the presence of an external static magnetic field are reported. The Korteweg-de Vries-Burgers (KdVB) equation is derived by the reductive perturbation technique (RPT). The wave solution gives shock waves depending on various parameters as quantum diffraction parameter (β), electron and positron Fermi temperatures, and densities of the system species. Amplitude, polarity, speed, and width of wave solutions are remarkably modified by species densities, kinematic viscosity, and the Bohm potential. Existence of stable oscillation of ion-acoustic waves (IAWs) is shown by using the concept of phase plane analysis. Stability of wave solution is analysed by examining the Bohm potential effect. In the absence of dissipation, phase plane of the considered plasma system is analysed to discuss the existence of periodic wave solution. The results of this study could be helpful for comprehension of the wave features in dense quantum plasmas, like white dwarfs, laboratory plasma as interaction experiments of intense laser-solid matter and microelectronic devices.

Solitary wave solutions of the ionic currents along microtubule dynamical equations via analytical mathematical method

In this article, a new generalized exponential rational function method (GERFM) is employed to extract new solitary wave solutions for the ionic currents along microtubules dynamical equations, which is very interested in nanobiosciences. In this article, the stability of the solutions is also studied. As a result, a variety of solitary waves are obtained with free parameters such as periodic wave solution and dark and bright solitary wave solutions. The solutions are plotted and used to describe physical phenomena of the problem. The work shows the power of GERFM. We found that the proposed method is reliable and effective and gives analytical and exact solutions.

On the exact and numerical solutions to a new (2 + 1)-dimensional Korteweg-de Vries equation with conformable derivative

The aim of this paper is to introduce a novel study of obtaining exact solutions to the (2+1) - dimensional conformable KdV equation modeling the amplitude of the shallow-water waves in fluids or electrostatic wave potential in plasmas. The reduction of the governing equation to a simpler ordinary differential equation by wave transformation is the first step of the procedure. By using the improved tan(φ/2)-expansion method (ITEM) and Jacobi elliptic function expansion method, exact solutions including the hyperbolic function solution, rational function solution, soliton solution, traveling wave solution, and periodic wave solution of the considered equation have been obtained. We achieve also a numerical solution corresponding to the initial value problem by conformable variational iteration method (C-VIM) and give comparative results in tables. Moreover, by using Maple, some graphical simulations are done to see the behavior of these solutions with choosing the suitable parameters.