sine-Gordon Expansion Research Articles

Temporal wave dynamics, phase portrait and qualitative analysis of the time-dependent (2+1)-dimensional Zakharov-Kuznetsov equation

In this article, we analyze the effect of time-dependent coefficients and the complex wave dynamics of the (2+1)-dimensional Zakharov-Kuznetsov (ZK) equation. This equation provides a detailed, insightful, and realistic description of space physics, plasma physics, controlled fusion, and nonlinear sciences. The wave solutions are established using the generalized Kudryashov, modified simple equation, and modified sine-Gordon expansion techniques and are illustrated by graphical depictions, which provide valuable insight into understanding the complex dynamics of waves across different physical systems. Exact solitary wave solutions offer a dependable approach to investigating the behavior of a system under particular conditions and facilitating a comprehensive understanding of its dynamics. We also conduct a stability analysis and present the phase portrait of the solutions, which are useful in various fields, including physics, plasma physics, chemistry, biology, economics, and sociology. We ascertain that the profiles of 3D and 2D soliton-shaped waves are significantly affected by dynamic changes in coefficients, wave velocity, and associated model parameters. This research could help clarify the dynamics of intricate systems, paving to a better understanding and analysis of the temporal aspects of various phenomena.

Novel optical soliton solutions to nonlinear paraxial wave model

In this paper, we primarily focus on the nonlinear paraxial wave equation in Kerr media, a generalization of the nonlinear Schrödinger equation utilized to characterize the dynamics of optical beam propagation. By using three potent analytical methods, namely, the Sine-Gordon expansion method, the functional variable method, and the Bernoulli ([Formula: see text])-expansion method, numerous novel soliton solutions are derived. These solutions, comprising hyperbolic, trigonometric and exponential functions, represent significant additions to the field of optics. Furthermore, we elucidate the physical characteristics of these novel optical soliton solutions by presenting a series of three-dimensional (3D) and two-dimensional (2D) graphs with appropriate parameter values.

A view of optical soliton solution of the coupled Schrödinger equation with a different approach

The goal of this study is to investigate to optical soliton solution of the nonlinear coupled space-time Schrödinger equation using the Beta derivative and Sine-Gordon Expansion Method. All calculations in this study are made using some software program and the solutions obtained are substituted in the equations. New soliton solutions have been found using the suggested method for solving these problems. The solutions obtained have important areas of use in the fields of mathematical physics, in the field of quantum physics, optic and engineering.

Computational approach and convergence analysis for interval-based solution of the Benjamin–Bona–Mahony equation with imprecise parameters

PurposeTo provide a new semi-analytical solution for the nonlinear Benjamin–Bona–Mahony (BBM) equation in the form of a convergent series. The results obtained through HPTM for BBM are compared with those obtained using the Sine-Gordon Expansion Method (SGEM) and the exact solution. We consider the initial condition as uncertain, represented in terms of an interval then investigate the solution of the interval Benjamin–Bona–Mahony (iBBM).Design/methodology/approachWe employ the Homotopy Perturbation Transform Method (HPTM) to derive the series solution for the BBM equation. Furthermore, the iBBM equation is solved using HPTM to the initial condition has been considered as an interval number as the coefficient of it depends on several parameters and provides lower and upper interval solutions for iBBM.FindingsThe obtained numerical results provide accurate solutions, as demonstrated in the figures. The numerical results are evaluated to the precise solutions and found to be in good agreement. Further, the initial condition has been considered as an interval number as the coefficient of it depends on several parameters. To enhance the clarity, we depict our solutions using 3D graphics and interval solution plots generated using MATLAB.Originality/valueA new semi-analytical convergent series-type solution has been found for nonlinear BBM and interval BBM equations with the help of the semi-analytical technique HPTM.

Analytical solution of the (2+1)-dimensional Zoomeron equation by rational sine-Gordon Method

The current study is about the solution of the Zoomeron equation, one of the important models of mathematics and physics. In this study, the rational Sine-Gordon expansion method (RSGEM) is used to obtain various analytical solutions of the model. Compared to other methods, this method is quite effective and the desired results were obtained. Although there are many analytical solutions to the model used in the literature, we present rational type solutions for the first time with this method. We obtained rational hyperbolic function solutions, and also classified all soliton solutions (kink-like, kink, singular kink, anti-kink, dark, bright). In addition, geometric representations of the solutions in two-, and three-dimensional space and contour shape are made with the Mathematica software program.

Rational Sine-Gordon expansion method to analyze the dynamical behavior of the time-fractional phi-four and (2 + 1) dimensional CBS equations

This study uses the rational Sine-Gordon expansion (RSGE) method to investigate the dynamical behavior of traveling wave solutions of the water wave phenomena for the time-fractional phi-four equation and the (2 + 1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equation based on the conformable derivative. The technique uses the sine-Gordon equation as an auxiliary equation to generalize the well-known sine-Gordon expansion. It adopts a more broad strategy, a rational function rather than a polynomial one, of the solutions of the auxiliary equation, in contrast to the traditional sine-Gordon expansion technique. Several explanations for hyperbolic functions may be produced using the previously stated approach. The approach mentioned above is employed to provide diverse solutions of the time-fractional phi-four equation and the (2 + 1) dimensional CBS equations involving hyperbolic functions, such as soliton, single soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, and others. The RSGE approach enhances our comprehension of nonlinear processes, offers precise solutions to nonlinear equations, facilitates the investigation of solitons, propels the development of mathematical tools, and is applicable in many scientific and technical fields. The solutions are graphically shown in three-dimensional (3D) surface and contour plots using MATLAB software. All screens display the absolute wave configurations in the resolutions of the equation with the proper parameters. Furthermore, it can be deduced that the physical properties of the found solutions and their characteristics may help us comprehend how shallow water waves move in nonlinear dynamics.

A novel analytical approach to the Benjamin–Ono equation

This article examines some travelling wave solutions to the second-order Benjamin-Ono equation by using an analytical scheme via the sine-Gordon expansion technique. The Benjamin-Ono equation is similar to the KdV equation and it describes internal waves in fluids in a deep layer. We achieved some travelling wave solutions, including hyperbolic functions. All the obtained solutions were graphically analyzed based on their physical properties. As a result, the mentioned method is an effective one that provides analytical solutions for strongly non-linear partial differential models.

Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution

This paper presents the fractional generalized q-deformed sinh-Gordon equation. The fractional effects of the temporal derivative of the proposed model are studied using a conformable derivative. The analytical solutions of the governing model depend on the specified parameters. The resulting equation is studied with two integration architectures: the sine-Gordon expansion method and the modified auxiliary equation method. These strategies extract hyperbolic, trigonometric, and rational form solutions. For appropriate parametric values and different values of fractional parameter α, the acquired findings are displayed via 3D graphics, 2D line plots, and contour plots. The graphical simulations of the constricted solutions depict the existence of bright soliton, dark soliton, and periodic waves. The considered model is useful in describing physical mechanisms that possess broken symmetry and incorporate effects such as amplification or dissipation.

Existence and uniqueness solution analysis of time-fractional unstable nonlinear Schrödinger equation

The time-fractional unstable nonlinear Schrödinger (NLS) equations capture the time evolution of disturbances within media, tailored for describing phenomena in unstable media to help model and understand the intricate dynamics of systems prone to instability, the behavior of disturbances in complex and unstable environments and many more. In this study, we investigate the exact and analytical solutions of time-fractional unstable nonlinear Schrödinger (NLS) equations with the time-fractional β-time derivative, a powerful technique known as the sine-Gordon expansion (SGE) method is used. Non-locality behaviors that cannot be modeled using classical calculus are included in this equation. It discusses the relationships between waves of different frequencies or the same frequency but with distinct polarizations having a diverse set of key applications in nonlinear optics. We begin by using a fixed-point argument to prove the existence and uniqueness of solutions to the time-fractional problem under consideration. The partial differential equation of fractional order is transformed into an ordinary differential equation using the complex transformation. Numerous exact solutions are achieved with all the arbitrary parameters. The specific values of the obtained solutions' associated parameters are analyzed graphically with 3D plots as multiple soliton types of different wave solutions with the help of symbolic software. Understanding that the features of the solutions are far from intuitive because it is based on the parameter selection from the figures is difficult. The main objective of this paper is to investigate some fresh and further general solitary wave solutions to the suggested equation that properly explain the above-stated phenomena.

Exact soliton solutions and the significance of time-dependent coefficients in the Boussinesq equation: theory and application in mathematical physics

This article effectively establishes the exact soliton solutions for the Boussinesq model, characterized by time-dependent coefficients, employing the advanced modified simple equation, generalized Kudryashov and modified sine–Gordon expansion methods. The adaptive applicability of the Boussinesq system to coastal dynamics, fluid behavior, and wave propagation enriches interdisciplinary research across hydrodynamics and oceanography. The solutions of the system obtained through these significant techniques make a path to understanding nonlinear phenomena in various fields, surpassing traditional barriers and further motivating research and application. Significant impacts of the coefficients of the equation, wave velocity, and related parameters are evident in the profiles of soliton-shaped waves in both 3D and 2D configurations when all these factors are treated as variables, which are not seen in the case for constant coefficients. This study enhances the understanding of the significant role played by nonlinear evolution equations with time-dependent coefficients through careful dynamic explanations and detailed analyses. This revelation opens up an interesting and challenging field of study, with promising insights that resonate across diverse scientific disciplines.

On the extraction of complex behavior of generalized higher-order nonlinear Boussinesq dynamical wave equation and (1+1)-dimensional Van der Waals gas system

<p>In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper.</p>

Chaotic behavior and construction of a variety of wave structures related to a new form of generalized q-Deformed sinh-Gordon model using couple of integration norms

<abstract><p>The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica.</p></abstract>

Extraction of new solitary wave solutions in a generalized nonlinear Schrödinger equation comprising weak nonlocality.

This article delves into examining exact soliton solutions within the context of the generalized nonlinear Schrödinger equation. It covers higher-order dispersion with higher order nonlinearity and a parameter associated with weak nonlocality. To tackle this equation, two reputable methods are harnessed: the sine-Gordon expansion method and the [Formula: see text]-expansion method. These methods are employed alongside suitable traveling wave transformation to yield novel, efficient single-wave soliton solutions for the governing model. To deepen our grasp of the equation's physical significance, we utilize Wolfram Mathematica 12, a computational tool, to produce both 3D and 2D visual depictions. These graphical representations shed light on diverse facets of the equation's dynamics, offering invaluable insights. Through the manipulation of parameter values, we achieve an array of solutions, encompassing kink-type, dark soliton, and solitary wave solutions. Our computational analysis affirms the effectiveness and versatility of our methods in tackling a wide spectrum of nonlinear challenges within the domains of mathematical science and engineering.

The Modulation Instability Analysis and Analytical Solutions of the Nonlinear Gross−Pitaevskii Model with Conformable Operator and Riemann Wave Equations via Recently Developed Scheme

In this manuscript, we focus on the application of recently developed analytical scheme, namely, the rational sine-Gordon expansion method (SGEM). Some new exact solutions of Riemann wave system and the nonlinear Gross−Pitaevskii equation (GPE) by using this method are extracted. This method is based on the general properties of the SGEM which uses the fundamental properties of trigonometric functions. Many novel analytical solutions such as dark, bright, mixed dark–bright, hyperbolic, and periodic wave solutions are successfully extracted. Physical meanings of solutions are simulated by the various figures in 2D and 3D along with the contour graphs. Strain conditions of the existence are also reported in detail. Finally, modulation instability analysis of the nonlinear GPE is studied in detail.

On the complex properties of the first equation of the Kadomtsev-Petviashvili hierarchy

Abstract This work studies the first equation of the Kadomtsev-Petviashvili (KP) hierarchy. The sine-Gordon expansion method (SGEM) and the rational SGEM (RSGEM) are applied to the governing model. RSGEM is the developed version of SGEM. New complex travelling wave solutions, logarithmic and complex function properties are obtained. Several simulations such as 2D, 3D and contour surfaces of the obtained results are plotted. Physical meanings of these solutions are also reported. Strain conditions are also extracted.

Plenteous specific analytical solutions for new extended deoxyribonucleic acid (DNA) model arising in mathematical biology

In this paper, the generalized Kudryashov (GK) approach and the sine-Gordon expansion approach are used for constructing new specific analytical solutions of the deoxyribonucleic acid model, which include the well-known bell-shaped soliton, kink, singular kink, periodic soliton, contracted bell-shaped soliton and anti-bell-shaped soliton. The efficacy of these strategies demonstrates their utility and efficiency in addressing a wide range of integer and fractional-order nonlinear evolution problems. The physical relevance of the demonstrated results has been proven using three-dimensional forms. It is interesting to mention that the solutions achieved here using the provided methods are extra-extensive and may be used to explain the internal interaction of the deoxyribonucleic acid model originating in mathematical biology. The suggested approach was utilized to get exact traveling wave solutions for fractional nonlinear partial differential equations appearing in nonlinear science.

Characteristics of New Stochastic Solitonic Solutions for the Chiral Type of Nonlinear Schrödinger Equation

The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE). This model outlines the energy characteristics of quantum physics’ fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the CNLSE through multiplicative noise effects. This method accurately described a variety of solitary behaviors, including bright solitons, dark periodic envelopes, solitonic forms, and dissipative and dissipative–soliton-like waves, showing how the solutions changed as the values of the studied system’s physical parameters were changed. The stochastic parameter was shown to affect the damping, growth, and conversion effects on the bright (dark) envelope and shock-forced oscillatory wave energy, amplitudes, and frequencies. In addition, the intensity of noise resulted in enormous periodic envelope stochastic structures and shock-forced oscillatory behaviors. The proposed technique is applicable to various energy equations in the nonlinear applied sciences.

Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations with Beta-Derivatives via the sine-Gordon Expansion Method

The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)- dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems into nonlinear ordinary differential equations with integer orders. There is then no further demand for any normalization or discretization in the calculation process. The exact explicit solutions to the problems obtained with the SGEM are written in terms of hyperbolic functions. The exact solutions are new and published here for the first time. The effects of varying the fractional order of the beta-derivatives are studied through numerical simulations. 3D, 2D, and contour plots of solutions are shown for a range of values of fractional orders. As parameter values are changed, we can identify a kink-type solution, a bell-shaped solitary wave solution, and an anti-bell shaped soliton solution. All of the solutions have been carefully checked for correctness and could be very important for understanding nonlinear phenomena in beta partial differential equation models for systems involving the interaction of a Riemann wave with a long wave and interactions of two long waves with distinct dispersion relations.

On Survey of the Some Wave Solutions of the Non-Linear Schrödinger Equation (NLSE) in Infinite Water Depth

In this work, we use two different analytic schemes which are the Sine-Gordon expansion technique and the modified exp -expansion function technique to construct novel exact solutions of the non-linear Schrödinger equation, describing gravity waves in infinite deep water, in the sense of conformable derivative. After getting various travelling wave solutions, we plot 3D, 2D and contour surfaces to present behaviours obtained exact solutions.