Trigonometric Function Solutions Research Articles

The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model

Abstract In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

The exact solutions for the fractional Riemann wave equation in quantum mechanics and optics

In this paper, the fractional Riemann wave equation with M-truncated derivative (FRWE-MTD) is considered. The Jacobi elliptic function method and the modified extended tanh function method are applied to acquire new elliptic, rational, hyperbolic, and trigonometric functions solutions. Moreover, we expand some earlier studies. The obtained solutions are important in explaining some exciting physical phenomena, since the Riemann wave equation is used in various fields, including quantum mechanics, optics, signal processing, and general relativity. Also, this equation is used to describe the propagation of waves in various dispersive systems, where wave motion is affected by the medium through which it travels. Several 3D and 2D graphs are shown to demonstrate how the M-truncated derivative affects the exact solutions of the FRWE-MTD.

Investigation of the wave solutions of two space–time fractional equations in physics

In this study, two fractional differential equations describing the characteristics of three-dimensional nonlinear ion-acoustic waves (IAWs) in plasmas are considered with the conformable fractional derivative (CFD). 3+1-dimensional space–time fractional Jimbo–Miwa (JM) equation and 3+1-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov (mKdV-ZK) equation are reduced into ordinary differential equations and an investigation into the exact solutions is carried out through the modified exponential function method (MEFM). These exact wave solutions are beneficial in interpreting the interior structure and characteristics of physical phenomena modeled by the aforementioned nonlinear evolution equations (NLEEs) arising in magnetized plasma. Various analytical methods were applied to such equations. Therefore, it is vital to determine the proper technique for the present problem. The proposed scheme has demonstrated a different spectrum of wave structures. The portrait of the exact solutions permits the exploration of plasmas, plasma models, and optics. The software system Mathematica is employed and the rational, exponential, trigonometric, and hyperbolic function solutions are acquired. The graphical results are presented for the physical nature of the solutions. The results are compared with the integer-order models.

A Dynamical Analysis and New Traveling Wave Solution of the Fractional Coupled Konopelchenko–Dubrovsky Model

The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software.

Dimensionless dynamics: Multipeak and envelope solitons in perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

In this work, the dimensionless form of the improved perturbed nonlinear Schrödinger equation with Kerr law of fiber nonlinearity is solved for distinct exact soliton solutions. We examined the multi-wave solitons and rational solitons of the governing equation using the logarithmic transformation and symbolic computation using an ansatz functions approach. Multi-wave solitons in fluid dynamics describe the situation in which a fluid flow shows several different regions (or peaks) of high concentration or intensity of a particular variable (e.g., velocity, pressure, or vorticity). Multi-wave solitons in turbulent flows might indicate the existence of several coherent structures, like eddies or vortices. These formations are areas of concentrated energy or vorticity in the turbulent flow. Understanding how these peaks interact and change is essential to comprehending the energy cascade and dissipation in turbulent systems. Furthermore, a sub-ordinary differential equation approach is used to create solutions for the Weierstrass elliptic function, periodic function, hyperbolic function, Chirped free, dark-bright (envelope solitons), and rational solitons, as well as the Jacobian elliptic function, periodic function, and rational solitons. Also, as the Jacobian elliptic function's' modulus m approaches values of 1 and 0, we find trigonometric function solutions, solitons-like solutions, and computed chirp free-solitons. Envelope solitons can arise in stratified fluids and spread over the interface between layers, such as layers in the ocean with varying densities. Their research aids in the management and prediction of wave events in artificial and natural fluid settings. In fluids, periodic solitons are persistent, confined wave structures that repeat on a regular basis, retaining their form and velocity over extended distances. These structures occur in a variety of settings, including internal waves in stratified fluids, shallow water waves, and even plasma physics.

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.

Riccati–Bernoulli sub-ode method-based exact solution of new coupled Konno–Oono equation

Nonlinear partial differential equations (NLPDEs) have emerged as a major focus of study in a variety of nonlinear science disciplines. This is because in general, mathematical, physics and engineering problems may be stated by NLPDEs. It may be noted that a specific kind of exact/analytical solution called traveling wave solutions for NLPDEs has great significance. In this regard, this paper addresses the traveling wave solution to the new coupled Konno–Oono equation (NCKOE), a set of NLPDE. A traveling wave transformation and the Riccati–Bernoulli sub-ode method (RBSOM) are used here to retrieve the exact/analytical traveling wave solution of the NCKOE. By using the above-mentioned approach, NLPDEs can be updated toward a collection of algebraic equations. Further, different types of solitary wave solitons, Trigonometric function solutions, Hyperbolic soliton solutions and dark bright solitons to the NCKOE have been successfully retrieved. Importantly, the newly derived solutions adhere to the main equation when they are inserted into the governing equations. Furthermore, through an appropriate selection of parameters, three-dimensional and two-dimensional figures are presented for physical illustration of the obtained solutions. These visual representations serve to showcase the effectiveness, conciseness and efficiency of the applied techniques.

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.

Optimal System, Symmetry Reductions and Exact Solutions of the (2 + 1)-Dimensional Seventh-Order Caudrey–Dodd–Gibbon–KP Equation

In this paper, the (2+1)-dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of (1+1)-dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation.

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.

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.

New exact solutions to the generalized shallow water wave equation

In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this nonlinear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely [Formula: see text]-expansion method. Using the aforementioned method and computational software, we have obtained different kinds of periodic and singular solitary wave solutions of the generalized shallow water wave equation. The obtained solutions are exponential function and trigonometric function solutions. Using 2D and 3D plots of the wave solutions, the dynamic behaviors of the developed solutions are displayed. The retrieved solutions validated the effectiveness and robustness of the proposed technique.

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 Korteweg–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>

Investigation of soliton solutions to the truncated M-fractional (3+1)-dimensional Gross-Pitaevskii equation with periodic potential

<p>This research explores some modernistic soliton solutions to the (3+1)-dimensional periodic potential the Gross–Pitaevskii equation with a truncated M-fractional derivative plays a significant role in Bose–Einstein condensation, which describes the dynamics of the condensate wave function. The obtained results include trigonometric, hyperbolic trigonometric and exponential function solutions. Three techniques named: the $ \exp_a $ function method, the Sardar sub-equation method, and the extended $ (G'/G) $-expansion approach are employed to achieve a variety of new solutions for the governing model. More comprehensive information about the dynamical representation of some of the solutions is being presented by visualizing the 2D, 3D and contour plots. This work reveals a number of new types of traveling-wave solutions, such as the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular, and the singular soliton solutions. These novel solutions are not the same as those that were previously studied for this governing equation. The presented techniques demonstrate clarity, efficacy, and simplicity, revealing their relevance to diverse sets of dynamic and static nonlinear equations pertaining to evolutionary events in computational physics, in addition to other real-world applications and a wide range of study fields for addressing a variety of other nonlinear fractional models that hold significance in the fields of applied science and engineering.</p>

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.