Emad Kheder El-Shewy

This work explored the complicated coupled (1+1)‐dimensions Kuralay model, which is an integrable spin mode. The Riccati–Bernoulli subsidiary ordinary‐differential equations method has been applied to obtain general mathematical solutions in the form of rational, hyperbolic, trigonometric and exponential solutions. The dynamical properties of this solution forms in our physical model appears in the form of new novel nonlinear solutions as explosive cnoidal solutions, blow up solution, oscillatory, explosive envelope, periodic explosive envelope, stationary explosive, super form, dark and bright envelope solitons. For appropriate free physical parameter values, significant changes in the wave amplitudes of this solutions have been produced without change in phase. The obtained results by proposed method clarify the simplicity and efficacy of our method in examined nonlinear modes used in mathematical physics.

In this paper, we consider the Chiral nonlinear Schödinger equation (CNLSE), where the multiplicative noises term varies arbitrarily over time. This equation defines several edge states of Hall effect characteristics in quantum physics applications. We apply the sine-Gordon expansion method to produce some new stochastic solutions for the CNLSE. Some solitary and dissipative solutions were obtained in the form of rational, envelope and shock structures. We demonstrate how the multiplicative noise and model parameters affects the way the solutions behave. We provide some configurations for the both deterministic and stochastic solutions to illustrate their behaviour. It is known that noise dominates envelope growing, damping, and all wave propagation. As it is achieved, the results presented here are crucial to the development of quantum physics. The proposed methodology can be developed to solve more complex problems in applied science.

In this paper, we use the unified solver technique to investigate the effective irregular wave propagations in dispersive and dissipative modes via the (2 + 1)-dimensional Fokas model, which describe a simple extension of the nonlinear Schrödinger equation. Using a complex traveling wave transformation, the Fokas model is transformed into a nonlinear ordinary differential equation. The unified solver method subsequently produces several kinds of solitary wave solutions. More precisely, it provides solutions in trigonometric, hyperbolic, dissipative, rational, solitonic, and super soliton forms. Among the many benefits of the given technique are the reduction of complex computations and the concise presentation of important results. To demonstrate the wave structures for the dispersive and dissipative Fokas model, two-dimensional, three-dimensional, and contour plots of selected solutions are created utilizing the MATLAB software. We also demonstrate how physical parameters affect the acquired solutions’ behavior. The suggested method could be improved to address more complex applied science problems.

In this study, we used the Riccati–Bernoulli technique to generate new solitary structures for the model of the nonlinear two-dimensional Schrödinger equation (NTDNLSE) with a source term. These structures included rational, periodic, solitary, explosive, dissipative, and blow-up solitons. To demonstrate the shape of the obtained solutions for suitable free parameter values, a few 2D and 3D charts are displayed. The dynamical and phase portrait behaviors with the aid of model parameters controlled the type of obtained NTDNLSE solitary solutions. The properties of NTDNLSE structure solutions that depend on the Kerr, drift, and dispersion have been investigated. Many fluid and space plasma applications depend on the dynamical behaviors of these solutions. The results show how straightforward and quick this method is for constructing elaborate nonlinear wave patterns and rich exact solutions to nonlinear equations in mathematical physics.

In this article, we present a unified mathematical solver that is widely used in applied nonlinear differential equations based on the well-known tanh–coth method. We employ this solver to provide new solutions for the unstable and modified unstable nonlinear Schrödinger models. We also introduce He’s frequency technique to examine the dynamical behavior properties. The presented solutions include various mathematical structures, such as solitons, dissipative super-waves, bright and dark envelope waves, and explosive envelope waves. The behavior of the presented solutions for suitable free parameter values is depicted by two-dimensional and three-dimensional graphs. The proposed approach offers various benefits, such as eliminating time-consuming and difficult calculations. The results presented here can be utilized to characterize a wide range of physical phenomena in numerous disciplines of practical study, such as pulse propagation in optical fibers and quantum field. Finally, this study increases the current understanding of mathematical methods for solving nonlinear differential equations, which may be applied to other nonlinear wave equations in other scientific areas.

This article analyzes and examines the nonlinear Maccari’s equations with a noise term using the powerful sub-equation mathematical approach. Several significant solitary and rational solutions have been provided. These solutions are crucial in a number of fields, such as superfluid, optical fiber communication, and space-confined plasma waves. The effects of model parameters on the obtained solutions’ features have been discussed. The randomness parameter affects the envelope, solitonic structures, and energy properties. It was reported that by increasing the random parameter, Brownian isolated driving shock waves with random amplitude were produced. Finally, the method described here may be applied to many nonlinear systems and new energy trends in natural science.

For many nonlinear applications described by the dynamics of nonlinear Schrödinger equation with higher-order terms (HONLSE) such as nonlinear optics, space plasma physics molecular biology, astrophysics, quantum mechanics, superfluid, fluid mechanics, and fiber optics communications, a unique closed-form solution have been obtained using energy equation. In addition, some new solitary solutions HONLSE have been obtained via the unified solver method. The resultant solutions behave as breathers, super solitons, envelope breathers, blow up, localized super waves, periodical super shock, train super solitons, and shock structures. The modulations of Kerr nonlinear, chromatic dispersive, and wave packet drift parameters on the wave characteristics of the obtained solutions have been investigated. It was reported that the model parameters affect the amplitude, steepness, and width of the resultant structures. The provided solution can be used as a box solver for a variety of natural science systems described by distinct nonlinear equations.

In this work, the higher order electrostatic and solitonic EA energy plasma in auroral plasma has been investigated. The perturbed KdV form has been solved. The modulated soliton speed, energy of the cold beam electrons, and the associated higher fifth order electrostatic field have been mathematically derived. The effect of perturbed beam density as well as beam temperature parameters on the dressed electrostatic and energy properties has been discussed. The auroral plasma results in this work may be important in space applications.

A model of nonlinear electron acoustic electrostatic plasma waves having electron beams has been investigated. The structure of Korteweg–de Vries soliton energy and the corresponding electrostatic field have been discussed. The parametric regimes for wave existence of compressive and rarefactive structures have been examined via auroral zone data. The effects of densities and thermal temperature of electrons and beams on the nature of the electrostatic field and fluid energies have been taken into account.