Narrow Results

This key purpose of this study is to investigate soliton solution of the fifth-order Sawada–Kotera and Caudrey–Dodd–Gibbon equations in the sense of time fractional local $M$-derivatives. This important goal is achieved by employing the unified method. As a result, a number of dark and rational soliton solutions to the nonlinear model are retrieved. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behavior. The results demonstrate that the presented approach is more effective in solving issues in mathematical physics and other fields.

In this work, the new nonlinear $(1+1)$-dimensional Schrödinger (NLS) equation, which describes wave propagation in optical fibers, has been successfully studied using the $(G^{\prime }/G)$-expansion method and the Unified method (UM). These techniques have been applied for the first time in extracting new analytic solutions for the mentioned NLS equation. The obtained solutions exhibit various types of optical solitons, including singular solitons, solitary waves, periodic solitons, and wave collisions. These solutions have wide-ranging applications in various fields such as fiber optics, telecommunication systems, plasma physics, hydrodynamics, and nonlinear optics. The behavior of the derived solutions has been visually represented using three-dimensional, two-dimensional, and contour plots, considering appropriate choices of the involved parameters within specific time intervals. These plots provide insights into the dynamics and characteristics of the solutions. The effectiveness, reliability, and ease of application of the $(G^{\prime }/G)$-expansion method and the UM are demonstrated through the obtained solutions. These mathematical tools have proven to be powerful in solving nonlinear partial differential equations, offering valuable contributions to the field of mathematical physics, soliton theory and nonlinear sciences.

The aim of this work is to find some intriguing optical soliton solutions in ($2+1$) dimensions. These soliton solutions including rational, dark, periodic, and elliptic solitons are discovered using the unified technique and the fractional order Local M-derivative to address the temporal fractional Kundu–Mukherjee–Naskar equation. It is the modification of familiar Nonlinear Schrödinger equation and used to describe the bending of an optical solitonic beam in the domain of nonlinear fiber optics and communication system. The obtained solutions are suggested with relevant conditions for their existence and displayed against 3D graphics. Also, to observe and identify the effect of fractional-order parameter on constructed solutions is shown through 2D graphs. The findings highlight that the suggested approach is simple, efficient and successful in determining the exact solution of models in optics, engineering, and other nonlinear sciences.

In this paper, we explore the generalized (3 + 1)-dimensional modified Kadomtsev–Petviashvili equations with variable coefficients, which are usually used in the fields of ferromagnetism, magneto-optics, plasma physics and fluid mechanics. For the sake of uncovering more physical phenomena related to this system, we consider the single-traveling-wave polynomial solutions in the light of the unified method and the double-wave polynomial solutions in line with the generalized unified method. Remarkably, the solitary-, soliton- as well as elliptic-type solutions are all discussed in these two kinds of solutions. Furthermore, the physical explanations of the solutions are given graphically and analytically for different choices of the free parameters (especially of the nonlinear coefficients of the equations that related to the physical insights). By discussing the wave propagation of each solution that we procured from the perspectives of amplitude, shape, symmetry or periodicity, we are capable of realizing the inherent characteristics of this equation commendably and discovering the correlative physical world more efficiently.

In this paper, new solutions of the time-fractional Hirota–Satsuma coupled KdV equation model the intercommunication between two long waves that have well-defined dispersion connection received successfully by the unified method, the improved $F$-expansion method and the homogeneous balance method. In contrast, these methods are simple and efficient, and can obtain different exact solutions to this equation. By symbolic calculation, polynomial solutions, hyperbolic function solutions, trigonometric function solutions, rational function solutions, etc. are acquired. Furthermore, we plot and analyze some solutions.

This paper investigates different kinds of exact solutions of variable-coefficients Chiral Schrödinger equation $\text{(VCCNLSE)}$. In this equation, the fractional quantum Hall effect edge states are described. By the unified method, we successfully get the soliton, solitary and elliptic wave solutions. Through the $(G^{\prime }/G)$-expansion and the exponential expansion methods, we acquire different traveling wave solutions. Using the extended $F$-expansion method, we attain the Jacobi elliptic function solutions. So as to get more physical information about the exact solutions of VCCNLSE, $\text{3D, 2D}$, density and contour maps are described.