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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 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 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]