2024-11

Physica Scripta (Volume : 99)

Schrödinger's nonlinear equation is a fundamental model in fiber optics and many other areas of science. Using the Jacobi elliptic expansion function method, the time-fractional cubic-quartic nonlinear Schrödinger equation and cubic-quartic resonant nonlinear Schrödinger equation are investigated. By applying the effective Jacobi elliptic expansion function method, optical soliton solutions such as bright, dark, singular, periodic singular, exponential, and Jacobi elliptic function solutions have been obtained. The effect of the time-fractional derivative on the solutions is also revealed. Graphical representations are illustrated to showcase the physical properties of raised solutions, providing a comprehensive understanding of the solutions’ functionality.

2024-10

Nonlinear Dynamics (Volume : 112)

This work explores the generalized variable-coefficient (3+1)-dimensional Kadomtsev-Petviashvili equation, which surfaces in fluid and multi-component plasma research as an important application. Plasmas and fluids are currently attracting attention because of their capacity to facilitate a wide range of wave phenomena. A diversity of wave structures, including M-lump-like waves and multiple solitons are secured. Moreover, the various hybrid solutions are analyzed and discussed with the assistance of the Hirota bilinear method and the long wave technique. The mechanical features of solutions and collision-related aspects within a diversity of nonlinear systems are demonstrated by the results of this investigation. The extracted solutions are thoroughly analyzed to determine their physical significance. This is done by presenting a range of graphs that illustrate the dynamics of the solutions for specific parametric values. The reliability of the methods we have implemented is confirmed by our findings, which also indicate their potential for future use in identifying unique and varied solutions to nonlinear evolution equations experienced in the fields of engineering and mathematical physics.

2024-10

Journal of Mathematics (Issue : 1) (Volume : 2024)

This study analyzes the Hirota–Satsuma–Ito equation, which discusses the propagation of unidirectional shallow-water waves and the interactions between two long waves with different dispersion forms. For the proposed equation, the sine-Gordon expansion method has been considered. This method is derived from the sine-Gordon equation. Different types of solutions, namely, bright, periodic, and dark-bright soliton solutions, are derived. When these solutions are compared to other previously published research, to our knowledge, the study concludes that they are innovative, and this method was not applied to this equation. The validation of the obtained solutions is verified and plotted as three-dimensional figures to comprehend physical phenomena. With the proper parameter values, distinct graphs are created to convey the physical representation of specific solutions. The results of this paper show that the method effectively improves a system’s nonlinear dynamical behavior. This study will be useful to a wide range of engineers who specialize in engineering models. The findings show that the computational approach is successful, simple, and even applicable to complex systems.

2024-09

Physica Scripta (Volume : 99)

Optical solitons and traveling wave solutions for the higher-order dispersive extended nonlinear Schrödinger equation are studied. Ultrashort pulse propagation in optical communication networks is described by this equation. To find exact solutions to the model, the unified Riccati equation expansion method and the Jacobi elliptic function expansion method are successfully applied. The optical solutions includes various solitary wave solutions, such as dark, bright, combined dark-bright, singular, combined periodic, periodic, Jacobian elliptic, and rational functions. Three-dimensional and two-dimensional graphs of solutions are presented. Also, the dynamical behavior of waves and the impact of time on solutions by selecting appropriate parameters are illustrated.

2024-08

Advances in Mathematical Physics (Volume : 2024)

This article analyzes the dynamics of waves to a new higher dimensional Kadomtsev−Petviashvili equation. The higher dimensional Kadomtsev−Petviashvili equation and its expansions have attracted a great deal of scientific interest during the past few decades. Several nonlinear phenomena in a range of domains, like the dynamics of long waves with modest amplitudes in oceans and plasma physics, are studied using this family. In this study, we successfully apply the Hirota bilinear method with the adoption of several test strategies. A set of results like breather, two-wave, and lump periodic solutions are secured. To visually depict the output, a variety of graphs featuring distinct shapes are produced in response to appropriate parameter values. The computational complexities and results emphasize the transparency, effectiveness, and ease of the technique, indicating the method’s applicability to many kinds of both static and dynamic nonlinear equations regarding evolutionary phenomena in computational physics, as well as other practical domains and research fields.

2024-08

Chaos, Solitons & Fractals (Volume : 187)

The (3+1)-dimensional Boussinesq equation is under consideration. This equation is considered as a prominent mathematical model in physics with many practical applications. In a fluid medium, the studied model can accurately represent viscous flows containing a variety of fluids with interfaces and provides a reasonable distribution of turbulent stresses associated with mean velocity gradients. For analyzing the studied equation, we apply the Hirota method and discuss the variety of multiple solitons and M-lump solutions. To visually represent the results, a range of graphs with unique shapes are generated per the specified parameter values. The computational intricacies and outcomes underscore the technique’s efficacy, simplicity, and transparency, demonstrating its suitability for numerous types of static and dynamic nonlinear equations of evolutionary phenomena in computational physics, in addition to other research and practical domains. The physical properties of solutions and the collision-related components of various nonlinear physical processes are illustrated with these results.

2024-08

International Journal of Theoretical Physics (Volume : 63)

The main objective of this work is to construct novel optical soliton solutions for nematic liquid crystals with conformable derivative using the new Kudryashov approach, a method arising in plasma physics and fluid mechanics. The obtained optical soliton solutions such as W-shape, bell shape, singular, dark-bright, bright, dark, and periodic solutions are explored and expressed by the hyperbolic functions, the exponential functions, and the trigonometric functions to clarify the magnitude of the nematic liquid crystals model with conformable derivative. The resulting traveling wave solutions of the equation play an important role in the energy transport in soliton molecules in liquid crystals. This paper contributes to understanding the fantastic features of nematicons in optics and further disciplines. The kinetic behaviors of the real part, imaginary part, and the square of modulus soliton solutions are illustrated by two-dimensional (2D) and three-dimensional (3D) contours graphs choosing the suitable values of physical parameters. It can be noticed that the novel Kudryashov approach is a powerful tool and efficient technique to solve various types of nonlinear differential equations with fractional and integer orders. That will be extensively used to describe many interesting physical phenomena in the areas of gas dynamics, plasma physics, optics, acoustics, fluid dynamics, classical mechanics.

2024-07

Physica Scripta (Issue : 8) (Volume : 99)

We aim to investigates the nonlinear Schrödinger equation including time-fractional derivative in (3+1)-dimensions by considering cubic and quantic terms The modified Sardar sub-equation method is used that lead to the discovery of a unique class of optical solutions. To transform the suggested nonlinear equation into an ordinary differential equation, we applied wave transformations, resulting in a set of nonlinear equations that offer diverse solution scenarios. The derived solutions encompass dark, wave, bright, mixed dark-bright, bell-shape, kink-shape, and singular soliton solutions. To enhance our understanding of the dynamic behavior exhibited by these solitons under varying time parameter values, visual simulations through a variety of graphs is presented. Furthermore, a comprehensive comparison is conducted, exploring a range of values for the conformable fractional order parameter. This comparison aims to highlight on the influence of fractional order variations on the solutions, contributing valuable insights into the nuanced dynamics of the system. Overall, this study serves to advance our understanding of nonlinear processes, and its potential applications in real-life phenomena. In the field of nonlinear optics, this equation can describe the propagation of optical pulses in nonlinear media. It helps in understanding the behavior of intense laser beams as they propagate through materials exhibiting nonlinear optical effects such as self-focusing, self-phase modulation, and optical solitons.

2024-07

Journal of Nonlinear Optical Physics & Materials (Volume : 33)

In this paper, several new optical soliton solutions of the time-fractional generalized resonant nonlinear Schrödinger equation, relevant to pulse propagation in optical fibers, are constructed using the extended simplest equation approach. The new exact solutions are expressed in terms of hyperbolic functions, trigonometric functions, and rational functions. Further, two-dimensional, threedimensional, and contour graphs of king-type, wave, dark-bright, bright, and bell-shaped optical soliton solutions are plotted to elucidate the significance of the time-fractional generalized resonant nonlinear Schrödinger equation using appropriate values of physical parameters. The dynamic behavior of the present solutions, incorporating the effect of the conformable fractional derivative, is depicted through two-dimensional graphs. The derived optical solutions are considered novel and have presumably not been previously reported in the literature.

2024-06

Optical and Quantum Electronics (Volume : 56)

This paper dedicates to study the optical soliton solutions for the time-fractional perturbed Fokas-Lenells equation using the novel modified Sardar sub-equation approach. Several novel optical solutions include a class of categories, comprising mixed dark-bright, bell-shape, singular, and wave optical solutions are constructed. Further, the magnitude of the time-fractional perturbed Fokas-Lenells model by investigating the impact of the conformable fractional parameter and the effect of the time parameter on the novel optical solutions is analyzed. The current type of Sardar sub-equation approach is a general form of several approachs, such as the tanh-function extension approach, the modified Kudryashov approach, the improved Sardar sub-equation method, and others. It can be claimed that the current optical soliton solutions are novel and have not existed in the literature. The results of this study are expected to shed light on the field of soliton theory in nonlinear optics. Time-fractional perturbed Fokas–Lenells soliton solutions could be applied to model and understand the behavior of signals in optical communication systems and potentially improving data transmission.

2024-06

Modern Physics Letters B (Volume : 38)

The (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation which consists of the KdV equation and the SK equation is the subject of investigation in this study. The studied equation has rich physical meaning in nonlinear waves. The KdV- type equations hold great importance as a prototypical representation of an infinite-dimensional system that is completely integrable and exactly solvable in the context of nonlinearity. The KdV equation is utilized to describe shallow water waves in a density-stratified ocean, which exhibit weak and nonlinear interactions with long internal waves. The Hirota bilinear method has been used with the support of various test functions. For the purpose of analyzing the governing equation, numerous solutions are secured, including breathers and two-wave solutions. Breather waves refer to solitary waves that exhibit both partial localization and periodic structure in either space or time. Breathers serve crucial functions in nonlinear physics and have been observed in various physical domains, including optics, hydrodynamics, and quantized superfluidity. To visually represent the results, a range of graphs with unique shapes are generated in accordance with the specified parameter values. The computational intricacies and outcomes underscore the technique’s efficacy, simplicity and transparency, demonstrating its suitability for numerous types of static and dynamic nonlinear equations pertaining to evolutionary phenomena in computational physics, in addition to other research and practical domains. The physical properties of solutions and the collision-related components of various nonlinear physical processes are illustrated with these results.

2024-05

Modern Physics Letters B (Volume : 38)

This paper explores a specific class of equations that model the propagation of optical pulses in dual-core optical fibers. The decoupled nonlinear Schrödinger equation with properties of M fractional derivatives is considered as the governing equation. The proposed model consists of group-velocity mismatch and dispersion, nonlinear refractive index and linear coupling coefficient. Different types of solutions, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted by using the integration methods known as fractional modified Sardar subequation method and modified F-expansion method. Optical soliton propagation in optical fibers is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. 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. Furthermore, hyperbolic, periodic and exponential solutions are generated. A fractional complex transformation is applied to reduce the governing model into the ordinary differential equation and then by the assistance of balance principle the methods are used, depending upon the balance number. Also, we plot the different graphs with the associated parameter values to visualize the solutions behaviours with different parameter values. The findings of this work will help to identify and clarify some novel soliton solutions and it is expected that the solutions obtained will play a vital role in the fields of physics and engineering.

2024-05

Modern Physics Letters B (Volume : 38)

Here, we investigate the impact of space-time reverse (STR) problems, a result of parity-time symmetry in optics and quantum mechanics, on soliton propagation in optical fibers. The STR problems are characterized by the existence of a field and its reverse. The research introduces a new classification of two scenarios: non-interactive and interactive fields and reverse fields. The solutions for the generalized Fokas–Lenells equation (gFLE) with STR and third-order dispersion are derived. To tackle this, adaptive transformations for the field and its reverse are introduced, employing a unified method. In the non-interactive scenario, both exact and approximate solutions are found. However, in the interactive case, only exact solutions are discovered. This work reveals that the presence of the field and its reverse unveils new soliton structures, including bright–dark envelope solitons and right and left envelope-solitons. In the non-interactive case, the field displays a right envelope-soliton, while the reverse field exhibits a left envelope-soliton (or vice versa). The study hypothesizes that the presence of a reverse field might impede soliton propagation in optical fibers. The research also includes an analysis of modulation instability (MI), determining that MI is initiated when the coefficient of Raman scattering exceeds a critical value. Furthermore, the study examines the modulated wave gain and explores global bifurcation through phase portrait by constructing the Hamiltonian function.

2024-04

Optical and Quantum Electronics (Volume : 56)

This research paper’s primary goal is to find fresh approaches to the Hirota–Maccari system. This system explains the dynamical features of the femto-second soliton pulse in single-mode fibers. The bright soliton, dark soliton, dark-bright soliton, dark singular, bright singular, periodic soliton, and singular solutions are developed utilizing the modified auxiliary equation technique. To make the physical significance of each unique solution clearer, it is mapped in both 2D and 3D. The primary Hirota–Maccari system is being verified by all new solutions, and the constraint condition is also provided. The obtained optical solitons may be important for the analysis of nonlinear processes in optic fiber communication and signal processing.

2024-03

Optical and Quantum Electronics (Volume : 56)

The main goal of this paper is to study the higher-order dispersive extended nonlinear Schrödinger equation, which demonstrates the propagation of ultrashort pulses in optical communication networks. In this study, both the sinh-Gordon expansion method and the generalized exponential rational function method are used to offer some novel optical solutions. These optical soliton solutions are dark soliton, bright soliton, singular, periodic, and dark-bright soliton solutions. The obtained optical soliton solutions are presented graphically in 2D and 3D to clarify the behavior of solutions more effectively. The constraint conditions are also used to verify the exitances of the new analytical solutions. Moreover, all solutions compared to solutions obtained previously are new, and all the new wave solutions have verified Eq. (1) after we substituted them into the studied equation. In the future, these novel soliton solutions will be very helpful in developing fluid dynamics, biomedical issues, dynamics of adiabatic parameters, industrial research, and many other areas of science. To our acknowledgment, the presented optical solutions are novel, and also beforehand these methods have not been applied to this studied equation.

2024-01

Optical and Quantum Electronics (Volume : 56)

This article focuses to extract the optical solitons in fiber optics modeled by (1 + 1)-dimensional coupled nonlinear Schrödinger equation (NLSE) by the assistance of truncated M-fractional derivative. Circularly polarized waves in fiber optics are described by the studied equation. The NLSEs have attracted greater attention due to their ability to effectively elucidate a diverse array of intricate physical phenomena and their capacity to exhibit deeper dynamical patterns through localised wave solutions. The solutions are obtained using the modified Sardar sub-equation method (MSSEM), which is a recently developed integration technique. Various types of optical pulses, including bright, dark, combo, and singular soliton solutions, are obtained. Considering appropriate parameters, a variety of graph shapes are sketched to describe the graphical presentation of the calculated outcomes. The findings indicate that the chosen technique is effective in improving nonlinear dynamical behaviour. We anticipate that many engineers who use engineering models will find this study to be of interest. The results validate the simplicity, effectiveness, and universality of the selected computational approach, even when applied to complex systems.

2024-01

Optical and Quantum Electronics (Volume : 56)

The optical soliton solutions of the time-fractional higher-order nonlinear Schrödinger equation in the presence of the Kudryashov nonlinear refractive index are studied in this paper using the new Kudryashov approach and Bernoulli’s equation approach. The current model specializes in discerning the propagation of optical soliton pulses within optical fbers. Obtaining the optical solutions for this model, particularly the sextic power, is an essential yet challenging task. To generate the nonlinear ordinary diferential equation, we insert the complex wave transformations into the present time-fractional nonlinear Schrödinger equations. Here, a system of linear equations in polynomial form using the proposed approach is acquired. By solving the linear system of equations, various solution sets were generated, each containing diferent values for the parameters of the studied equation. Additionally, the process yielded distinct approaches for solving the problem. Several novel optical soliton solutions are constructed for the time-fractional Schrödinger equation with the Kudryashov nonlinear refractive index, and the obtained soliton solutions satisfy the model. The visual representations of the obtained solution functions through contour, three-dimensional, and two-dimensional depictions in various simulations are shown, as presented in the fgures. The results suggest that the employed methods are efcient and powerful tools to be applied to various diferential equations with fractional and integer orders.

2024-01

Optical and Quantum Electronics (Volume : 56)

In this manuscript, under consideration model is the decoupled nonlinear Schrodinger equation (NLSE) arising in dual-core optical fibers. 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. We have secured the optical pulses by the assistance of modern integration tool like the modified extended tanh expansion method. The optical solitons are expressed in the forms of dark, singular and combined optical soliton 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. Moreover, we have secured the hyperbolic, and periodic solutions. The used method not only provides previously extracted solutions but also secures new solutions. Given suitable parameter values, multiple graphs with different shapes are drawn to represent the output in a visual manner. The findings of this study demonstrate that the selected methodologies are efficacious in enhancing nonlinear dynamical phenomena. It is anticipated that a considerable number of engineers that utilize engineering models will find this study to be of interest. The results show that the chosen techniques are effective, easy to implement, and applicable to complex systems in a variety of fields, particularly optical fibers filed. The results suggest that the system possesses a potentially abundant presence of soliton structures.

2023-11

Results in Physics (Volume : 54)

In this paper, the generalized (3+1)-dimensional nonlinear Sasa-Satsuma model with conformable fractional derivative using the new Kudryashov approach is considered to find a class of novel exact solutions in optical fibers. The acquired new solutions are extract by the hyperbolic functions and exponential function which are assorted as dark, bright, singular, mixed dark-bright, dark-bright, bell-shape, and periodic optical soliton solutions. The contour, three-dimension, two-dimension of various forms of the novel optical solutions are sketched to determine the prominence of the time-fractional generalized (3+1)-dimensional nonlinear Sasa-Satsuma model. In addition, to show the magnitude of the conformable fractional derivative the effect of the conformable fractional order derivative on a class of the new optical solutions are depicted via illustrative graphs. Finally, we found that the present technique is an accurate tool to investigate the analytic solutions of the fractional differential equations. The proposed Sasa-Satsuma model can be applied to the transmission of optical fibers’ ultra-fast pulses.

2023-11

Optical and Quantum Electronics (Volume : 55)

A key objective of the paper is to study the dynamical system for the two types of Sasa–Satsuma equations, namely; Sasa–Satsuma equation and higher-order Sasa–Satsuma equation. A Sasa–Satsuma equation is used to describe the propagation of femtosecond pulses through optical fiber systems. The bifurcation and chaotic characteristic of the Sasa–Satsuma equation and higher-order Sasa–Satsuma equation that arises in fluid dynamics and nonlinear optics are studied. For both models, by using the theory of planar dynamical system the bifurcation and chaotic characteristic of the Sasa–Satsuma equation and higher-order Sasa–Satsuma equation that arises in fluid dynamics and nonlinear optics are studied. For a better understanding of these dynamical behaviors, phase portraits in 2D and 3D figures are dawn. For both equations, the equilibrium points and their effects on the bifurcation behavior are analyzed. Moreover, from the presented results, both models have different dynamical behavior.

2023-10

The European Physical Journal Plus (Volume : 138)

The mechanism of pattern formations has been widely studied and for different types of Reaction-Diffusion equations. This phenomenon has a wide range of applications in the fields of biology, chemistry, engineering, etc. In this paper, we have studied the pattern formation for a Reaction–Diffusion model with nonlinear reaction terms; this equation is different from RDM which has been studied before, and which derived from the interaction between Turing stationery and wave instability. Next, we study the possible traveling wave solution for our RDM and their stability close to the steady states. We discretize the system of Reaction-diffusion equations in one dimension using Semi-Implicit second-order difference method and we investigate the different types of travelling wave solutions (TWS). A finite element package namely COMSOL Multiphysics is used to show some types of pattern formations and for two types of initial conditions. The initial conditions are chosen to investigate the type of spots that can be formulated from the interaction. In parallel, we have proved theoretically the regions where those pattern formations can be found depending on the value of the diffusion coefficients and wave number

2023-10

Results in Physics (Volume : 54)

This article discusses the behavior of specific dispersive waves to new (3+1)-dimensional Hirota bilinear equation (3D-HBE). The 3D-HBE is used as a governing equation for the propagation of waves in fluid dynamics. The Hirota bilinear method (HBM) is successfully applied together with various test strategies for securing a class of results in the forms of lump-periodic, breather-type, and two-wave solutions. Solitons for nonlinear partial differential equations (NLPDEs) can be identified via the well-known mathematical methodology known as the Hirota method. However, this requires for bilinearization of nonlinear PDEs. The method employed provides a comprehensive explanation of NLPDEs by extracting and also generating innovative exact solutions by merging the outcomes of various procedures. To further illustrate the impact of the parameters, we also include a few numerical visualizations of the results. These findings validate the usefulness of the used method in improving the nonlinear dynamical behavior of selected systems. These results are used to illustrate the physical properties of lump solutions and the collision-related components of various nonlinear physical processes. The outcomes demonstrate the efficiency, rapidity, simplicity, and adaptability of the applied algorithm.

2023-09

Results in Physics (Volume : 53)

In the present work, we employed a novel modification of the Sardar sub-equation approach, leading to the successful derivation of several exact solutions for the time-fractional Ginzburg–Landau equation with Kerr law nonlinearity. These solutions encompass a range of categories, including singular, wave, bright, mixed dark-bright, and bell-shaped optical solutions. We demonstrate the dynamic behavior and physical significance of these optical solutions of the proposed model via several graphical simulations, including contour plots, three-dimensional (3D) graphs, and two-dimensional (2D) plots. Furthermore, we investigate the magnitude of the time-fractional Ginzburg–Landau equation by analyzing the influence of the conformable fractional order derivative and the impact of the time parameter on the newly constructed optical solutions. The proposed technique is a generalized form that incorporates various methods, including the improved Sardar sub-equation method, the modified Kudryashov method, the tanh-function extension method, and others. To the best of our knowledge, these solutions are novel and have not been reported in the literature. Moreover, the present method is efficient and robust for analyzing applied differential equations in plasma physics and nonlinear optics.

2023-09

Results in Physics (Volume : 53)

The Boussinesq equation is a partial differential equation that describes the behavior of waves in shallow water. In this paper, we address some new dynamical behaviors to the (3+1)-dimensional Boussinesq equation, which are not constructed beforehand. Various solutions namely: multi-soliton, multi-M-lump, and the hybrid soliton solutions are reported. New explored features of equation are presented graphically to better analyze the gained solutions. For different period of time multi-soliton, multi-lump solutions are plotted. The results have important applications in oceanography, geophysics, and fluid dynamics, and is used to study the behavior of waves in complex three-dimensional domains, particularly in situations where the nonlinear effects are strong.

2023-09

Results in Physics (Volume : 53)

The purpose of this research paper is to investigate the (3+1)-dimensional Hirota bilinear equation that arises in nonlinear waves in fluid dynamics, plasma physics and shallow water waves. We use Hirota’s bilinear approach and a long-wave method to explore the dynamical features of the suggested equation. Multiple M-lump waves mixed with soliton solutions are constructed. Also, the breather wave and its interaction with one-soliton and M-lump waves are derived. These solutions are believed to play a role in understanding the dynamical aspects of the (3+1)-dimensional Hirota bilinear equation.

2023-08

International Journal of Modern Physics B

An analysis is carried out to investigate the influence of the magnetic field over Carreau fluid flow and heat transfer in a liquid film on an unsteady stretching surface. The governing equations are converted to a system of boundary value problem (BVP) by using the similarity transforms. The differential equations are solved numerically using the modified Laplace decomposition method (MLDM) and the obtained results are compared with the existing schemes. An excellent agreement between them illustrates the accuracy of MLDM to the suggested model. The influence of the dimensionless governing parameters like magnetic field number, Prandtl number, unsteadiness parameter, Weissenberg parameter, and radiative number on velocity and temperature profiles is discussed in tabular form and shown through illustrative graphs.

2023-08

Optik (Volume : 290)

In this work, we introduce a novel truncated M proportional (T-MP) derivative and consider a T-MP perturbed derivative nonlinear Schrodinger equation (PDNLSE). The PDNLSE is a nonlinear model which arises in nano optical fibers (photonic nanowires). While the DNLSE exhibits self-steepening (SS), the PDNLSE exhibits self-phase modulation (SPM) and Raman scattering (RS) effects, Here, the exact solutions of the PDENLSE are derived, here, by implementing the unified method. These solutions are displayed in graphs. It is found that a sufficient condition for a hyper-chaotic solution to hold is that an elliptic function solution exists. Non elliptic solutions may be, also, hyper-chaotic. It is worth mentioning that hyper chaotic may occur in economic and financial mathematics. Nonchaotic solutions exhibit many geometric structures, chirped solitons, and rhombus-shaped and M-shaped solitons. It is found that modulation instability triggers when the coefficient of the third-order dispersion exceeds a critical value. Further, the global bifurcation is investigated via phase portrait by constructing the Hamiltonian function.

2023-08

Results in Physics (Volume : 52)

This article pays attention to the interaction of waves for the (2+1)-dimensional KdV equation arising in the diversity of fields with the properties of conformable fractional derivatives. The KdV equation is notably significant as a prototypical example of an exactly solvable nonlinear system (that is, an infinite-dimensional system that is completely integrable). In a density-stratified ocean, the KdV equation characterizes shallow water waves that interact weakly and nonlinearly with long internal waves. The Hirota bilinear method (HBM) is successfully adopted with different test approaches. Different kinds of solutions like lump-periodic, breather-type, and two-wave solutions, have been obtained. The method used adequately describes NLPDEs since it both provides solutions that were previously confirmed and generates fresh exact solutions by combining the results of several operations. We also plot the graphs using the corresponding parameter values to demonstrate the graphical representation of selected solutions. These findings demonstrate the beneficial effects of the approach in improving the system’s nonlinear dynamical behavior. The outcomes demonstrate the efficiency, swiftness, ease, and adaptability of the used algorithm, even when applied to intricate systems.

2023-07

Journal of Nonlinear Optical Physics & Materials (Volume : 32)

In this paper, the extended simplest equation method is utilized to construct the novel exact optical solitons solutions of the perturbed time fractional Chen–Lee–Liu equation with conformable fractional derivative. The acquired optical solitons and other solutions are expressed via the rational functions, the trigonometric functions, and the hyperbolic functions; in addition to guaranteeing the existence of the acquired results, the constrain conditions are provided. Furthermore, to elucidate the magnitude of the proposed equation, various obtained solutions are plotted via two-dimensional (2D) and three-dimensional (3D) graphs using appropriate values of parameters. It is observed that the present technique is a powerful tool and efficient for finding the analytical solution for nonlinear differential equations of integer and fractional orders.

2023-06

International Journal of Modern Physics B (Volume : 37)

This paper discusses the wave propagation to the nonlinear Sharma–Tasso–Olver–Burgers (STOB) equation which is used as the governing model in different fields. Natural phenomena are typically complex and nonlinear, defying simple linear superposition. Researchers have been studying a wide range of natural phenomena in depth, and nonlinear science has gradually become a part of people’s consciousness. One of the most significant research questions in nonlinear science centers around the nonlinear evolution equation and its precise solution. We have secured different shapes of the solitary wave solutions including kink-type, shock-type and combined solitary wave solutions with the assistance of recently developed integration tool, namely the new extended direct algebraic method (NEDAM). Additionally, the solutions for the hyperbolic, exponential and trigonometric functions are retrieved. Moreover, based on a comparison of our results to those that are well known, the study indicates that our solutions are innovative. Using proper parameters in numerical simulations and physical explanations, it is possible to demonstrate the significance of the results. The results of this research can improve the nonlinear dynamic behavior of a system and indicate that the methodology employed is adequate. It is proposed that the offered method can be utilized to support nonlinear dynamical models applicable to a wide variety of physical situations. We hope that a wide spectrum of engineering model professionals will find this study to be beneficial.

2023-05

Results in Physics (Volume : 49)

This study deals the dynamics of waves to the conformable fractional (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equations. The (2+1)-dimensional NNV equations are the isotropic Lax integrable extension of the (1+1)-dimensional Korteweg–de Vries equations. Fractional differential models (FDMs) from the corresponding integer order model can describes more complex behavior and even cover all properties of integer order model. By the usage of different test approaches, the Hirota bilinear method (HBM) has been successfully applied. The Hirota method is a well-known and reliable mathematical tool for finding the soliton solutions of fractional nonlinear partial differential equations in many fields. However, it demands bilinearization of nonlinear fractional PDE. A class of results in the shapes of lump-periodic, breather-type and two wave solutions have been extracted. Numerical visualizations of the results are also used to demonstrate the implications of the fractionality and parameters. It is acceptable to assume that many experts in engineering models will learn from this research. The results prove the used algorithm is effective, quick, easy, and flexible in its application to various systems.

2023-05

Optical and Quantum Electronics (Issue : 7) (Volume : 55)

In this paper, the extended simplest equation technique is considered to construct various exact optical solutions to the time-fractional nonlinear Schrodinger equation with second-order spatiotemporal and group velocity dispersion coefficients. The acquired novel optical soliton solutions are illustrated by the hyperbolic functions, the rational functions, and the trigonometric functions. The singular, dark, bright, mixed bright, dark–bright, and wave soliton solutions of the proposed model are successfully constructed. Further, to clarify the magnitude of the present nonlinear time-fractional Schrodinger model several solutions of the new exact optical solutions are plotted via two-dimensional and three-dimensional graphs using suitable values of physical parameters. The results acquired illustrate that the utilized technique is simple and quite efficient for exploring exact soliton solutions for different differential equations of fractional and integer orders arising in optics and applied mathematics. The novel optical solutions can assist researchers with an interest in plasma physics to unravel the mystery of numerous nonlinear phenomena that arise in various plasma models.

2023-05

Alexandria Engineering Journal (Volume : 74)

In this article, the Hirota bilinear method is utilized to reveal multi-fusion solutions to the Broer-Kaup-Kupershmidt (BKK) system in (2 + 1)-dimension. The Broer-Kaup equations are a pair of linked nonlinear partial differential equations which plays a great role in nonlinear physics. The solutions are one-, two-, three-, and four-soliton resonant wave solutions. On the other hand, the new modified Sardar sub-equation method (SSEM) is employed to reach complex wave solutions in the form of topological, non-topological, and other wave solutions. By selecting suitable values for parameters, the dynamics of some of the reported solutions are graphed. The reported solutions are novel and have not previously been reported in the present literature. The solutions may be useful in explaining the physical meaning of various nonlinear physical models.

2023-03

Results in Physics (Volume : 47)

In this study, the (m+ 1/G′)-expansion technique and the extended rational sine-cosine approach are applied to investigate the (2+ 1)-dimensional integrodifferential equation, namely the Jaulent-Miodek evolution equation (JMEE). Via these powerful approaches, some wave solutions to the JMEE evolution equation are successfully constructed. Using suitable values of the parameters, we plot the 2-and 3-dimensional graphs of the reported solutions. The two utilized approaches are powerful and efficient mathematical tools that secure important wave solutions. The reported solutions in this study may be used in explaining the physical features of various nonlinear physical models.

2023-03

Optik (Volume : 280)

This paper uses a new Kudryashov approach to construct various novel exact optical solutions of the time-fractional Fokas system with conformable fractional derivatives. The acquired optical solutions are distributed into several categories of bell shape, singular, exponential, dark, bright, dark-bright, and periodic soliton solutions. The graphical simulations of several graphs are given for a better understanding of the effect of the fractional order derivative and the dynamical behavior of the present optical solutions with various values of time parameter. In order to highlight the significance of the Fokas system, a class of the acquired solutions are given using suitable parameter values on three-dimensional (3D), two-dimensional (2D), and contour graphs. As a result, this technique is productive and practically effective for understanding nonlinear problems in engineering, physics, and mathematics.

2023-03

CFD Letters (Issue : 5) (Volume : 15)

The purpose of this study is to investigate the boundary layer of Carreau fluid and heat transfer over an exponentially stretching plate derived in a vertical porous with variable surface thermal flux. The partial differential equations that represent the momentum equation and heat equation are commuted into nonlinear ODEs by applying similarity transformations and results found numerically. The impact of several emerging dimensionless parameters labelled the Weissenberg number (We), the power-law index (), Velocity slip (), Thermal jump (), and Prandtl number () on the velocity profile and heat transfer on the boundary layer are showed in detail. In more detail, also the influence of physical parameters on local skin friction and Sherwood number are studied. The shooting method with the explicit technique is used to find the solution and all results are illustrated graphically and numerically. We noted that by increasing power index, radiation parameter and velocity slip, the velocity profile increases, and the temperature profile decreases. Furthermore, it is deduced that rising the thermal radiation parameter reduces the local Nusselt number.

2023-03

Results in Physics (Volume : 48)

In this study, we focus on the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation in fluid dynamics, which is useful for modeling weakly dispersive waves transmitted in quasi media and fluid mechanics. As a general matter, this paper examines the gBKP equation including variable coefficients of time that are widely employed in plasma physics, marine engineering, ocean physics, and nonlinear sciences to explain shallow water waves. Using Hirota’s bilinear approach, one-, two, and three-soliton solutions to the problem are constructed. By employing a long-wave method, 1-M-, 2-M, and 3-M-lump solutions are derived. In addition, interaction phenomena of one-, and two-soliton solutions with one-M-lump wave are revealed. Moreover, an interaction solution between a two-M-lump wave and a one-soliton solution is also offered. the straight lines that M-lump waves travel among them are derived. We believe that our findings will help improve the dynamical properties of (3+1)-dimensional BKP-type equation.

2023-02

Optical and Quantum Electronics (Issue : 5) (Volume : 55)

We use the (m+1/G')-expansion method in reaching the exact solution of the (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq, Newel-Whitehead-Segel, and Zeldovich equations. New solutions in form of the kink, complex and singular solutions are reported. On the other hand, the Adomian decomposition method is employed to find approximate solutions to the the (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq equation. The three-dimensional figures and their corresponding contour plots for the reported solutions are drawn. Also, a table is presented for the approximate solutions. The reported results may be useful in studying physical features of various nonlinear mathematical models.

2023-02

Optical and Quantum Electronics (Volume : 55)

The modified exponential function method is used to obtain some new analytical solutions of the nonlinear Gerdjikov–Ivanov equation with the M-fractional operator. The novel obtained solutions are expressed in hyperbolic, trigonometric, and exponential function forms. Moreover, the instability modulation and gain spectra of the Gerdjikov–Ivanov equation are also analyzed. Constraints conditions are utilized to verify the existence of the solutions. Presented solutions are novel, satisfy and verify the M-fractional Gerdjikov–Ivanov equation.

2023-02

Optical and Quantum Electronics (Volume : 55)

In this paper, the extended simplest equation technique is utilized to construct the novel exact optical soliton solutions of time fractional Fokas system with conformable fractional derivative. The acquired optical soliton solutions such as dark, bright, dark-bright, periodic, and singular solutions are constructed and expressed via the rational functions, the trigonometric functions, and the hyperbolic functions to elucidate the magnitude of the proposed equation. The dynamical behaviors of the square of modulus solutions, real and imaginary parts are depicted via two dimensional and three dimensional contours graphs using appropriate values of parameters. It is observed that the present method is an efficient and powerful tool for solving nonlinear differential equations of integer and fractional orders.

2023-02

Chaos, Solitons and Fractals (Volume : 169)

One of the most popular ways to solve wave equations and explain nonlinear physical phenomena is with nonautonomous soliton solutions. This work aims to analyze the (3+1)-dimensional variable-coefficient Kudryashov–Sinelshchikov model, which explains pressure waves in liquid with flocs. Fluid dynamics, a subfield of fluid mechanics used in physics and engineering, is the study of how liquids and gases flow. There are many subfields within it, including aerodynamics and hydrodynamics. There are several uses for fluid dynamics, namely quantifying forces and moments. Using Hirota direct method, we auspiciously provide multiple solitons and M-lump solutions to this equation. We use specified input variables to accentuate the physical traits of the results obtained via two-, three-dimensional, and contour graphics since doing so is significant. The findings are applied to exemplify the physical traits of lump solutions and the collision-related elements of various nonlinear physical processes.

2023-02

Frontiers in Heat and Mass Transfer (Issue : 7) (Volume : 20)

In this paper. We study the behavior of heat transfer of Casson fluid at the magnetohydrodynamic stagnation point with thermal radiation over a continuous moving sheet. The appropriate similarity transfer is used to transfer the governing differential equations into the ordinary differential equation and then solved by the collocation method based on spline function. The quasi-linearization technique is utilized to approximate the non-linear equation of the model with a system of linear equations subsequently the collocation method based on the spline function is employed to validate the solution of these equations. The obtained results are investigated with the existing literature by direct comparison. The influence of distinct physical parameters on the velocity and temperature profiles are depicted through tables and illustrative graphs.

2023-02

Nonlinear Dynamics (Volume : 111)

Lump solutions are a prominent option for numerous models of nonlinear evolution. The intention of this research is to explore the variable coefficients Kadomtsev–Petviashvili equation. We auspiciously provide multiple soliton and M-lump solutions to this equation. Additionally, the presented results are also supplied with collision phenomena. Owing of its essential role, we employ appropriate parameter values to emphasis the physical characteristics of the provided results using 3D and contour charts. The outcomes of this work convey the physical characteristics of lump and lump interactions that occur in many dynamical regimes.

2023-01

AIMS Mathematics (Issue : 2) (Volume : 8)

This paper is presented to investigate the exact solutions to the modified Zakharov-Kuznetsov equation that have a critical role to play in mathematical physics. The tan(ϕ(ζ)/2)-expansion, (m+G′(ζ)/G(ζ))-expansion and He exponential function methods are used to reveal various analytical solutions of the model. The equation regulates the treatment of weakly nonlinear ion-acoustic waves in a plasma consisting of cold ions and hot isothermal electrons throughout the existence of a uniform magnetic field. Solutions in forms of W-shaped, singular, periodic-bright and bright are constructed.

2023-01

AIMS Mathematics (Issue : 3) (Volume : 8)

This paper is devoted to analysis the behavior of heat transfer of Casson-Carreau fluid at the magnetohydrodynamic (MHD) stagnation point with thermal radiation over a continuous moving sheet. The suitable similarity transform is utilized to transfer the governing differential equations into a system of differential equations and then solve the converted non-linear system by the collocation technique based on the B-spline function (CTBS) and Runge-Kutta method (RK). The quasi-linearization technique is utilized to approach the non-linear equations of the model to a system of linear equations and used CTBS to acquire the solution of the system of linear equations. The obtained results are investigated with the present literature by direct comparison. It is found that an increment in the value of the Weissenberg number decreases the velocity profile and enhances the temperature profile for Casson and Carreau fluids. Conversely, increasing the values of the magnetic parameter, shrinking parameter, and Casson fluid parameter improve the velocity profile and depreciate the thermal distribution. Further, the temperature profile declines with an improvement in radiation parameter and Prandtl number for Casson and Carreau fluids. The influence of distinct physical parameters on the velocity and temperature profiles are depicted via tables and illustrative graphs.

2023-01

Results in Physics (Volume : 45)

One of the most effective ways to understand nonlinear quantum systems is with lump solutions. The objective of this study is to acquire more about the (3+1)-dimensional soliton equation. We successfully present this equation with various solitons and M-lump solutions. We adopt specific parameter values to accentuate the physical features of the provided exact solutions through 3D and contour plots as doing so is of extreme significance. The submitted results indicate the physical qualities of lump-and-lump interaction events in various nonlinear physical processes.

2022-12

Communications in theoretical physics (Issue : 1) (Volume : 74)

Lump solutions are one of the most common solutions for nonlinear evolution equations. This study aspires to investigate the generalized Hietarintatype equation. We auspiciously provide multiple M-lump waves. On the other hand, collision phenomena to multiple M-lump waves with soliton wave solutions are also provided. During the collision, the amplitude of the lump will change significantly over the processes, whereas the amplitude of the soliton will just minimally alter. As it is of paramount importance, we use suitable values of parameter to put out the physical features of the reported results through three dimensional and contour graphics. The results presented express physical features of lump and lump interaction phenomena of different kinds of nonlinear physical processes. Further, this study serve to enrich nonlinear dynamics and provide insight into how nonlinear waves propagate.

2022-12

Optical and Quantum Electronics (Issue : 1) (Volume : 55)

We aim to determine the analytical solutions for the Davey–Stewartson (RDS) equations in (2+1)-dimensions using the extended sine-Gordon equation expansion method. It is a natural two-dimensional form of the nonlinear Schrödinger equation. As a first step, we change the imaginary (2+1)-dimensional RDS model to a system of nonlinear ordinary differential equations, then we apply the sine-Gordon method. With the use of Wolfram Mathematica software, we try to find novel solutions to the resulting system of the nonlinear differential equation. Constructed solutions are drawn in 2D and 3D-dimensions graphs to more realize their characteristics.

2022-09

Nonlinear Dynamics (Issue : 1) (Volume : 110)

Our work aims to investigate the vcBLMPE in (3+1)-dimensions (3D-vcBLMPE) that characterizes wave propagation in incompressible fluids. In real-world issues, nonlinear partial differential equations containing time-dependent coefficients are more relevant than those with constant coefficients owing to inhomogeneities of media and nonuniformities of boundaries. In shallow water, linearization of the wave formation needs more critical wave capacity criteria than in water depths, and the strongly nonlinear aspects are readily visible. By using symbolic computation, several nonautonomous wave solutions with different geometric structures are obtained. Each of the gained solutions is presented graphically based on various arbitrary coefficients to demonstrate and better comprehend their dynamical properties. As a comparison between the new results and the results previously reported, we have presented several completely new findings in this study.

2022-08

International Journal of Nonlinear Sciences and Numerical Simulation

In this work, the dynamical behaviors of the Jimbo–Miwa equation that describes certain interesting (3 + 1)-dimensional waves in physics but does not pass any of the conventional integrability tests are studied. One-, two-, and three-M-lump waves are constructed successfully. Interactions between one-M-lump and one-soliton wave, between one-M-lump and two-soliton wave as well as between two-M-lump and one-soliton solution are reported. Also, complex multi-soliton, solutions are offered. The simplified Hirota’s method and a long-wave method are used to construct these types of solutions. The velocity of a one-M-lump wave is studied. Straight Lines of travel for M-lump waves are also reported. To our knowledge, all gained solutions in this research paper are novel and not reported beforehand. Moreover, the gained solutions are presented graphically in three dimensions to better understand the physical phenomena of the suggested equation.

2022-08

Results in Physics (Volume : 42)

The aim of this work, is to construct some novel solutions for a new extension of the shallow water model in (3+1)-dimensions. Based on two methods namely; simplified Hirota’s method and a long-wave method a class of solutions are reported. Multiple soliton solutions in complex form, breather wave, and mixed breather-soliton solutions are constructed by using simplified Hirota’s method. To explore rational solutions and construct a class of physical interaction phenomena, a long-wave method is applied to N-soliton solutions. To our knowledge, the addressed solutions in this article are novel. Moreover, to better realize the physical meaning of obtained solutions, all solutions are plotted in three-dimension. The obtained solutions may be used in a wide variety of physical scenarios relevant to the environment, including modeling of floods and tsunamis as well as flow in rivers and open channels.

2022-06

Proceedings of the Institute of Mathematics and Mechanics (Issue : 1) (Volume : 48)

In this work, the perturbed Chen-Lee-Liu equation, which describes the propagation of an optical pulse in plasma and optical fiber is studied. The (m + 1/G′ )-expansion method is used for this purpose. As a result, bright-singular, dark-singular, dark and periodic optical soliton waves are constructed. Specific values for the parameters under conditions are also provided to display the pulse propagation of the found solutions.

2022-06

Chinese Journal of Physics (Volume : 77)

The aim of this work is a study of a generalized Kadomtsev–Petviashvili equation in (3+1)-dimensions that arise in fluid mechanics and plasma physics. We use a long-wave method to construct some new solutions for a gKP equation such as single-M-lump wave, double-M-lump wave, triple-M-lump wave, interaction phenomena of one-M-lump with one-soliton and two-soliton solutions, as well as an interaction between two-M-lump and one-soliton solutions are constructed. To our knowledge presented solutions are new and have never been described before. Moreover, obtained solutions are shown graphically to more understand its physical phenomena.

2022-03

Journal of Ocean Engineering and Science (Issue : 3) (Volume : 7)

The aim of this work is to analyze and explore the dynamics of two extensions of the Bogoyavlenskii–Schieff equation. The Hirota bilinear method is applied to the equations that arise in plasma physics. The N-soliton, fusion, rational solutions and breather solutions, as well as the interaction between M-lump and soliton solutions, are retrieved for both extended equations. The M-lump wave solutions are created from the soliton wave solutions, which are established via the Hirota bilinear method, by considering a longwave limit to the soliton wave solutions and presenting suitable conjugation conditions. The solutions for a given choice of constants are represented graphically to better understand the associated physical phenomena such as the propagation behaviors and the types of collisions.

2022-03

Journal of Ocean Engineering and Science (Issue : 3) (Volume : 7)

In this work, we use the (m+ 1/G′)-expansion method and the Adomian decomposition method to study the 3D potential Yu–Toda–Sasa–Fukuyama (3D-pYTSF) equation which has a good application in the two-layer liquid medium. For the first time, the (m+ 1/G′)-expansion and the Adomian decomposition methods are used to establish novel exact wave solutions and to study some numerical solutions for the 3D-pYTSF equation, respectively. Through using the analytical method, kink-type wave, singular solution and some complex solutions to the suggested equation are successfully revealed. The obtained wave solutions are represented with some figures in 3D and contour plots.

2021-12

International Journal of Computer Mathematics (Issue : 12) (Volume : 98)

In this work, the Date–Jimbo–Kashiwara–Miwa (DJKM) equation include time-dependent in (2+1)-dimensions that characterize the propagation of nonlinear dispersive waves in inhomogeneous media is studied. We construct rational solutions and the interaction physical phenomena between rational solution and multi-soliton wave to the time-dependent DJKM equation. The multi-soliton solutions are revealed by using the Hirota simple method, and then via the long-wave method, the M-lump solutions for two types of time-dependent are derived. Moreover, the interaction phenomena between rational solution and single-, double-soliton solutions are studied.

2021-09

Mathematical Methods in the Applied Sciences (Issue : 13) (Volume : 44)

In this literature, the dynamic characteristics of the Bogoyavlenskii–Schieff equation in (2 + 1)‐dimension that arises in plasma physics are studied. Several characteristics of multi‐soliton solutions, complex rogue wave, M‐lump solutions, fusion solutions, and interaction phenomena between M‐lump and soliton solutions also with a fusion solution are discussed. A logarithmic variable transform is used to convert the studied nonlinear equation to a Hirota trilinear form. For all solutions, three‐dimensional figures are presented to more understand its dynamic behaviors. All findings are recent, and no experts have reported them.

2021-09

Journal of Ocean Engineering and Science (Issue : 4) (Volume : 6)

In this study, we investigate the (2 + 1)-dimensional Korteweg-De Vries (KdV) equation with the extension of time-dependent coefficients. A symbolic computational method, the simplified Hirota’s method, and a long-wave method are utilized to create various exact solutions to the suggested equation. The symbolic computational method is applied to create the Lump solutions and periodic lump waves. Hirota’s method and a long-wave method are implemented to explore single-, double- and triple-M-lump waves, and interaction physical phenomena such as an interaction of single-M-lump with one-, two-soliton solutions, as well as a collision of double-M-lump with single-soliton waves. Furthermore, the simplified Hirota’s method is employed to explore complex multi-soliton solutions. To realize dynamics, the gained solutions are drawn via utilizing different arbitrary variable coefficients.

2021-08

Modern Physics Letters B (Issue : 30) (Volume : 35)

In this paper, the (2+1)-dimensional resonant Davey–Stewartson equations are solved by using two methods; namely, (m+1/G′)-expansion and (m+G′/G)-expansion methods. A wave transform is used to convert the (2+1)-dimensional resonant Davey–Stewartson (RDS) equations with M-derivative into a system of nonlinear ordinary differential equations. Different forms of solutions, such as dark, bright, singular and periodic singular solutions are successfully constructed. The obtained solutions are plotted in 3D for both M- derivative and classical derivative to more understand the effect of M-derivative on the studied equation.

2021-07

Discrete and Continuous Dynamical Systems series S (Issue : 7) (Volume : 14)

In this paper, three images of nonlinearity to the fractional Lakshmanan-Porsezian-Daniel model in birefringent fibers are investigated. The new bright, periodic wave and singular optical soliton solutions are constructed via the (m + G'/G)-expansion method, which are applicable to the dynamics within the optical fibers. All solutions are novel compared with solutions obtained via different methods. All solutions verify the conformable Lakshmanan-Porsezian- Daniel model and also, for the existence the constraint conditions are utilized. Moreover, 2D and 3D for all solutions are plotted to more understand its physical characteristics.

2021-06

Chinese Journal of Physics (Volume : 71)

In this research paper, the well-known simple Hirota’s method is employed to study the (2+1)-dimensional Sawad-Kotera equation. The logarithmic variable transformation is implemented on the proposed problem to construct the bilinear Hirota form. Based on its bilinear representation, the features of multi soliton solutions, M-lump waves, and the mixed 1-M-lump with one-soliton, and two-soliton solutions are explored. For one M-lump solution, the wave motion in the x and y directions are also studied. To better understand the physical phenomena of the gained solutions, three-dimensional graphics and their corresponding surfaces are also presented.

2021-05

AIMS Mathematics (Issue : 7) (Volume : 6)

In this paper, the modified auxiliary expansion method is used to construct some new soliton solutions of coupled Schrodinger-Boussinesq system that includes beta derivative. The new ¨ exact solution is obtained have a hyperbolic function, trigonometric function, exponential function, and rational function. These solutions might appreciate in laser and plasma sciences. It is shown that this method, provides a straightforward and powerful mathematical tool for solving the nonlinear problems. Moreover, the linear stability of this nonlinear system is analyzed.

2021-03

Modern Physics Letters B (Issue : 8) (Volume : 35)

In this paper, we consider the shallow water wave model in the (2+1)-dimensions. The Hirota simple method is applied to construct the new dynamics one-, two-, three-, N-soliton solutions, complex multi-soliton, fusion, and breather solutions. By using the quadratic function, the one-lump, mixed kink-lump and periodic lump solutions to the model are obtained. The Hirota bilinear form variable of this model is derived at first via logarithmic variable transform. The physical phenomena to this model are explored. The obtained results verify the proposed model.

2021-03

International Journal of Computer Mathematics (Volume : 98)

In this work, the Date–Jimbo–Kashiwara–Miwa (DJKM) equation include time-dependent in (2+1)-dimensions that characterize the propagation of nonlinear dispersive waves in inhomogeneous media is studied. We construct rational solutions and the interaction physical phenomena between rational solution and multi-soliton wave to the time-dependent DJKM equation. The multi-soliton solutions are revealed by using the Hirota simple method, and then via the long-wave method, the M-lump solutions for two types of time-dependent are derived. Moreover, the interaction phenomena between rational solution and single-, double-soliton solutions are studied.

2021-03

International Journal of Nonlinear Sciences and Numerical Simulation

In this research, we explore the dynamics of Caudrey–Dodd–Gibbon–Sawada–Kotera equations in (1 + 1)-dimension, such as N -soliton, and breather solutions. First, a logarithmic variable transform based on the Hirota bilinear method is defined, and then one, two, three and N -soliton solutions are constructed. A breather solution to the equation is also retrieved via N -soliton solutions. All the solutions that have been obtained are novel and plugged into the equation to guarantee their existence. 2-D, 3-D, contour plot and density plot are also presented.

2021-02

International Journal of Computer Mathematics (Issue : 8) (Volume : 98)

This paper aims at investigating the periodic wave solutions for the (3+1)-dimensional potential-Yu-Toda-Sasa-Fukuyama equation, from its bilinear form, obtained using the Hirota operator. Two major cases were studied from two different ansatzes. The 3D, 2D and density representation illustrating some cases of solutions obtained have been represented from a selection of the appropriate parameters. The modulation instability is employed to discuss the stability of got solutions. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics and so on.

2021-02

Journal of Ocean University of China (Issue : 1) (Volume : 20)

The present article deals with multi-waves and breathers solution of the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation under the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, solitary wave solutions, periodic and cross-kink solutions in which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the multi-waves and breathers, all cases the periodic and cross-kink solutions can be captured from the 1- and 2-soliton. The obtained solutions are extended with numerical simulation to analyze graphically, which results in 1- and 2-soliton solutions and also periodic and cross-kink solutions profiles. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on. We have shown that the assigned method is further general, efficient, straightforward, and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering. We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.

2021-01

Physica Scripta (Issue : 3) (Volume : 96)

We study dynamics of soliton waves, lump solutions and interaction solutions to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation, which possesses a Hirota bilinear form. Multi-soliton solutions, one-M-lump solutions, and physical interactions between solitons and 1-M-lump solutions are presented. By using a positive quadratic function, lump solutions and their interaction solutions with kink and solitary waves are also generated. To show dynamical properties and physical behaviors of the resulting solutions, 3D-plots and contour plots at different times are made and analyzed.

2020-11

Soft Computing (Issue : 25) (Volume : 24)

In this work, we attempt to construct some novel solutions of nematicons within liquid crystals including three types of nonlinearity namely Kerr, parabolic, and power law, using the generalized exponential rational function method. The investigation of nematic liquid crystals, using the proposed method, shows that there is diversity between the solutions gained via this method with those obtained via different methods. Further, we use the constraint conditions to guarantee the existence of the solutions. The W-shaped surfaces, dark soliton, bright soliton, singular soliton, period singular soliton, periodic waves, and complex solutions of the studied equations are successfully constructed. Moreover, some obtained solutions are drawn to a better understanding of the characteristics of nematicons in liquid crystals.

2020-10

Communications in Theoretical Physics (Issue : 11) (Volume : 72)

Investigating the dynamic characteristics of nonlinear models that appear in ocean science plays an important role in our lifetime. In this research, we study some features of the paired Boussinesq equation that appears for two-layered fluid flow in the shallow water waves. We extend the modified expansion function method (MEFM) to obtain abundant solutions, as well as to find new solutions. By using this newly modified method one can obtain novel and more analytic solutions comparing to MEFM. Also, numerical solutions via the Adomian decomposition scheme are discussed and favorable comparisons with analytical solutions have been done with an outstanding agreement. Besides, the instability modulation of the governing equations are explored through the linear stability analysis function. All new solutions satisfy the main coupled equation after they have been put into the governing equations.

2020-10

Konuralp Journal of Mathematics (Issue : 2) (Volume : 8)

In this manuscript, we have applied the sine-Gordon expansion method and the Bernoulli sub-equation method to seek the traveling wave solutions of the (2+1)-dimensional time-fractional partial Zoomeron equation. The exact solutions of the Zoomeron equation that are obtained by the sine-Gordon method are plotted in 3D figures, as well as the effects of the fractional derivative α are illustrated in 2D figures, while the exact solutions of the Zoomeron equation that are obtained by the Bernoulli sub-equation method are plotted in 3D figures and contour plot. Bright solutions, kink soliton, singular soliton solution, and complex solutions to the studied equation are constructed. Also, different values of the fractional parameter α are tested to study the effect of the parameter. We conclude that these methods are sufficient for seeking the exact solutions.

2020-08

Results in Physics (Volume : 19)

In this work, N-soliton waves, fusion solutions, mutiple M-lump solutions and the collision phenomena between one-M-lump and one-, two-soliton solutions to the (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation are successfully revealed. A class of one-, two-, three-soliton, one-, two-fusion solutions are derived via the Hirota bilinear method and 1-M-lump, 2-M-lump solutions are constructed via the long-wave method. Moreover, physical collision phenomenon of 1-M-lump with one-, two-soliton solutions and also, with fusion solutions are successfully presented. The velocity of the 1-M-lump wave in x- and y-direction are also studied.

2020-06

Advances in Mathematical Physics (Volume : 2020)

The multiple Exp-function method is employed for searching the multiple soliton solutions for the new extended (3+1)-dimensional Jimbo-Miwa-like (JM) equation, the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation, the generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation, and a variable-coefficient extension of the DJKM (vDJKM) equation, which contain one-soliton-, two-soliton-, and triple-soliton-kind solutions. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.

2020-02

Physica Scripta (Issue : 3) (Volume : 95)

In this paper, we use the modified auxiliary expansion method to seek some new solutions of the paraxial nonlinear Schrodinger equation. The solutions have a hyperbolic function, trigonometric function, exponential function, and rational function forms. The linear stability analysis of paraxial NLSE is also presented to study the modulational instability (MI). Two cases when the instability modulation becomes to occur are investigated. Depending on MI cases, the MI gain spectrum are also investigated and presented graphically.All solutions are new and verified the main equation of the paraxial wave equation. Moreover, the constraint conditions for the existence of soliton solutions are also showed

2020-01

Applied Sciences (Issue : 1) (Volume : 10)

In this paper, the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation in parabolic law media are investigated to obtain the dark, singular, bright-singular combo and periodic soliton solutions. Two powerful methods, the (m+G/G') improved expansion method and the exp(−ϕ (ξ)) expansion method are utilized to construct some novel solutions of the governing equations. The obtained optical soliton solutions are presented graphically to clarify their physical parameters. Moreover, to verify the existence solutions, the constraint conditions are utilized.

2020-01

Pramana - Journal of Physics (Issue : 35) (Volume : 94)

The purpose of this study is to find some novel soliton solutions of Fokas–Lenells (FL) equation where the perturbation terms are taken into account with nonlinearity. The sine-Gordon expansion method (SGEM) and the (m+(G′/G))-expansion method are used in this context. The dark, bright, dark–bright and singular optical soliton solutions are successfully obtained. Moreover, the constraint conditions for guaranteeing the existence of solutions are also given.

2019-11

Frontiers in Physics (Volume : 7)

In this paper, we use the modified exponential function method in terms of Kf(x) instead of ef(x)and the extended sinh-Gordon method to find some new family solution of the M-fractional paraxial non-linear Schrödinger equation. The novel complex and real optical soliton solutions are plotted in 2-D, 3-D with a contour plot. Moreover, the dark exact solutions, singular soliton solutions, kink-type soliton solution, and periodic dark-singular soliton solutions for M-fractional paraxial non-linear Schrödinger equation are constructed. We guarantee that all solutions are new and verified the main equation of the M-fractional paraxial wave equation. For existence, the constraint condition is also added.

2019-03

Heat Transfer Research (Issue : 7) (Volume : 50)

In this article, the magnetohydrodynamic (MHD) thermal boundary layer of a Carreau flow of Cu–water nanofluids over an exponentially permeable stretching thin plate is investigated numerically. Internal heat source/sink is also taken into account. Aft er gaining the system of leading equations, the appropriate transformations have been first employed to come across the fitting parallel conversions to alter the central governing equations into a suit of ODEs and then the renovated system of ODE along with appropriate boundary conditions is numerically solved by the shooting method with fourth-order Runge–Kutta technique. The consequences of the relevant factors of physical parameters on velocity and temperature of merging water (H2O) and nanoparticles (Cu) have been exemplified through graphs.

2019-01

Journal of Magnetics (Issue : 4) (Volume : 23)

Effects of the uniform magnetic field on aqueous magneto-Nanofluid confined in a porous domain with wall stretching/shrinking non-linearly is analyzed via this communication. The problem is modeled using continuity, momentum and energy equation along with linear thermal radiation. The effects of physical quantities are observed for Cu, Al₂O₃, TiO₂ and Ag particles in water. The coupled boundary layer PDE’s are reduced into the system of ODEs by utilizing similarity transformation and solved using shooting and Runga-Kutta fourth order technique. Stability of the obtained results are also analyzed. The results are displayed through graphs. It is observed that the momentum boundary layer is thicker when silver particles are introduced in water. Whereas, temperature profile has the minimum value for silver nanoparticles and maximum for Titanium dioxide. Also, in case of shrinking sheet dual solutions are obtained along with smallest Eigen values.

2018-11

JP Journal of Heat and Mass Transfer (Issue : 4) (Volume : 15)

The fluid flow and heat transfer behavior in a Maxwell liquid film over a stretching surface with thermal radiation and viscous dissipation is presented in this paper. The governing nonlinear thermal boundary layer that covers the physical problem is formulated and transformed into a system of higher order nonlinear ordinary differential equations using similarity transformation. The resultant systems of equations are solved numerically using Explicit Runge-Kutta scheme technique along with shooting technique. The effects of various thermophysical interesting parameters on the velocity profile, thermal energy and nanoparticle concentration, skin friction, Nusselt number and Sherwood number are discussed and analyzed graphically and numerically. Favorable comparisons with previous published papers have been done with an excellent agreement.

2017-06

International Journal of Applied and Computational Mathematics (Issue : 1) (Volume : 3)

In this paper, three dimensional incompressible Casson fluid flow past a linear stretching porous plate under the magnetic field effect is analyzed using differential transformation method (DTM) and numerical method. For increasing the accuracy of DTM, Padé approximation is applied. Comparison betweenDTM-Padé and numerical method shows that Padé with the order [15,15] can be an exact and high-efficiency procedure for solving these kinds of problems. The influence of the Casson fluid parameter (β), Prandtl number (Pr), a magnetic parameter (M), stretching parameter (c) and porous parameter (λ) on non-dimensional temperature and velocity profiles are investigated. The results indicated for the Casson fluid flow that increasing the Hartmann number make a decrease in the velocity boundary layer thicknesses.

2017-06

Advances and Applications in Fluid Mechanics (Issue : 4) (Volume : 20)

This research is presented to analyze and investigate the magnetohydrodynamic Casson flow of liquid film over the stretching surface. Continuity and Navier-Stokes equation that covers problem is transformed to the nonlinear ordinary differential equation, next arising governing equations are solved numerically using ADMPade in order to investigate variations of interesting parameters on the velocity profile, finally all physical parameters are presented graphically.

2017-05

International Journal of Advanced and Applied Sciences (Issue : 7) (Volume : 4)

In this research, Carreau-Casson Fluids flow under the effect of energy transfer with internal heat source/sink and radiation over a stretching sheet are being analyzed and investigated. Shooting method with the help of 4-order Runge-Kutta (RK4) integration technique is applied to the governing equation of fluid flow and heat equation. The effect of dimensionless governing parameters on velocity, thermal profiles along with the friction factors and local Nusselt numbers are showed graphically and numerically. Different physical interesting parameters on the fluid velocity and heat equation are described visually and numerically.

2016-12

International Journal of Advanced and Applied Sciences (Issue : 1) (Volume : 4)

In this research, we analyze heat transfer of MHD boundary layer flow of Casson fluid. Strong nonlinear ordinary differential equations are solved numerically using Shooting method with fourth-order Runge-Kutta (RK4) integration technique. Variations of interesting different parameters on the velocity are shown graphically.

2016-04

International journal of pure and Applied Mathematics (Issue : 2) (Volume : 107)

In this article, the flow of liquid film over an unsteady elastic stretching surface is analyzed. Similarity transformations are used to transform the governing equations to a nonlinear ordinary differential equation. The differential equation reformulated to a system of Volterra integral equations and solved analytically using the new technique of numerical solution called homotopy perturbation method. The results of the proposed method are compared with previously published work and the results are found to be in an excellent agreement. Also, discussed and presented graphically the effects of various parameters Darcy number and unsteadiness parameter.

2014-05

Al-Rafidain Journal of Computer sciences and Mathematics, (Issue : 2) (Volume : 11)

In this paper, the stability and dynamics of thin liquid films flowing down on an inclined plane are investigated by using an integral approximation. The strong non-linear evolution equations are derived by the integral approximation with a specified velocity profile. The evolution equations are used to study the linear stability for liquid films. As a result, an output of this research, we showed that the effect of inclination of films is an unstable factor.

2014-01

IOSR Journal of Engineering (IOSRJEN) (Issue : 1) (Volume : 4)

In this paper, we study the traveling wave solutions of a reaction-diffusion system with a slow reaction and a slow diffusion for one component. We use a semi-implicit method and finite element method (COMSOL software) for solving this system. We compare both methods and we found an excellent agreement between the solutions.

2013-09

International Journal of Mathematical Archive (Issue : 9) (Volume : 4)

This research is dedicated for analyzing the stability of thin liquid film in the presence of heat and surfactant, the fluid flow of thin liquid films represents by Navier-Stokes equation and equation of continuity in two dimensional forms as shown in figure (1). We find that the stability of films occur when the Prandtl and Schmidt numbersare greater than zero,otherwise the film become unstable