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In this paper, we consider the conformable resonant nonlinear Schrödinger equation (CRNLSE) incorporating Kerr law nonlinearity and provide some new analytical solutions. Three analytical methods such as the Exp-function method, modified exp ( - Φ ( η ) ) -expansion function method, and exp ( - Φ ( η ) ) -expansion function method are utilized to handle this problem. These methods yield various innovative and fascinating solutions, including trigonometric, exponential, periodic, bright, singular, dark, and rational solutions, as well as their constraint conditions. Moreover the modulation instability (MI) of the consider model is also investigated. For each solution discovered, 2D, 3D, and contour plots are also sketched to clarify their physical configuration. The reported solutions enhance the previously established results. The solutions demonstrate that these methods are efficient and effective for locating traveling wave solutions and could be a beneficial tool for handling other complex nonlinear partial differential equations (NLPDEs) emerge in diversified scientific fields such as hydrodynamics, nonlinear optics, nonlinear fibers and plasmas. Graphic abstract

Amid growing environmental apprehensions, an increasing consumer demographic actively pursues genuine and transparent information traceability, aiming to bolster their commitment to sustainable practices. This study aims to investigate how consumer preferences for traceability, sustainability, and sustainable brand reputation of products influence the optimal decisions and selling quantity strategies adopted by firms operating in heterogeneous smart sustainable supply chains (SSCs). Utilizing both centralized and decentralized scenarios, this research constructs differentiated duopoly models, classifying participating entities as either boundedly rational or naive. Applying a differential game approach for an infinite time horizon, the study assesses the entities’ steady optimal decisions as well as the performances of the sustainable product across two scenarios. A cournot duopoly game is employed to examine the selling quantity strategies of differentiated smart SSCs. We leverage various tools for numerical simulations, including 1-D bifurcation diagrams, the largest Lyapunov exponent (LLE), phase diagrams, and basins of attraction. The outcomes of our investigation underscore the substantial influence of consumer preferences for sustainability and traceability on the transparency of information related to sustainable products. This influence reverberates across manufacturers and suppliers, where brand reputation emerges as a pivotal factor in shaping decision-making processes. Our study meticulously examines the repercussions of the profit-sharing ratio and the adjustment speed of boundedly rational entities.

This article presents bi-soliton, soliton, and shock solutions for the ion-acoustic wave (IAW) propagating in an unmagnetized nonplanar dusty plasma containing cold ions, neutral particles, and the electrons obeying κ -deformed Kaniadakis distribution. Applying the reductive perturbation method (RPM) to the basic equations, the nonplanar KdV–Burger (KdVB) equation is derived. It is found that there is a parametric domain for which the nonlinear coefficient vanishes, and naturally, the infinite divergence of the amplitude of the soliton arises. To describe the dynamic features of the wave quantities at or near the parametric domain, the nonplanar modified KdV–Burger (mKdVB) equation is constructed. The dissipation effect in the viscous plasma is expressed in the current study by a Burgers term, and the weighted residual method (WRM) is used to produce a solitary-type progressive wave solution for very small values of the Burgers term. Hirota’s bilinear formalism is employed for finding the multi-soliton solutions for the nonplanar mKdV system. However, the strong dissipation may cause the origination of a shock solution. An approximate analytical solution is also explored by means of WRM, through which the evolution of shock waves is determined. Finally, under the variation of various physical parameters, the evolution of different types of wave quantities is numerically examined. Graphical abstract

The present review provides an overview of the basic theory of sputtering with recent models, focusing in particular on sputtered atom energy distribution functions. Molecular models such as Monte-Carlo, kinetic Monte-Carlo, and classical Molecular Dynamics simulations are presented due to their ability to describe the various processes involved in sputter deposition at the atomic and molecular scale as required. The sputter plasma, the sputtering mechanisms, the transport of sputtered material and its deposition leading to thin film growth can be addressed using these molecular simulations. In all cases, the underlying methodologies and some selected mechanisms are highlighted. Graphic abstract

Rate coefficients for vibrational energy transfer are calculated for collisions between molecular nitrogen and hydrogen in a wide range of temperature and of initial vibrational states ( v ≤ 40 for N 2 and w ≤ 10 for H 2 ). These data are needed for the modelling of discharges in N 2 /H 2 plasma or of atmospheric and interstellar medium chemistry in different temperature ranges. The calculations were performed by a mixed quantum-classical method, to recover quantum effects associated with the vibrational motion, on a refined potential energy surface. The obtained rates present striking discrepancies with those evaluated by first-order perturbation theories, like the SSH (Schwartz, Slavsky, Herzfeld) theory, which are often adopted in kinetic modelling. In addition, we present a detailed, though preliminary, analysis on the performance of different Machine Learning models based on the Gaussian Process or Neural Network techniques to produce complete datasets of inelastic scattering rate coefficients. Eventually, by using the selected models, we give the complete dataset (i.e., covering the whole vibrational ladder) of rate coefficients for the N 2 ( v ) + H 2 ( 0 ) ⟶ N 2 ( v - Δ v ) + H 2 ( 0 ) , Δ v = 1 , 2 , 3 processes. Graphical abstract

The nonlinear dynamics of a quasi-1D BEC loaded in a double-well potential is studied. The beyond-mean-field corrections to the energy in the form of the Lee–Huang–Yang term are taken into account. One-dimensional geometry is considered. The problem is described in the scalar approximation by the extended Gross–Pitaevskii (EGP) equation with the attractive quadratic nonlinearity, due to the Lee–Huang–Yang correction, and the effective cubic mean-field nonlinearity describing the residual intra- and inter-species interactions. To describe tunneling and localization phenomena, a two-mode model was obtained. The frequencies of the Josephson oscillations are found and confirmed by the full numerical simulations of the EGP equation. The parametric resonance in the Josephson oscillations, when the height of the barrier is periodically modulated, is studied. The predictions of the dimer model, including the case of the one-dimensional Lee–Huang–Yang superfluid, have been proven. Graphical abstract

In this work, we study an electron subjected to a harmonic oscillator potential confined in a circle of radius r 0 and in the presence of a constant electric field. We obtain energies and eigenfunctions for three different confinement radii as a function of the electric field strength. We have used the linear variational method by constructing the trial function as a linear combination of two-dimensional confined harmonic oscillator wave functions. We calculate the radial standard deviation as a measure of the dispersion of the probability density. We also computed the Shannon entropy and Fisher information, in configuration and momentum spaces, as localization-delocalization measures for three different confinement radii and as a function of the electric field strength. We find that Shannon entropy and Fisher information are more reliable than variance in determining electron location. The behaviour of Shannon entropy and Fisher information curves is shown to depend on each specific state under study. Graphical abstract