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In this work, we use two newly powerful and efficient techniques, the unified method and the unified Riccati equation expansion (UREE) method, to derive solitary wave solutions for the (1 + 1)-dimensional longitudinal wave equation (LWE), presenting a systematic approach to reveal the underlying dynamics. By incorporating mathematical analysis and numerical simulations, we investigate the behavior and properties of these solitary wave solutions in various fields of science. The derived solutions are entirely new, and these findings show the essential complexities of the longitudinal wave equation. These obtained solutions shed light on the fundamental mechanisms governing solitary waves in the longitudinal wave equation, contributing to a deeper understanding of nonlinear wave phenomena. We present effective visualizations of the dynamical wave structures using various graphs, such as 3D, 2D, and contour plots in the obtained solutions to improve our comprehensive understanding. Consequently, numerous types of wave profiles, including periodic wave, interaction between periodic wave and kink wave, W-shape wave, kink wave, interaction between breather wave and kink wave, traveling wave, mixed periodic, singular soliton, and some new types of soliton profiles were found. Consequently, this work presents a comprehensive study of solitary wave solutions using the unified method and UREE method for the (1 + 1)-dimensional LWE equation. The outcomes of the current study manifest that the considered methods are significant and systematic in solving nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this work, we implement the bifurcation theory for the planar dynamical system and the complete discrimination system method of polynomials to the integrable Kuralay equations. Solitons generated by these equations are found to be relevant in diverse fields such as ferromagnetic materials, nonlinear optics, and optical fibers. Through specific wave transformations, it undergoes conversion into an ordinary differential equation (ODE), which is subsequently transformed into a planar dynamical system associated with a one-dimensional Hamiltonian function. As per the qualitative theory of the planar dynamical system, phase portraits of the Hamiltonian system are plotted and used to construct some new traveling wave solutions. Numerical examination reveals diverse nonlinear structures in the analytical solutions, encompassing solitary waves, kink waves, and periodic wave profiles. Additionally, the integrable Kuralay equation employs the complete discrimination system method of polynomial for the first time, yielding solutions expressed in trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. Visual representations of the derived solutions include 3-D, 2-D, and contour plots. The reliability and effectiveness are affirmed through the numerical graphs of the solutions. Furthermore, we conducted a numerical investigation into the chaotic and quasiperiodic behavior of the perturbed system by introducing a specific periodic force into the primary system. [ABSTRACT FROM AUTHOR]

The exact solutions of the (2 + 1) dimensional Broer–Kaup–Kupershmidt (BKK) system which has been recommended to model the nonlinear and dispersive long gravity waves traveling along with the two horizontal directions in the shallow water of uniform depth were obtained. Firstly, the given system was reduced to an ordinary differential equation (ODE) with the help of the wave transformations. Then, the reduced ODE was solved with the help of two methods which are called the modified (G ′ / G) -expansion method and new extended generalized Kudryashov method. We checked the results with the Maple software and plotted 3D, contour and 2D plots of some obtained solutions. As a result, we obtained exact solutions that are different from each other and have not been obtained before. Results can enhance the nonlinear dynamical behavior of a given system and demonstrate the effectiveness of the employed methodology. Results will be beneficial to a large number of engineering model specialists and useful for understanding the wave motions. [ABSTRACT FROM AUTHOR]

In this paper, we used the generalized exponential rational function approach to extract a collection of different optical wave solutions to the highly nonlinear Schrödinger equation in (3+1) dimensions, which illustrates the formation of ultra-short optical pulses in highly nonlinear media. The applied approach is straightforward and robust, and it can extract various types of optical soliton solutions into a single framework. Dark and bright solitons, singular-form solitons, periodic waves, mixed-form solitons, and rational, exponential, and complex solutions are among the outcomes that are significant to diverse applied scientific applications in nonlinear optics, and nonlinear sciences. Finally, several soliton solutions in 3D, 2D and contour graphics, as well as the interactions of evolutionary waves, are presented to better understand the changing dynamics of soliton wave solutions in the model under consideration. The findings of this study are more beneficial, useful, and favorable in the real-world applications of nonlinear optics, optical communications, and optical fibers. [ABSTRACT FROM AUTHOR]

This paper analyzes the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation, which appears in the fields of applied magnetism and nanophysics. By taking advantage of two mathematical integration approaches, namely, the modified generalized exponential rational function method and the extended tanh function method, a variety of exact optical soliton solutions are obtained for the governing cmKdV equation. These acquired soliton solutions are determined in terms of hyperbolic, exponential, and trigonometric function types. By choosing suitable values of parameters, some 3D, 2D, and contour plots are portrayed with the aid of symbolic computation in Mathematica to visualize the underlying dynamics of the generated solutions. These solutions include doubly soliton, multi-soliton, singular periodic soliton, anti-bell-shaped soliton, and hyperbolic structures. Moreover, the modulation instability of the governing equation is also investigated by using the linear stability analysis. The results presented in this paper are novel and are reported for the first time in the literature. Again, modulation instability analysis was carried out on the governing model for the first time. Thus, the results obtained demonstrate that the two new mathematical schemes are quite concise and effective and can be useful in understanding the dynamical behaviors of many other nonlinear physical models appearing in nonlinear optics, nanophysics, and so many other areas of nonlinear sciences. [ABSTRACT FROM AUTHOR]

In this work, we use the generalized Riccati equation mapping method (GREMM) to seek solitary and periodic wave solutions to the generalized complex coupled Schrödinger–Boussinesq equations, which is a more useful model for describing the stationary propagation of coupled nonlinear upper-hybrid and magnetosonic waves in magnetized plasma. The GREMM approach is applied to compute a variety of soliton and other solutions in distinctive forms such as trigonometric, hyperbolic, singular, and interaction solutions in addition to rational. The main feature of this work is the achieved soliton solutions, which were not before constructed and are more general. All of the obtained solutions are validated by re-inserting them into the governing system with the help of mathematica. Moreover, 3D and 2D combined line profiles, as well as their contour graphics, are plotted to depict the dynamical behaviors of the retrieved solutions using numerical simulations. We exhibit how the wave profiles are impacted with varying time. Computational results demonstrate that applied research methodology and analytical results enrich the dynamics of the governing model. [ABSTRACT FROM AUTHOR]

In this research, we concentrated on the dispersive long wave system in two horizontal directions for dispersive nonlinear waves on the shallow water of an open sea or a wide channel of finite depth. We investigated this governing system by using two different methodologies, namely the generalized exponential rational function (GERF) method, and the new modified generalized exponential rational function (MGERF) method. The GERF method was first introduced by Ghanbari and Inc (Eur. Phys. J. Plus 133:-142, 2018) for finding the soliton solutions for highly nonlinear partial differential equations (NLPDEs). This technique is very reliable and straightforward and reduces the NLPDEs into ordinary differential equations (ODEs) under the wave transformation. Being motivated by the GERF technique, we proposed a newly modified generalized exponential rational function (MGERF) method under wave transformation. We obtained a diverse set of solutions involving trigonometric forms, hyperbolic forms, rational forms, and so on, which have a broad application spectrum in fields such as plasma physics, nonlinear optics, optical fibers, and nonlinear sciences by utilizing these methods. Due to the presence of various arbitrarily chosen constants, these solutions exhibit extensive and rich dynamical behavior. Based on the dynamical behaviors, we discovered that the soliton solutions were collisions of solitons, breather-like solitons, line-form solitons, multi-solitons, solitary waves, lump-form solitons, and other forms. Consider a nonlinear system with dispersive and dispersion terms, a nonlinear Schrödinger equation, and fractional nonlinear evolution equations, which will yield additional interesting and more achievable results. Finding solutions to these equations in solitary wave solution forms will be a difficult task. [ABSTRACT FROM AUTHOR]

In this paper, we use two efficient mathematical approaches to obtain a variety of soliton solutions to the (3+1)-dimensional Schrödinger equation: the generalized Riccati equation mapping method and the newly proposed modified generalized exponential rational function method. These techniques extracted standard, illustrative, rich dynamical structures, and further comprehensive soliton solutions and traveling wave solutions involving hyperbolic form, trigonometric form, and exponential form. The obtained results have been verified by placing them back into the mentioned nonlinear partial differential equation via symbolic computation in Mathematica. Thereafter, the graphical demonstrations of some attained solutions are discussed for a better understanding of the physical phenomenon. We have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the generalized Riccati equation mapping method and modified generalized exponential rational function techniques for extracting soliton solutions to nonlinear partial differential equations are efficient, compatible, and reliable in nonlinear sciences, optical fibers, and engineering. [ABSTRACT FROM AUTHOR]

In this work, we studied the perturbed Chen–Lee–Liu (CLL) equation by taking advantage of a very reliable and efficient approach, namely, the generalized Riccati equation mapping method for describing propagation pulses in optical fiber. To find the soliton solutions to the perturbed CLL equation, we first apply the traveling wave transformation to reduce the considered Schrödinger equation to a system of an ordinary differential equation in the context of the real and imaginary parts of the perturbed CLL equation. Thereafter, after taking some conditions on the involved parameters, we obtained the optical soliton solutions for the governing equation. To understand the behavior of attained solutions, we discussed the real, imaginary, and absolute parts of solutions with 2-dimensional and 3-dimensional plots. The dynamical behavior of the obtained solutions demonstrates that they are solitary wave solitons, mixed periodic solitons, anti-bell shape solitons, solitary wave with lump solitons, singular type solitons, periodic wave solitons, and so on. [ABSTRACT FROM AUTHOR]

The primary goal of this work is to derive newly constructed invariant solutions, conservation laws, and modulation instability in the context of the (2+1)-dimensional cubic nonlinear Schrödinger's equation (cNLSE), which explains the phenomenon of soliton propagation along optical fibers. The nonlinear Schrödinger equations and their various formulations hold significant importance across a wide spectrum of scientific disciplines, especially in nonlinear optics, optical fiber, quantum electronics, and plasma physics. In this context, we utilize Lie symmetry analysis to determine the vector fields and assess the optimality of the governing equation. Upon establishing the optimal system, we derive similarity reduction equations, thereby converting the system of partial differential equations into a set of ordinary differential equations. Through the simplification of these ordinary differential equations, we are able to construct several optically invariant solutions for the governing equation. Furthermore, through the utilization of the generalized exponential rational function (GERF) approach, we have derived additional intriguing closed-form solutions. Conservation laws are derived for the governing equation by utilizing the resulting symmetries introduced by the Ibragimov scheme. Furthermore, modulation instability and gain spectrum are derived for this equation to understand the correlation between nonlinearity and dispersive effects. To provide a comprehensive and insightful portrayal of our findings, we have created three-dimensional (3D) visualizations of these solutions, which reveal the periodic waves and the solitary wave structures. The form of cubic nonlinear Schrödinger's equation discussed in this article and the optical solutions obtained have never been studied before. Also, these attained solutions can be beneficial to study analytically the identical models arising in fluid dynamics, birefringent fibers, plasma physics, and other optical areas. [ABSTRACT FROM AUTHOR]

In this paper, new soliton solutions with Atangana–Baleanu (AB) fractional for ion sound and Langmuir waves (ISALWs) are presented. An extended sub-equation method (ESEM) is used in symbolic software to deal with this governing model to obtain various soliton solutions in terms of trigonometric functions, hyperbolic functions, and other functional forms. Brand-new and general solitary wave solutions were obtained for the governing model. The behavior of the soliton solutions was also examined with the three-dimensional (3D) graphics. The dynamical behavior of the obtained solutions shows that they are various varieties of soliton shapes, including bell-shaped, single-soliton-shaped, and mixed-type solitons. This approach is thought to be a promising one for handling a variety of AB fractional evolution systems to validate the effectiveness of the ESEM. [ABSTRACT FROM AUTHOR]