In this work, the non-integrable nonplanar (cylindrical and spherical) damped Kawahara equation (ndKE) is solved and analyzed analytically. The ansatz method is implemented for analyzing the ndKE in order to derive some high-accurate and more stable analytical approximations. Based on this method, two-different and general formulas for the analytical approximations are derived. The obtained solutions are applied for studying the distinctive features for both cylindrical and spherical dissipative dressed solitons and cnoidal waves in a complex plasma having superthermal ions. Moreover, the accuracy of the obtained approximations is numerically examined by estimating the global maximum residual error. Also, a general formula for the nonplanar dissipative dressed solitons energy is derived in detail. This formula can recover the energy of the nonplanar dissipative dressed solitons, the planar dressed solitons, the planar damped dressed solitons, and the nonplanar dressed solitons. Both the suggested method and obtained approximations can help a large sector of authors interested in studying the nonlinear and complicated phenomena in various fields of science such as the propagating of nonlinear phenomena in physics of plasmas, nonlinear optics, communications, oceans, and seas.

This work examines the Painlevé integrability of a (3 + 1)-dimensional Boussinesq-type equation. Using the Mathematica program, we rigorously establish Painlevé's integrability for the suggested problem. By utilizing Hirota's bilinear technique, we obtain the dispersion relations and phase shifts, which enable us to derive multiple soliton solutions. In addition, we systematically derive a wide range of lump solutions using the Maple symbolic computation. The investigation extends to encompass a variety of exact solutions with distinct structural features, including kink, periodic, singular, and rational solutions. This comprehensive analysis illustrates the profound richness of the model's dynamics and its potential to elucidate diverse nonlinear wave phenomena across various physical contexts. Therefore, the results that we will obtain play a vital role in understanding the mechanism of generation and propagation of many mysterious phenomena that arise in various scientific fields, including plasma physics, fluid mechanics, and the propagation of waves on the surfaces of seas and oceans to optical fibers.

This work develops two higher-dimensional extensions for both Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. We investigate the Painlevé integrability of each couple of the aforementioned two models. We show that the Painlevé integrability fails for one equation of each couple but holds true for the x-derivative of this model. We examine multiple soliton solutions for the integrable extensions of these two models by utilizing the bilinear form. The outcomes will contribute to a deep understanding of the propagation mechanism of the propagation and interaction of multi-solitons in a variety of nonlinear media, including sea waves, optical fibers, and plasma physics.

In this work, two new (3 + 1)-dimensional integrable wave equations are investigated. The complete Painlevé integrability of the two suggested equations will be investigated using Mathematica. We employ the method of Hirota to formally derive two sets of multiple soliton solutions for the two suggested models. Additionally, using symbolic computation with Maple, we provide a variety of lump solutions for the two suggested models. Other exact solutions of distinct structures, such as periodic, singular, and many other physical nonlinear structures, will be determined. We should mention here that the proposed two new models will assist many authors that are working in the field of fluids and plasma physics, in understanding the scenarios of the nonlinear waves that arise in different physical systems. Also, this study will contribute to understanding the nature of nonlinear waves that arise in the seas and oceans.

This study analyzes the family of one of the most essential fractional differential equations due to its wide applications in physics and engineering: the multidimensional fractional linear and nonlinear diffusion equations. The Caputo fractional derivative operator is used to treat the time-fractional derivative. To complete the analysis and generate more stable and highly accurate approximations of the proposed models, three extremely effective techniques, known as the direct Tantawy technique, the new iterative transform technique (NITM), and the homotopy perturbation transform method (HPTM), which combine the Elzaki transform (ET) with the new iterative method (NIM), and the homotopy perturbation method (HPM), are employed. These reliable approaches produce more stable and highly accurate analytical approximations in series form, which converge to the exact solutions after a few iterations. As the number of terms/iterations in the problems series solution rises, it is found that the derived approximations are closely related to each problem’s exact solutions. The two- and three-dimensional graphical representations are considered to understand the mechanism and dynamics of the nonlinear phenomena described by the derived approximations. Moreover, both the absolute and residual errors for all generated approximations are estimated to demonstrate the high accuracy of all derived approximations. The obtained results are encouraging and appropriate for investigating diffusion problems. The primary benefit lies in the fact that our proposed plan does not necessitate any presumptions or limitations on variables that might affect the real problems. One of the most essential features of the proposed methods is the low computational cost and fast computations, especially for the Tantawy technique. The findings of the present study will be valuable as a tool for handling fractional partial differential equation solutions. These approaches are essential in solving the problem and moving beyond the restrictions on variables that could make modeling the problem challenging.