This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable $(G^{\prime }/G,1/G)$-expansion and generalized exp($-\varphi \left(\xi \right)$)-expansion techniques are presented to construct the ion-acoustic wave results of a (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) model. As a result, the novel soliton and other wave solutions in a variety of forms, including kink- and anti-kink-type breather waves, dark and bright solitons, kink solitons, and multi-peak solitons, etc., are attained. With the help of software, the solitary wave results (that signify the electrostatic potential field), electric and magnetic fields, and quantum statistical pressures are also constructed. These solutions have numerous applications in various areas of physics and other areas of applied sciences. Graphical representations of some of the obtained results, and the electric and magnetic fields as well as the electrostatic field potential are also presented. These results demonstrate the effectiveness of the presented techniques, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields.

In this paper, we investigate the soliton solution of the Zakharov–Kuznetsov (Z–K) equation using a modified unified method. The Z–K equation pertains to the evolution of quasi- (1D) shallow-water waves under conditions, where viscosity and surface tension effects can be disregarded. By employing the newly modified unified method, we derive analytical solutions employing hyperbolic, trigonometric, rational, and exponential functions. Interestingly, several of these solutions are novel and have not been described earlier. These distinct wave solutions hold noteworthy applications in various fields, including applied sciences and engineering. By setting particular values for the solution parameters, we reveal new graphical patterns that escalate our interpretation of the physical behavior within this model. Computational efforts and outcomes underscore the potency and robustness of the suggested technique, suggesting its potential applicability to various nonlinear models appearing in mathematical physics and diverse scientific domains. Moreover, this technique can be utilized to solve other intricate Z–K equations in the field of mathematical physics.

In modern world, most of the optimization problems are nonconvex which are neither convex nor concave. The objective of this research is to study a class of nonconvex functions, namely, strongly nonconvex functions. We establish inequalities of Hermite-Hadamard and Fejér type for strongly nonconvex functions in generalized sense. Moreover, we establish some fractional integral inequalities for strongly nonconvex functions in generalized sense in the setting of Riemann-Liouville integral operators.

In this regard, some exact traveling wave solutions of the (4 + 1)-dimensional Fokas equation are newly presented. The aforementioned exact traveling wave solutions have been successfully attained with a new approach known as the extended generalized Riccati equation mapping method. As a consequence, traveling wave solutions will be gained in the form of bright soliton solutions, periodic solutions, breather soliton solutions, lump soliton solutions, and others. Moreover, the stability analysis and conserved quantities of the aforementioned solutions are also carefully discussed. The results are quite useful in the field of applied science.

The propagation of ultra-short light pulses in a monomode optical fiber is governed by a higher-order nonlinear Schrödinger equation, which constitutes its foundation. In this study, we constructed different optical solitons and wave solutions based on the modified generalized Riccati equation mapping approach for this dynamical model. The proposed approach finds applications in engineering, mathematics, physics, fiber optic communications, nonlinear optics, plasma physics, quantum optics and computing, biophysics, and condensed matter physics. To bridle the explanatory power, we analytically described the existence conditions pertaining to the above-mentioned soliton types by varying the relevant model parameters: bright and dark solitons, periodic solitons, singular solitons, peaked solitons, and peakon solitons. Then, solution profiles were graphically illustrated, portraying their dynamics and structural characteristics. The stability of the solutions was then cross-checked through modulation instability analysis as a crude means to check their accuracy and robustness. Thus, the model proposed above is effective.