Hyperbolic Function Solutions Research Articles

A new version of trial equation method for a complex nonlinear system arising in optical fibers

In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics.

New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

Extraction of optical wave structures to the coupled fractional system in magneto-optic waveguides

In this article, truncated M-fractional coupled nonlinear Schrodinger equation (NLSE) with quadratic–cubic nonlinearity is under observation. The studied model is composed of chromatic dispersion, magneto-optic parameter and inter-modal dispersion. The NLSE is the most significant physical model to explain the fluctuations of optical soliton proliferation. The NLSEs have become more popular because of the clarity with which they explain a wide range of complex physical phenomena and the depth with which they display dynamical patterns via localized wave solutions. Optical soliton propagation in magneto-optic is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. The optical solitons are secured in the forms of bright, dark, singular and combo solitons. In addition, hyperbolic, periodic and exponential function solutions have been recovered. The modified Sardar subequation and enhanced modified extended tanh-expansion approaches recently developed integration tools are adopted in this study for securing the solutions. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. The effect of parameters have been observed by allotting suitable values and sketching the different shapes of the graphs.

Dynamical analysis of optical soliton solutions for CGL equation with Kerr law nonlinearity in classical, truncated [formula omitted]-fractional derivative, beta fractional derivative, and conformable fractional derivative types

The study of optical soliton solutions plays a vital role in nonlinear optics. The foremost area of optical solitons research encompasses around optical fiber, telecommunication, meta-surfaces and others related technologies. The aim of this work is to integrate optical soliton solutions of the complex Ginzburg-Landau (CGL) model with Kerr law nonlinearity, also showing the effect of diverse fraction derivative and comparing it with the classical form. Here, the local derivative is used as the conformable wisdom known as the truncated M-fractional derivative, beta fractional derivative, and conformable fraction derivative. We also deliberated on some assets satisfied by the derivative. The CGL model is useful to describe the light propagation in optical communications, optical transmission, and nonlinear optical fiber. Under the right circumstances, the affectionate unified scheme is implemented for the complex Ginzburg-Landau model to generate the optical wave pattern. For α1=2α2, the unified scheme generates the solution of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme provides some novel optical solitons such as periodic waves, periodic with rogue waves, breather waves, different types of periodic rogue waves, a singular soliton solution, and rogue with the periodic wave for the special value of the free parameters. For α2=-ρ+6α18, the unified scheme generates the solutions of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme offers some fresh optical solitons such as periodic rogue waves, multi-rogue waves, double periodic waves, and periodic waves. In numerical argument, the wave patterns are offered with 3-D and density plots. To test the stability of the obtained solutions, we show diverse fractional forms such as beta time fractional, and conformable time fractional derivative and compare these fractional derivatives with their classical form in 2D plots. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of the obtained solutions, emphasizing their distinct features and depictions using unified technique. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations.

Qualitative analysis and explicit solutions of perturbed Chen–Lee–Liu equation with refractive index

The main purpose of this article is to study the qualitative analysis and explicit solutions of perturbed Chen–Lee–Liu equation with refractive index. By utilizing wave transformation, two-dimensional dynamics in a plane are obtained. Using the qualitative theory of planar dynamical systems, a series of planar phase portraits are presented. By analyzing the bifurcations of these phase portraits and the characteristics of equilibrium points, some explicit solutions such as Jacobian function solutions and hyperbolic function solutions are constructed. These can further understand the dynamic behavior of perturbed Chen–Lee–Liu equation and their wave propagation.

New exact solutions of the (3+1)-dimensional double sine-Gordon equation by two analytical methods

The (3+1)-dimensional double sine-Gordon equation plays a crucial role in various physical phenomena, including nonlinear wave propagation, field theory, and condensed matter physics. However, obtaining exact solutions to this equation faces significant challenges. In this article, we successfully employ a modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods to construct new analytical solutions to the double sine-Gordon equation. These solutions can be divided into four categories like trigonometric function solutions, hyperbolic function solutions, exponential solutions, and rational solutions. Our key findings include a rich spectrum of soliton solutions, encompassing bright, dark, singular, periodic, and mixed types, showcasing the (3+1)-dimensional double sine-Gordon equation ability to model diverse wave behaviors. We uncover previously unreported complex wave structures, revealing the potential for complex nonlinear interactions within the (3+1)-dimensional double sine-Gordon equation framework. We demonstrate the modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods effectiveness in handling higher-dimensional nonlinear partial differential equations, expanding their applicability in mathematical physics. These method offers enhanced flexibility and broader solution categories compared to conventional approaches.

Comparison of Approximate Analytical and Numerical Solutions of the Allen Cahn Equation

Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.

Unveiling novel dynamics in Q-deformed Sinh-Gordon model: exploring explicit wave solutions and stability analysis

The new generalized Q-deformed Sinh-Gordon equation is investigated in this article. Numerous physical interpretations and applications exist for the said model in several scientific fields. Therefore, for analyzing the underlying physical system, it is imperative to look at the traveling wave behavior displayed by this equation. To complete our thrust three innovative integration architectures including the modified sub-equation method, the G′/(bG′+G+a) -expansion method and G′G2 -expansion method have been utilized to create precise closed form traveling wave solutions. These methods provide solutions in the form of hyperbolic function solutions, trigonometric function solutions and rational solutions. The obtained solutions are discussed with 3D surface plots and 2D line plots to visualize and support the theoretical results. Furthermore, stability for the model is also discussed. This paper discusses a major development as well as gradual improvements to previous works on the current model. Moreover, we discuss the stability analysis of the studied equation and the outcomes are visually depicted in graphical form. To authenticate the solutions provided in this manuscript, each solution undergoes a meticulous validation process.

Mathematical modeling and component generalization of (2+1)-dimensional Schwarz–Kortweg–de Vries model in shallow water waves

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of [Formula: see text]-dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF’s parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising.

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

Abundant solitons for the generalized Hirota–Satsuma couple KdV system with an efficient technique

The generalized Hirota–Satsuma Coupled Kortwedge–de Varies (GHSC–KdV) system is considered for investigation to study the exact solutions by using the extended generalized G′/G-expansion technique. The GHSC–KdV system describes the interaction of two long waves with different dispersion relations. The dynamical structures of the obtained solutions are solitary-type waves, multi-soliton waves, and singular–kink type waves. It contains information about the hyperbolic, trigonometric, and rational function solutions for the above system. Furthermore, the derived solutions of the system that comprises arbitrary functional parameters and other constant parameters can be used to enhance the advanced behavior of physical situations.

Exact solutions of Shynaray-IIA equation (S-IIAE) using the improved modified Sardar sub-equation method

In this paper, we present an innovative approach to acquire the exact solutions of the Shynaray-IIA equations (S-IIAE), by using the improved modified Sardar sub-equation method (IMSSEM). The S-IIAE are nonlinear and coupled partial differential equations that arise in various fields of physics and engineering such as optical fibers and ferromagnetic materials. The IMSSEM is applied to S-IIAE; we successfully derived exact solutions that accurately described the wave propagation behavior of the system under consideration. The obtained solutions include rational, trigonometric, and trigonometric hyperbolic function solutions. The obtained solutions are concise and offer a deeper insight into the dynamics and characteristics of the S-IIAE. Moreover, some of the new solutions to S-IIAE are plotted in different dimensions through which bright, anti-kink and bright solitary wave structures are established. The results of the study also indicated that the proposed method is a valuable approach for achieving analytical solutions to a wide range of nonlinear partial differential equations.

Traveling wave solution of (3+1)-dimensional negative-order KdV-Calogero-Bogoyavlenskii-Schiff equation

<abstract><p>We explored the (3+1)-dimensional negative-order Korteweg-de Vries-alogero-Bogoyavlenskii-Schiff (KdV-CBS) equation, which develops the classical Korteweg-de Vries (KdV) equation and extends the contents of nonlinear partial differential equations. A traveling wave transformation is employed to transform the partial differential equation into a system of ordinary differential equations linked with a cubic polynomial. Utilizing the complete discriminant system for polynomial method, the roots of the cubic polynomial were classified. Through this approach, a series of exact solutions for the KdV-CBS equation were derived, encompassing rational function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. These solutions not only simplified and expedited the process of solving the equation but also provide concrete and insightful expressions for phenomena such as optical solitons. Presenting these obtained solutions through 3D, 2D, and contour plots offers researchers a deeper understanding of the properties of the model and allows them to better grasp the physical characteristics associated with the studied model. This research not only provides a new perspective for the in-depth exploration of theoretical aspects but also offers valuable guidance for the practical application and advancement of related technologies.</p></abstract>

New soliton solutions of the conformal time derivative generalized $ q $-deformed sinh-Gordon equation

<abstract><p>In this article, our main purpose was to study the soliton solutions of conformal time derivative generalized $ q $-deformed sinh-Gordon equation. New soliton solutions have been obtained by the complete discrimination system for the polynomial method. The solutions we obtained mainly included hyperbolic function solutions, solitary wave solutions, Jacobi elliptic function solutions, trigonometric function solutions and rational function solutions. The results showed abundant traveling wave patterns of conformal time derivative generalized $ q $-deformed sinh-Gordon equation.</p></abstract>

Chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion

The main object of this paper is to study the bifurcation, chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. A traveling wave transformation is used to simplified the perturbed stochastic nonlinear Schrödinger equation into ordinary differential equation. The dynamic behavior of two-dimensional planar dynamical systems and their perturbed systems are studied, and bifurcation, phase portrait, and Poincaré section are presented. Furthermore, traveling wave solutions included Jacobian function solutions, trigonometric function solutions and hyperbolic function solutions are constructed.

Applications of complete discrimination system approach to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation with optical soliton and bifurcation analysis

In this research, the complete discriminant system (CDS) of polynomial method (CDSPM) will be applied to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) with a particular emphasis on the introduction of various optical solitons and wave structures. Our analysis of the CQNLSE-AC illustrates the importance of CDSPM, and we give dynamic results, such as critical conditions and bifurcation points for solutions. Additionally, we determine several types of optical soliton solutions, including Jacobian elliptic function (JEF), hyperbolic function, and trigonometric function solutions, and also convert JEF into solitary wave (SW) solutions.

Abundant dynamical solitary waves through Kelvin-Voigt fluid via the truncated M-fractional Oskolkov model

This effort explores the dynamic behavior of solitary wave fronts using the Oskolkov model in a truncated M-fractional derivative form. This effort has been performed through the application of modified extended tanh and a novel form of modified Kudryashov techniques. All obtained solutions manifest in the form of sinusoidal, hyperbolic, and constant basis function solutions within the fractional model. Within this framework, the applied techniques yield various solitary wave structures, including bright and dark bell solitons, kinks, anti-kinks, linked lumps, and collisions of kinks. Additionally, they produce singular kink solitons and lump waves. The effects of fractionality and the coefficient of the highest-order derivative are depicted in Figs. 1 through 11. Moreover, all figures are organized to illustrate the properties of the innovative soliton wave propagations. The studies have also revealed that the fractional Oskolkov model can support both fundamental and higher-order solitons, each characterized by distinct properties and highly efficient behaviors. Various dynamical solitary waves obtained here can be described transmission waves through Kelvin-Voigt fluid with real happenings as it represented by fractional form.

The new wave structures to the perturbed NLSE via Wiener process with its wide-ranging applications

This article extracts stochastic soliton waves for the perturbed nonlinear Schödinger’s equation (PNLSE) forced by multiplicative noise through the Itô sense by utilizing two unified solver methods. The presented solutions involve three types: rational function, trigonometric function, and hyperbolic function solutions. These stochastic solutions are critical for studying numerous complicated phenomena in heat transfer, new physics, and many other fields of applied science. We demonstrate the effect of multiplicative noise on the solution of the stochastic PNLSE, which have never been studied before. The study and acquired solutions clarify that the unified solver technique is sturdy and efficient. Finally, several 2D and 3D graphs for selected solutions are shown.

Further advanced investigation of the complex Hirota-dynamical model to extract soliton solutions

The complex Hirota-dynamical model (CHDM) has applications in the study of plasma physics, the investigation of fusion energy, astrophysical research, and space studies. The CHDM may also be used to investigate turbulent flows to study shocks and other nonlinear phenomena, and light waves venturing through the fibers. Nowadays, plasma physics, fusion energy, astrophysical research, and space studies are very interesting topics in the modern research. So, we need to shed light on this model as a good application in these fields. For deep investigation of these physical problems, we need to find their analytical solutions. In this study, we explore a variety of soliton solutions with different geometrical structures for the CHDM via the double variable expansion method. By means of this method, we have obtained three types of soliton solutions, namely, hyperbolic, trigonometric, and rational function solutions. The graphical interpretation of these solutions gives us some popular shapes such as singular-periodic, kink, bell, and singular shapes. The performed method is an efficient technique to execute and provides reliable analytical soliton solutions which are very important to further advanced investigation of the mention equation.

The Dynamical study and analysis of diverse optical soliton solutions to Chen-Lee-Liu equation by implementing complete discrimination system for polynomial method

Optical solitons are discovered with Chen-Lee-Liu equation (CLLE). CLLE explains the transportation of optical pulses in fibre optics. The goal of the current study is to examine various analytical soliton solutions of CLLE via complete discrimination system for polynomial (CDSP) approach. These consists of elliptic, periodic, irrational, hyperbolic and exponential function solutions. This article also explores the Hamiltonian and topological properties of CLLE. The phase portraits of dynamical system provides the evidence of presence and reliability of topological properties of solutions. CDSP technique is not only suitable for obtaining soliton solutions but for also for conducting qualitative analysis. We also create 3D and 2D graphs by utilising the appropriate values of related parameters to give the physical interpretation of derived chirped solutions.