In this study, we investigate a nonlinear Schrödinger equation relevant to the evolution of optical beams in weakly nonlocal media. Utilizing the modified F-expansion method, we construct a variety of novel soliton solutions, including dark, bright, and wave solitons. These solutions are illustrated through comprehensive graphical simulations, including 2D contour plots and 3D surface profiles, to highlight their structural dynamics and propagation behavior. The effects of the temporal parameter on soliton formation and evolution are thoroughly analyzed, demonstrating its role in modulating soliton shape and stability. To further explore the system’s dynamics, chaos and sensitivity theories are employed, revealing the presence of complex chaotic behavior under perturbations. The outcomes underscore the versatility and richness of the present model in describing nonlinear wave phenomena. This work contributes to the theoretical understanding of soliton dynamics in weakly nonlocal nonlinear optical systems and supports advancements in photonic technologies. This study reports a novel soliton structure for the weak nonlocal cubic–quantic NLSE and also details the comprehensive chaotic and sensitivity analysis that represents the unexplored dynamical behavior of the model. This study further demonstrates how the underlying nonlinear structures, along with the novel solitons and chaotic dynamics, reflect key symmetry properties of the weakly nonlocal cubic–quintic Schrödinger model. These results enhanced the theoretical framework of the nonlocal nonlinear optics and offer potential implications in photonic waveguides, pulse shape, and optical communication systems.

To increase the speed of information flow and storage capacity in electronic devices laser can be used to carry information instead of electric current. Since the photon is faster than electrons, one expects information to be transmitted very fast through the internet when photons replace electrons. This requires searching for chips that act as capacitors, inductors or resistors. To do this Maxwell's equation for the electric field intensity beside the electron equation of motion were used. The electron is assumed to vibrate naturally inside a frictional medium in the presence of a local electric and magnetic fields. These equations have been used to find a useful expression for the absorption coefficient. The absorption coefficient was found to be dependent on the laser and natural frequencies beside the coefficient of friction in addition to the internal electric and magnetic fields. These parameters can be fine-tuned to make the chip act as a capacitor, inductor or resistor. The laser intensity decreases when the absorption coefficient inecreases. Thus, the absorption coefficient acts as an electic resistor. Therefore, if the absorption coefficient inecreases upon decreasing the frequency the chip acts as a capacitor. But when the absorption coefficient inecreases when the laser frequency inreases the chip acts as an inductor. In the case that the absorption coefficient inecreases with the concentration of the carriers it acts in this situation as a resistor. For magnetic materials with magnetic flux density that cancels the frictional force, when the laser frequency is equal nearly to the atom's natural frequency the material acts as an inductor. But when the frictional force is low with the internal and external electric fields in phase, the material acts as a capacitor. However, it acts as a resistor for negligible natural frequency, when no electric dipoles exist and when the internal magnetic field force balance the frictional force.

In this work, we introduce a new four-parameter distribution, called the integrated linear–Weibull (ILW) model, constructed by embedding a dynamic linear component within the Weibull framework. The ILW distribution is capable of capturing a wide variety of symmetric and asymmetric density shapes and accommodates diverse failure-rate behaviors. We derive several of its key mathematical and statistical properties, including moments, extropy, cumulative residual entropy, order statistics, and their asymptotic forms. The mean residual life function and its reciprocal relationship with the failure rate are also obtained. An algorithm for generating random samples from the ILW distribution is provided, and model identifiability is examined numerically through the Kullback–Leibler divergence. Parameter estimation is carried out via maximum likelihood, and a simulation study is conducted to assess the accuracy of the estimators; the results show improvements in estimator performance as sample size increases. Finally, three real datasets involving failure-time observations and one describing hydrological and epidemiological data are analyzed to demonstrate the practical usefulness of the ILW model. In these applications, the proposed model exhibits competitive or superior performance relative to several existing lifetime distributions based on standard model selection criteria and goodness-of-fit measures.

Maxwell's electric equation and Schrödinger quantum equation are used to find useful expressions for the probability of electrons existence and the electrons transport. The first expression which uses special relatively also shows that the probability of the electrons existence is higher for lower energy states compared to higher energy states where the probability is low. This means that electrons transport takes place from higher energy states to lower states. This conforms to the laws of chemistry where the atom tends to be stable by forcing electrons to occupy lower energy states. The second expression used the momentum velocity relation instead of special relatively. This expression shows also electrons transport from higher to lower states. It also shows that the probability of the electrons existence is higher for lower electrons density regions and lower for higher electrons density regions. This means that the electrons transport is from higher density regions to lower density regions. This conforms to the diffusion law.