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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 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]