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We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained.

An extended version of (3+1)-dimensional non-linear Schrödinger equation that has a cubic–quintic nonlinear component under the stochastic effects is examined in this investigation. Several stochastic exact solutions of this model is acquired through the application of the improved modified extended tanh-function scheme (IMETFS). This method offers a practical and effective approach to finding precise solutions to several kinds of nonlinear partial differential equations. In addition, these solutions include stochastic soliton solutions (bright, singular, combo dark-singular), and exact solution such as singular periodic, Jacobi elliptic function, Weierstrass elliptic doubly periodic solution, rational, and exponential functions. Since it is the first study of its sort to examine multiplicative white noise’s impacts in this particular setting, it offers fresh insights and innovative research approaches for the field’s future studies. The work adds much to our understanding of soliton theory and how it relates to optical fiber technology while illuminating hitherto unknown facets of multiplicative white noise. To illustrate the impact of the noise, a few recovered solutions with varying noise strengths are given graphically as examples.

This study examines the analytic wave solutions of a fourth-order (2+1)-dimensional nonlinear Schrödinger equation with higher-order odd and even terms using the modified Sardar sub-equation method. The modified Sardar sub-equation method has simple calculations, high accuracy, low computational effort and provides variety of solution forms. The results show a wide range of solutions including bright, dark and singular solitons as well as hyperbolic, periodic, singular periodic, and exponential solutions. Assigning suitable values to free parameters allows for the graphic representation of the dynamical wave structures of some analytical solutions using two- and three-dimensional graphics.

The higher order nonlinear Schrödinger equation (NLSE) describes ultrashort pulse propagation in optical fibres. In this paper, we discuss analytically the non-linear schrödinger equation with higher order dispersion terms up to sixth order and sextic power law of nonlinearity with the help of the improved modified extended tanh-function scheme. New families of solutions, such as dark solitons, singular solitons, rational solutions, periodic solutions and exponential type solutions, have been extracted. Extracted solutions will be useful in the development of the communication systems. Furthermore, some solutions are graphically depicted to demonstrate their physical nature by using distinct values of the parameters.

In this article, stochastic solutions of Radhakrishnan–Kundu–Lakshmanan equation (RKLE) with multiplicative noise are investigated. Studying is conducted with the aid of the modified extended direct algebraic integration technique which depended on the extended Riccati equation. To aid in our evaluation, a graphic representation of noise’s impact on the retrieved solutions is presented using MATLAB software packages.

In the present paper, the improved modified extended tanh-function method is employed to get new exact traveling wave solutions for the doubly dispersive model. Several types of solutions are obtained using the proposed method. Theses solutions are bright, dark and bright-dark combo soliton solutions. Also, hyperbolic type solutions, singular periodic wave solutions, Jacobi elliptic function solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions are obtained. Some of these solutions are reported in this paper for first time, namely, dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic solutions. Furthermore, 3D graphs of some solutions are introduced for illustrating the physical interpretation.

Optical soliton solutions in a magneto-optical waveguide and other exact solutions are investigated for the coupled system of the nonlinear Biswas–Milovic equation with Kudryashov’s law using the extended F-expansion method. Various types of solutions are extracted, such as dark soliton solutions, singular soliton solutions, a dark–singular combo soliton, singular combo soliton solutions, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions. The obtained different types of wave solutions help in obtaining nonlinear optical fibers in the future. Furthermore, some selected solutions are described graphically to demonstrate the physical nature of the obtained solutions. The results show that the current method gives effectual and direct mathematical tools for resolving the nonlinear problems in the field of nonlinear wave equations.

The coupled higher-order nonlinear Schrödinger equations (CHNLSEs) are investigated in this paper. This model mimic the actuation of the solitons through multimode fibers. The technique of improved modified extended tanh function is used to conduct the study. Many exact solutions are achieved like bright solitons, dark solitons and singular solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, hyperbolic and Jacobi elliptic solutions are raised. Using 3D and 2D graphs, the physical behavior of the produced solutions is illustrated. [ABSTRACT FROM AUTHOR]

This study is focusing on the integrable (3+1)-dimensional equation that combines the potential Kadomtsev-Petviashvili (pKP) equation with B-type Kadomtsev-Petviashvili (BKP) equation, also known as the pKP-BKP equation. The idea of combining integrable equations has the potential to produce a variety of unexpected outcomes such as resonance of solitons. This article provides a wide range of alternative exact solutions for the pKP-BKP equation in three dimensional form, including dark solitons, singular solitons, singular periodic solutions, Jacobi elliptic function (JEF) solutions, rational solutions and exponential solution. The improved modified extended (IME) tanh function method is employed to investigate these solutions. All of the obtained solutions for the investigated model are presented using the Wolfram Mathematica program. To further help in understanding the solutions' physical characteristics and dynamic structure, the article provides visual representations of some derived solutions using 2D representation in addition to the 3D graphs via symbolic computation. This article aims to use a potent strategy using a powerful scheme to derive different solutions with various structures. Additionally, the results greatly improve and enhance the literature's solutions to a combined pKP-BKP equation and allow deep understanding of the nonlinear dynamic system through different exact solutions. [ABSTRACT FROM AUTHOR]

We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained. [ABSTRACT FROM AUTHOR]

In the current work, we investigate the optical solitons and exact solutions for the conformable time fractional generalized Radhakrishnan–Kundu–Lakshmanan (GRKL)equation with the polynomial law of nonlinearity which represent the mathematical model of pulse propagation in a nonlinear medium in optical fibers. The modified extended direct algebraic method is presented to get the exact solutions for this model. Many varieties of traveling wave solutions are established like, solitons (bright, dark, singular, combo singular-dark), combo periodic solutions, periodic solutions, Jacobi elliptic solutions, exponential solutions and Weierstrass elliptic solutions. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. [ABSTRACT FROM AUTHOR]

In this article, the modified extended direct algebraic method is applied for the perturbed highly dispersive nonlinear Schrödinger equation with conformable fractional derivative and sextic-power law refractive index. Various types of solutions are extracted such as bright solitons, dark solitons, combo bright-dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions and rational solutions. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. [ABSTRACT FROM AUTHOR]

This work explores the effect of fractional derivative on the fourth-order nonlinear Schrödinger equation with Kerr law nonlinearity, a highly significant equation in the study of wave propagation in dispersive media. By employing the improved modified extended tanh-function method, a variety of optical soliton solutions are derived. These solutions including dark solitons, bright solitons, and singular solitons. Moreover, singular periodic solutions and exponential solutions are raised. These solutions offer valuable insights into the dynamic behavior of nonlinear wave phenomena. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order. Bright and dark solitons, pivotal components of our findings, play a critical role in fiber optics by facilitating the transmission of high-power optical signals with exceptional attributes such as shape preservation. These properties eliminate the need for external pulse compression, simplifying the design and operation of optical systems. The outcomes of this study contribute in advancing our knowledge of wave propagation in dispersive media and have practical implications for the development of efficient and robust optical communication technologies. [ABSTRACT FROM AUTHOR]

The use of optical fiber for communication has grown at a rapid pace due to the demands of the modern information era. In this paper, the modified extended direct algebraic method is implemented to explore optical solitons and other exact wave solutions for the coupled system of perturbed highly dispersive complex Ginzburg–Landau equation with polynomial nonlinearity law which describe the transmission of solitons in birefringent fibers. The obtained solutions include bright solitons, dark solitons, singular solitons and combo dark-singular solitons. Additionally, Jacobi elliptic function solutions, exponential solutions, and singular periodic solutions are also offered. To ensure the existence of the obtained soliton solutions, we set some constraints on the parameters. In addition, some selected solutions are visually shown to demonstrate the physical properties of the exact solutions. [ABSTRACT FROM AUTHOR]

One of the natural consequences of soliton propagation is a birefringence phenomenon occurring due to bending, twisting or rough handling of fibres, which produce a bit of detrimental effect. In this research, we find optical soliton solutions and other exact solutions for the high-order coupled nonlinear Schrödinger system which describe the transmission of solitons in birefringent fibers. The study is conducted with the aid of the improved modified extended tanh function method. A variety of distinct traveling wave solutions are furnished. The obtained solutions include dark solitons, bright solitons, and singular solitons. Additionally, hyperbolic solutions, periodic solutions, singular periodic solutions, rational solutions, exponential solutions, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic type solutions are also offered. To help readers physically grasp the acquired solutions, graphical representations of some of the extracted solutions are provided. [ABSTRACT FROM AUTHOR]

In this work, the dynamics for coupled system of dispersive nonlinear Schrödinger equation with Kudryashov's law of nonlinearity are studied with the aid of improved modified extended tanh function method. This model is used to mimic the wave propagation through magneto-optic wave guides. Various types of solutions are extracted for the proposed model. These solutions including dark solitons, bright solitons and singular solitons, Weierstrass elliptic and singular periodic solutions. Moreover, the graphical illustrations of some solutions are introduced to demonstrate the nature of them. [ABSTRACT FROM AUTHOR]

In this study, the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation (KP-BBME) is examined. KP-BBM is used as a water wave model to mimic the wave propagation for fluid flows and to describe bidirectional propagating water wave surface. Studying is conducted by applying the modified extended mapping method to construct various and novel solutions for the proposed model. These solutions including {dark, bright, and singular} solitons, Weierstrass elliptic, exponential and singular periodic solutions. The extracted solutions confirmed the efficacy and strength of the current technique. To illustrate the physical characteristics of the established solutions, 3D, 2D and contour plots are depicted for many selected solutions. [ABSTRACT FROM AUTHOR]

In this work, we investigate the optical solitons and other waves through magneto-optic waveguides with Kudryashov’s law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation factors using the modified extended mapping approach. Many classifications of solutions are established like bright solitons, dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions, rational wave, solutions, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic function solutions. Some of the extracted solutions are described graphically to provide their physical understanding of the acquired solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate soliton solutions and other exact solutions of the highly dispersive perturbed nonlinear Schrödinger equation having Kudryashov's arbitrary form with sextic-power law of refractive index and generalized non-local laws. Studying is conducted by applying the modified extended direct algebraic method, which offers several types of solutions. The extracted solutions including (combo bright-dark, singular, dark, bright) solitons, exponential solutions, rational solutions and singular periodic solutions. The extracted solutions confirmed the effectiveness and strength of the current technology. To illustrate the properties of some solutions, graphic representations of those solutions are given. This study contributes to the understanding of nonlinear wave phenomena and showcases the applicability of the modified extended direct algebraic method in obtaining exact solutions for complex nonlinear equations. The optical solitons produced in respect to this form have never been explored by the proposed technique before, and the results have never been published. [ABSTRACT FROM AUTHOR]

The improved modified extended tanh‐function approach was used to study optical stochastic soliton solutions and other exact stochastic solutions for the nonlinear Schrödinger‐Hirota equation with multiplicative white noise. The derived solutions include stochastic bright solitons, stochastic singular solitons, stochastic periodic solutions, stochastic singular periodic solutions, stochastic exponential solutions, stochastic rational solutions, and stochastic Jacobi elliptic doubly periodic solutions. Constraints on the parameters were taken into account to ensure the existence of the obtained stochastic soliton solutions. Additionally, selected solutions were presented graphically to illustrate the physical characteristics of the stochastic solutions. In this paper, we used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs. [ABSTRACT FROM AUTHOR]

The study focuses on the combination of three mathematical models, namely nonlinear Schrödinger's equation, Lakshmanan–Porsezian–Daniel model, and Sasa–Satsuma model, which is called the concatenation model. The study investigates optical solitons and other exact solutions using a modified extended mapping approach. The solutions derived include bright solitons, dark solitons, combo bright-dark soliton solutions, singular solitons, periodic and singular periodic solutions, exponential and rational solutions, and Jacobi elliptic solutions. The study considers constraints on the parameters to ensure the existence of the obtained soliton solutions. Selected solutions are depicted graphically to demonstrate their physical nature. [ABSTRACT FROM AUTHOR]

The generalised Schrödinger equation is a nonlinear equation that is frequently encountered in modern physics study. Additionally, it is an important equation for researching problems with light pulse transmission in nonlinear dispersive optical medium. Schrödinger–Hirota model is very important in communication systems because its refer to motion of solitary wave in optical fiber. In the ongoing study, the modified extended direct algebraic integration technique is applied to study the generalized Schrödinger–Hirota equation. Based on this method, various types of solutions, such as bright solitons, dark solitons, singular solitons, singular periodic solutions, exponential solutions, Weierstrass elliptic solutions and complexiton solutions are extracted. The extracted solutions confirmed the efficacy and strength of the current technique. To illustrate the physical characteristics of the established solutions, many selected solutions are visually depicted. [ABSTRACT FROM AUTHOR]

This manuscript investigates the cubic–quartic optical solitons in magneto-optical waveguides and other traveling wave solutions to coupled Biswas–Milovic equations with Kudryashov's nonlinear form. The solutions are extracted using the extended F-expansion method. Jacobi elliptical solutions, (bright–dark–singular) soliton solutions, hyperbolic solutions, periodic solutions, rational solutions, and exponential solutions are presented. moreover, a graphical illustration of a few selected results are present to clarify the potency and nature of the raised solutions. [ABSTRACT FROM AUTHOR]

In this work, we consider a concatenation model that is a concatenated version of the familiar nonlinear Schrödinger's equation, Lakshmanan–Porsezian–Daniel equation and the Sasa–Satsuma equation. With the help of the improved modified extended tanh-function method (IMETFM), new families of solutions are extracted for the proposed model. Bright solitons, dark solitons, combo bright-dark solitons, singular solitons, periodic solutions, exponential solutions, rational solutions and Jacobi elliptic solutions are obtained. The IMETFM enables to recover a wide spectrum of solutions to the governing model. The extracted solutions confirmed the efficacy and strength of the current technique. Moreover, the graphs for some solutions are presented to demonstrate their physical nature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the generalized Kundu-Eckhaus equation (KEE) with extra-dispersion, which describes the propagation of the ultra-short femtosecond pulses in an optical fiber. Many exact solutions are constructed with the aid of improved modified extended tanh-function method and a new transformation. As a result, we have found a variety of new families of exact traveling wave solutions including bright solitons, dark solitons, singular solitons, Weierstrass elliptic doubly periodic solutions, Jacobi elliptic solutions, periodic solutions and rational solutions. Physical interpretation for some of the obtained solutions are illustrated in Figures. [ABSTRACT FROM AUTHOR]

In this paper, we discussed analytically the nonlinear Schrödinger's equation having Kudryashov's quintuple power law of refractive index together with dual form of generalized nonlocal nonlinearity with the aid of extended F-expansion method. As a result, we have found a variety of new families of exact traveling wave solutions including bright soliton solutions, dark soliton solutions, singular soliton solutions, periodic solutions, Jacobi elliptic function solutions, rational solutions and exponential solutions. In this work, periodic solutions, rational solutions and exponential solutions are obtained to the governing model for the first time. For physical description of our newly obtained solutions, we have expressed them graphically to explain more efficiently the behavior of different shapes of solutions. [ABSTRACT FROM AUTHOR]

In our study we implement an improved modified extended tanh-function (IMETF) method for constructing new solitons and other types of solutions with NLSE having Kudryashov's generalized nonlinear refractive index. Bright, dark and singular optical soliton solutions have been recovered for the model by this approach. Moreover, 3D and contour graphs are plotted for more illustration. [ABSTRACT FROM AUTHOR]

Optical soliton solutions in a magneto-optical waveguide and other exact solutions are investigated for the coupled system of the nonlinear Biswas–Milovic equation with Kudryashov's law using the extended F-expansion method. Various types of solutions are extracted, such as dark soliton solutions, singular soliton solutions, a dark–singular combo soliton, singular combo soliton solutions, Jacobi elliptic solutions, periodic solutions, combo periodic solutions, hyperbolic solutions, rational solutions, exponential solutions and Weierstrass solutions. The obtained different types of wave solutions help in obtaining nonlinear optical fibers in the future. Furthermore, some selected solutions are described graphically to demonstrate the physical nature of the obtained solutions. The results show that the current method gives effectual and direct mathematical tools for resolving the nonlinear problems in the field of nonlinear wave equations. [ABSTRACT FROM AUTHOR]

Stochastic partial differential equations have wide applications in various fields of science and engineering. This paper addresses the optical stochastic solitons and other exact stochastic solutions through birefringent fibers for the Biswas–Arshed equation with multiplicative white noise using the modified extended mapping method. This model contains many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Stochastic bright soliton solutions, stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, stochastic singular soliton solutions, stochastic periodic solutions, stochastic rational solutions, stochastic Weierstrass elliptic doubly periodic solutions, and stochastic Jacobi elliptic function solutions are extracted. The constraints on the parameters are considered to guarantee the existence of these stochastic solutions. Furthermore, some of the selected solutions are described graphically to demonstrate the physical nature of the obtained solutions. [ABSTRACT FROM AUTHOR]

Optical solitons is one of the fastest growing areas of research in the field of nonlinear fiber optics. In the current work, the modified extended direct algebraic method is implemented to obtain highly dispersive solitons for the perturbed nonlinear Schrödinger's equation (NLSE) without self-phase modulation. Bright and singular soliton solutions are obtained. Also, singular periodic solutions, hyperbolic-type solutions, and exponential-type solutions are extracted. Under specific parameter values, 2D, 3D and contour graphs are introduced to visualize the model and demonstrate their accurate physical behaviors. [ABSTRACT FROM AUTHOR]

This paper is an wide account of bright solitons and periodic solutions for twin-core couplers and multiple-core couplers with polynomial law of nonlinearity and nonlinear perturbation terms using the Jacobi elliptic function expansion method. The restrictions on the parameters are considered to guarantee the existence of the obtained solitons. Moreover, graphs of some solutions are presented to show their features. [ABSTRACT FROM AUTHOR]

Our paper studies the optical solitons for the Kundu–Eckhaus (KE) equation with quintic nonlinearity and Raman effect. By applying the improved modified extended tanh-function method, many soliton solutions can be obtained such as bright soliton solutions, dark soliton solutions, and the singular soliton solution. In addition, we can obtain various types of solutions, namely, singular periodic solutions, exponential solutions, rational solutions, Jacobi elliptic solutions and Weierstrass elliptic doubly periodic solutions. Moreover, some selected solutions are illustrated graphically to show the physical nature and the characteristics of the obtained solutions. [ABSTRACT FROM AUTHOR]

In the present paper, the improved modified extended tanh-function method is employed to get new exact traveling wave solutions for the doubly dispersive model. Several types of solutions are obtained using the proposed method. Theses solutions are bright, dark and bright-dark combo soliton solutions. Also, hyperbolic type solutions, singular periodic wave solutions, Jacobi elliptic function solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions are obtained. Some of these solutions are reported in this paper for first time, namely, dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic solutions. Furthermore, 3D graphs of some solutions are introduced for illustrating the physical interpretation. [ABSTRACT FROM AUTHOR]

In this paper, the modified simple equation method and the extended simplest equation method are applied to secure optical solitons and other solutions for Radhakrishnan-Kundu Lakshmanan equation. Bright solitons, dark solitons and singular solitons are obtained. Also, periodic solutions, singular periodic solutions, rational-type solutions and hyperbolic solutions are reported. Moreover, for the physical illustration, some of the obtained solutions are represented graphically. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system of nonlinear Schrödinger equation involving the parabolic-nonlocal law of refractive index are considered. The extended simplest equation method is applied to obtain many new exact traveling wave solutions in magneto-optic waveguides. With the aid of this method, we obtain dark solitons, bright-dark solitons and singular solitons. Also, hyperbolic type solutions, trigonometric type solutions and singular periodic solutions are obtained. Furthermore, 3D plots and their 2D plots are presented for more illustration of some selected solutions. [ABSTRACT FROM AUTHOR]

In this work, the modified extended direct algebraic method is implemented to nonlinear Schrödinger equation with anti-cubic nonlinearity describing the dynamics of optical solitons in magneto-optic waveguides. Bright solitons, bright-dark combo solitons, singular solitons, dark solitons, hyperbolic solutions, rational solutions, Weierstrass solutions, Jacobi elliptic function solutions, exponential solutions, periodic solutions, singular periodic solutions are extracted. Moreover, for more physical illustration of the extracted solutions, three and two dimensional plots for the solutions are presented. [ABSTRACT FROM AUTHOR]

In this study, we rearrange the non-linear Schrödinger equation (NLSE) into a dual-mode structure, which expands it, and then we investigate the geometrical evaluation and representation of this novel model. Explicit, exact solutions are obtained by implementing the improved modified extended tanh-function (IMETF) approach. The study's findings have important ramifications for how solitons propagate in nonlinear optics. The resultant solutions can be found in several forms: singular, bright, rational, exponential, Jacobi elliptic function (JEF), Weierstrass elliptic doubly periodic solution, and singular periodic solutions. By comparing our obtained traveling wave solutions with existing literature, we demonstrate their novelty and substantive contribution to current research. The success of our approach suggests its potential applicability to address various nonlinear challenges in different fields, particularly within soliton theory, since the analyzed model appears in a multitude of applications. Besides, in order to better comprehend some of these found solutions behaviors, we showed their outlines in 3D and 2D graphs. [ABSTRACT FROM AUTHOR]