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One of the most important physical models for describing the dynamics of optical soliton propagation in the theory of optical fibers is the nonlinear Schrödinger equation (NLSE). Since there are so many uses for ultrafast signal routing systems and brief light pulses in telecommunication, optical soliton propagation in nonlinear optical fibers is a subject of intense current interest. The conventional Khater approach and the extended direct algebraic method are used in this research to study a generalized unstable space–time fractional NLS model in mathematical physics. This leads to the discovery of solutions for the bright, dark, periodic, rational and elliptic functions. To illustrate the physical nature of the nonlinear model, the contour plots in 3D, 2D and 2D are produced by assigning the appropriate values to the arbitrary constants. Additionally, consideration is given to the stability analysis of the solutions to the governing equation. In the current era of communications network technology and nonlinear optics, the applied strategy appears to be a more potent and effective way for producing precise optical solutions to a number of various modern models of recent generations.

This study investigates some novel solitary wave solutions of the perturbed Chen–Lee–Liu ([Formula: see text]) equation to explain the physical and dynamic behavior of the pulse in optical fiber. The perturbed [Formula: see text] equation is one of the icon models that have been derived from the well-known Schrödinger equation. Two analytical techniques are used to obtain novel solitary wave solutions. Then, using a well-known numerical scheme, these solutions are looked at objectively to figure out why they work. Several graphs in different styles are used to show how the pulse waves in an optical fiber are analyzed and how accurate the analysis is. The scientific novelty of this research is given by explaining the comparison figure out of our results that have been published in some recently published research papers.

In this study, three models of the nonlinear Schrödinger equation are investigated to see whether there is any novel traveling wave solutions’ structures. “The generalized Korteweg-de Vries, (2+1) dimensional Ablowitz-Kaup-Newell-Segur, and the Maccari models ”are among the systems examined. The kinetic energy operator is impacted by mass’s location and time dependence, and different representations of the kinetic energy operator in general. The innovative soliton wave solutions for the studied models are built using the generalized Riccati expansion technique. Some unique figures represent the numerical simulations of the derived solutions. Checking for accuracy and re-entering results into the models is part of validating and improving these approaches (Mathematica 13.1). AMS classification: 35J10; 35D30; 35R10; 65N15; 35Q51.

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

In this paper, the generalized exponential (GExp) method has been employed to construct novel solitary wave solutions of the nonlinear fractional biological population (FBP) model. This model is used to demonstrate the relation of the population with deaths and births. Many novel traveling wave solutions have been formulated in distinct forms such as exponential, hyperbolic and trigonometric forms. These solutions have been explained in three different axes. The first axis is plotting them in their three optional (real, imaginary and absolute value), the second axis is handling these solutions for constructing the requested conditions for applying trigonometric quintic B-spline (TQBS) scheme. The second one determines the accuracy of the obtained analytical solutions by showing the error’s value between the analytical and numerical solutions. At the same time, the third one is comparing our analytical and numerical solutions, which have recently been published that explain the paper’s contribution and novelty.

This paper investigates novel solitary wave solutions of the unidirectional Dullin–Gottwald–Holm model and employs the modified Khater (MKhat) method for studying the dynamical characterization of the prorogation of waves in shallow water. There are various solution types obtained such as kink, periodic, cone, anti-kink, etc. The accuracy of these solutions is checked by implementing He’s variational iteration technique. The analytical and numerical solutions are numerically simulated through 3D, 2D and contour plots for a visual explanation of the shallow water waves’ propagation and the match between both kinds of solutions. Additionally, the interaction between solutions is explained by some stream plots to show the local direction of the vector field at each point and a roughly uniform density throughout the property, which indicates no background scalar field. The novelty of the study’s solutions is explained by comparing it with the previously published articles.

This paper analyzes the analytical and numerical solutions’ structure of the combined mKdV equation and KdV equation (mKdV+KdV equation) using the Khater II (Khat. II) method and three accurate B-spline numerical schemes. ExCBS, SBS and TQBS numerical schemes are the numerical systems used. The handled model describes many distinct phenomena such as wave propagation of bounded particles with a harmonic force in a one-dimensional nonlinear lattice, propagation of ion-acoustic waves of small amplitude without Landau damping in plasma physics, and propagation of thermal pulse through a single sodium fluoride crystal in solid physics. Numerous examples show the relationship between quick and slow soliton, which generates phase shift. This phase shift is shown in a contour map to show the modest and colossal energy density along the path of fast and slow colliding solitons. Calculating the difference between analytical and numerical solutions shows whether they match spline-connected and distribution graphs.

In this study, the integrability conditions on mixed derivative nonlinear Schrödinger equations are the focus of this work. A. Rangwala mathematically defined these effects and dubbed this form the Rangwala–Rao equation ([Formula: see text]) in 1990. Using innovative soliton wave solutions and their interactions, we hope to better understand how dispersion affects the electric field and pulse propagation in optical fibers. Generalized Khater (GKhat.) provides unique solitary wave solutions to the [Formula: see text] problem. The pulses’ dynamical behavior through optical fibers is seen in these numerical simulations. The originality of the paper’s conclusions may be seen by contrasting our findings with those of other researchers.

This study investigates the dynamical attitude of a nonlinear elastic circular rod’s longitudinal oscillation with lateral inertia and finite radius. This model was derived in 1986 by Wei and Gui-tong with a fourth-order nonlinear mixed derivative. The axial symmetry of this model has been thought through by using cylindrical coordinates. Furthermore, the strain and kinetic energy in the length unit of the rod have been determined. Two recent computational (extended Fan-expansion (EFE) and generalized rational (GR)) techniques are employed to construct some novel solitary wave solutions. The soliton wave solutions are obtained using Mathematica 13 software and are given with the distinct physical properties of trigonometric, hyperbolic and rational solution species. The stability of the investigated model and the obtained solutions through the suggested two analytical schemes are tested. Putting different values of the parameters explains these solutions through some numerical simulations in two-dimensional, three-dimensional and contour plots.

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

This research paper investigates novel explicit wave solutions of the fractional Korteweg–de Vries (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. These models are used as gravity models in water and an interaction model between the long waves. The Atangana–Baleanu ([Formula: see text]) fractional operator is utilized for the first time to convert the fractional form of both models into nonlinear partial differential equations with an integer order. The extended simplest equation method is employed to construct some distinct types of solitary wave solutions such as exponential, rational, hyperbolic and trigonometric functions. For more illustration of our obtained solutions, some figures for them are given. The power and practical properties of the used method are tested.

In this research, a particular attention has been given on the fractional Fokas-Lenells equation via Atangana-Baleanu derivative operator that describe the propagation of short pulses in optical fibers. The integrability properties has been investigated while using the modified Khater method that we present in details. As a result, number of new soliton solutions are obtained along with constraints on some parameters that are introduced accordingly. For further details about our obtained solutions, some distinct types of solutions have been illustrated to explain more physical and dynamical behavior of the short wave pulses in the optical fibers.

This paper investigates the soliton wave on a free-moving fluid surface by studying the two-fluid model, which are the fourth-order Boussinesq and the modified Liouville equation. This study depends on one of the computational schemes to find exact and soliton wave solutions of these models. These solutions give novel physical properties of these waves, which enable their use in many fluid applications. In order to achieve our goal, the [Formula: see text]-expansion method is applied to these two models, and for more explanation of the physical properties of these models, some of the obtained solutions are sketched in different forms (two, three-dimensional and contour plots). Moreover, the obtained results are discussed for its novelty and how it changed from that achieved in the previous work.

New analytical wave solutions are investigated of the fractional nonlinear [Formula: see text]-dimensional Boiti–Leon–Manna–Pempinelli equation to explain more physical properties of the dynamical behavior of the incompressible fluid. A new fractional definition is used through the modified Khater method to get novel solitary wave solutions of this model. The stability property of the obtained solutions is tested to show the ability of our obtained solutions in using through the physical experiments. The novelty and advantage of the proposed method are illustrated by applying to this model. Some sketches are plotted to show more about the dynamical performance of this model.

This research paper applies the modified Khater method and the generalized Kudryashov method to the general Degasperis–Procesi (DP) equation, which is used to describe the dynamical behavior of the shallow water outflows. Some shock peakon wave solutions are obtained by using these methods. Moreover, some figures are sketched for these solutions to explain more physical properties of the general DP equation and to figure out the coincidence between different types of obtained solutions. The stability property by using the features of the Hamiltonian system is tested to some obtained solutions to show their ability for applying in the model’s applications. The obtained solutions were verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows their power and effectiveness for applying to many different forms of the nonlinear evolution equations with an integer and fractional order.