The propagation of electron-acoustic waves (EAWs) in an unmagnetized plasma, comprising $(r,q)$-distributed hot electrons, cold inertial electrons, and stationary positive ions, is investigated. Both the unmodulated and modulated EAWs, such as solitary waves, rogue waves (RWs), and breathers are discussed. The Sagdeev potential approach is employed to determine the existence domain of electron acoustic solitary structures and study the perfectly symmetric planar nonlinear unmodulated structures. Moreover, the nonlinear Schrödinger equation (NLSE) is derived and its modulated solutions, including first order RWs (Peregrine soliton), higher-order RWs (super RWs), and breathers (Akhmediev breathers and Kuznetsov–Ma soliton) are presented. The effects of plasma parameters and, in particular, the effects of spectral indices r and q, of distribution functions on the characteristics of both unmodulated and modulated EAWs, are examined in detail. In a limited cases, the $(r,q)$ distribution is compared with Maxwellian and kappa distributions. The present investigation may be beneficial to comprehend and predict the modulated and unmodulated electron acoustic structures in laboratory and space plasmas.

Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, i.e., solitary waves, to the evolution equation, two different methods are devoted for this purpose. In the first one, new hypotheses with Cole–Hopf transformation are employed to find general solitary wave solutions. In the second one, the ansatz method with hyperbolic sech algorithm are utilized to obtain a general solitary wave solution. The obtained solutions recover the solitary wave solutions to all one-dimensional KdV equations with a power law nonlinearity, such as the KdV equation with quadratic nonlinearity, the modified KdV (mKdV) equation with cubic nonlinearity, the super KdV equation with quartic nonlinearity, and so on. Furthermore, two different approaches with two different formulas for the Weierstrass elliptic functions (WSEFs) are adopted for deriving some general periodic wave solutions to the evolution equation. Additionally, in the form of Jacobi elliptic functions (JEFs), the cnoidal wave solutions to the KdV-, mKdV-, and SKdV equations are obtained. These results help many authors to understand the mystery of several nonlinear phenomena in different branches of sciences, such as plasma physics, fluid mechanics, nonlinear optics, Bose Einstein condensates, and so on.

This study extends the concept of cracking to self-gravitating, spherically symmetric compact objects in modified $$f(R, \varphi )$$ f ( R , φ ) theory of gravity, where R represents the Ricci scalar, and $$\varphi $$ φ is the scalar potential. In this regard, we consider spherically symmetric spacetime characterized with an anisotropic matter to detect the instabilities of self-gravitating compact objects via cracking and overturning. Further, we construct the general framework to observe the cracking and overturning points by applying the local density perturbation technique to the configuration governed by barotropic equation of state. The effectiveness of this approach is assessed by analyzing its results on the data of Her X-1, SAX J1808.4-3658, and 4U 1820-30 respectively. It is concluded that cracking points appear in the different interior regions of these three stars. Significantly, this study illustrates the effectiveness of the cracking approach by highlighting the regions sensitive to localized density disruptions, offering valuable insights into the structural behavior of compact stars within a modified gravity framework.

In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First, we introduce a novel method that combines the Laplace transform tool and the residual power series approach. We specifically give the specifics of how to apply the suggested approach to solve time-fractional nonlinear equations. Second, we use the FHE to evaluate the method’s efficacy and validity. Using 2D and 3D plots of the solutions, the derived and precise solutions are compared, confirming the suggested method’s improved accuracy. The results for nonfractional approximate and accurate solutions, as well as fractional approximation solutions for various fractional orders, are indicated in the tables. The relationship between the derived solutions and the actual solutions to each problem is examined, showing that the solution converges to the actual solution as the number of terms in the series solution of the problems increases. Two examples are shown to demonstrate the effectiveness of the suggested approach in solving various categories of fractional partial differential equations. It is evident from the estimated values that the procedure is precise and simple and that it can therefore be further extended to linear and nonlinear issues.

Kinetic Alfvén waves (KAWs) are investigated in an Oxygen–Hydrogen plasma with electrons following the behavior of $rq$-distribution in an upper ionosphere. We aim to study low-frequency and long wavelengths at 1700 kms in the upper ionosphere of Earth as detected by Freja satellite. The fluid model and reductive perturbation method are combined to obtain the evolutionary wave equations that can be used to describe both fractional and non-fractional KAWs in an Oxygen–Hydrogen ion plasma. This procedure is used to obtain the integer-order Korteweg–de Vries (KdV) equation and then analyze its solitary wave solution. In addition, this study is carried out to evaluate the fractional KdV (FKdV) equation using a new approach called the “Tantawy technique” in order to generate more stable and highly accurate approximations that will be utilized to accurately depict physical events. This investigation also helps locate the existence regions of the solitary waves (SWs), and in turn displays that the characteristics of KAWs are affected by a number of physical factors, such as the nonthermal parameters/spectral indices “r”, “q”, and obliqueness (characterized by $l_z$). Depending on the parameter governing the distribution, especially the nonthermality of inertialess electrons, the $rq$-distribution of electrons has a major impact on the properties of KAWs.

In this article, some fractional Hardy-Leindler-type inequalities will be illustrated by utilizing the chain law, Hölder’s inequality, and integration by parts on fractional time scales. As a result of this, some classical integral inequalities will be obtained. Also, we would have a variety of well-known dynamic inequalities as special cases from our outcomes when α = 1 .