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In this work, we proposed a new modified generalized Riccati equation mapping approach to successfully extract several analytical soliton solutions for the (2+1)-dimensional nonlinear Schrödinger (NLS) equation with the help of symbolic computation works in Mathematica.The (2+1)-dimensional NLS equation is used in many fields, including plasma physics, nonlinear optics, and quantum electrodynamics. The main objective of the present work is to develop an effective methodology for solving highly nonlinear evolution equations that are influenced by the enhancement of a previously known method. The approach under consideration is a newly improved version of the classic generalized Riccati equation mapping. By taking advantage of this newly proposed method, we produced a wide range of closed-form solutions, including new optical solitons, traveling waves, and soliton-like solutions, all of which are crucial for nonlinear optics, optical fibers, and the physical propagation of electromagnetic waves. One may clearly argue that the novel method is highly effective and successful in finding exact solutions to nonlinear evolution equations. Moreover, we obtained a variety of new families of soliton-like wave solutions. By using the mathematical software Mathematica, we also created 2D, 3D, and contour graphics for some of the reported solutions by choosing suitable parameter values. [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 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 work, we use two extremely efficient methods, generalized Kudryashov (GK) and generalized Riccati equation mapping (GREM), to extract various optical wave soliton solutions to the (2+1)-dimensional Davey-Stewartson Fokas (DS-Fokas) system, which is an ideal model for nonlinear pulse propagation in monomode optical fibers. The employed methods are very efficient and robust mathematical approaches for solving various nonlinear models of a variety of nonlinear Schrödinger's equations (NLSEs) in mathematical physics and sciences. Firstly, the traveling wave transformation converts the given nonlinear equation with a partial derivative into an ordinary differential equation (ODE). Then, different novel types of optical soliton solutions are attained using the abovementioned methods. Indeed, the obtained solutions are beneficial and important for explaining the physical phenomena of the DS-Fokas model. In the literature survey, we have found that these acquired solutions are very new and have not been discussed earlier. Some numerical simulations of the solutions have been performed for a better understanding of the results obtained with the use of symbolic computations. Furthermore, these obtained solutions are discussed graphically to attain a deep knowledge and vision of the mechanisms of all nonlinear phenomena. As a result, the wave profiles of the lump soliton, bell-shaped soliton, anti-bell-shaped soliton, periodic soliton, multisoliton, and other solitons have been displayed using three-dimensional (3D) contour plots and two-dimensional (2D) plots created with the computer software Mathematica 11.3. Moreover, the above-utilized techniques have been assumed to be essential tools for describing some nonlinear physical phenomena, namely, the nonlinear pulse propagations in monomode optical fibers, fluid mechanics, and plasma physics. [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 work, we investigate coupled space–time fractional nonlinear Schrödinger equation (FNSE) arising in physics. The FNSE can be utilized to explain non-relativistic quantum mechanical phenomena. With the aid of the conformable fractional derivative (CFD) and fractional complex transform (FCT), we implement the extensive direct algebraic approach (EDAA), the behaviors of some of the generated solutions are shown as 3D-graphics for various with different fractal orders. The optical soliton solutions that are bright periodic, kink bright and kink-bright periodic are among these precise solutions. The acquired results demonstrate the simplicity, effectiveness, and capacity to produce additional kinds of exact solutions of these proposed methods, which are useful in deciphering the intricate physical interpretation of space–time FNSE. [ABSTRACT FROM AUTHOR]