It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions.

The current study demonstrated and studied the existence of monotonic solutions, as well as the uniqueness of the solutions for a general and abstract form of a product of $n$-quadratic fractional integral equations of Hadamard-type in Orlicz spaces $L_{\phi }$. We utilized the analysis of the measure of non-compactness associated with Darbo's fixed-point theorem and fractional calculus to obtain the results.

In this paper, we have delved into the intricate dynamics of a discrete-time Hepatitis B virus (HBV) model, shedding light on its local dynamics, topological classifications at equilibrium states, and pivotal epidemiological parameters such as the basic reproduction number. Our analysis extended to exploring convergence rates, control strategies, and bifurcation phenomena crucial for understanding the behavior of the HBV system. Employing linear stability theory, we meticulously examined the local dynamics of the HBV model, discerning various equilibrium states and their topological classifications. Subsequently, we identified bifurcation sets at these equilibrium points, providing insights into the system's stability and potential transitions. Further, through the lens of bifurcation theory, we conducted a comprehensive bifurcation analysis, unraveling the intricate interplay of parameters that govern the HBV model's behavior. Our investigation extended beyond traditional stability analysis to explore chaos and convergence rates, enriching our understanding of the dynamics of the understudied HBV model. Finally, we validated our theoretical findings through numerical simulations, confirming the robustness and applicability of our analysis in real-world scenarios. By integrating biological and epidemiological insights into our mathematical framework, we offered a holistic understanding of the dynamics of HBV transmission dynamics, with implications for public health interventions and disease control strategies.

This article addressed the integrable and approximate solutions of Hadamard-type fractional Gripenberg's equation in Lebesgue spaces $L_1[1,e]$. It is well known that the Gripenberg's equation has significant applications in mathematical biology. By utilizing the fixed point (FPT) approach and the measure of noncompactness (MNC), we demonstrated the presence of monotonic integrable solutions as well as the uniqueness of the solution for the studied equation in spaces that are not Banach algebras. Moreover, the method of successive approximations was successfully applied and, as a result, we obtained the approximate solutions for these integral equations. To validate the obtained results, we provided several numerical examples.

Deriving sharp extinction criteria for realistic multi-stage Human Immunodeficiency Virus/Acquired Immunodeficiency Syndrome (HIV/AIDS) models under environmental uncertainty remains technically difficult, especially when both correlated continuous fluctuations and abrupt disruptive events are present. In such settings, classical deterministic thresholds may be misleading. Parameter regimes that predict persistence at the Ordinary Differential Equation (ODE) level can nevertheless exhibit extinction once stochastic effects are accounted for. In this work, we developed a new weighted infected-block $M$-matrix methodology to obtain an explicit, computable, almost-sure exponential extinction threshold for a six-compartment Itô-Lévy HIV/AIDS model with correlated diffusions and dual jump mechanisms. The proposed framework constructs positive weights directly from the infected-treatment transition block, and combines these weights with refined diffusion corrections and jump-compensator bounds to produce a precise extinction indicator that quantifies stochastic damping beyond deterministic invasion pressure. To support practical relevance, we calibrated the deterministic core to real monthly HIV-AIDS and ART data from Pakistan (2016–2021) via multi-start nonlinear least squares, computed all threshold objects from the fitted parameters, and then tuned the Itô-Lévy perturbations to match the extinction regime predicted by the theory. Numerical simulations validated the theoretical predictions and illustrated how sufficiently strong fluctuations and rare shocks can drive the system toward extinction even when ${\mathcal{R}}_0>1$ deterministically, thereby providing a data-informed tool to delineate extinction-persistence boundaries in complex stochastic HIV/AIDS dynamics.

In this paper, we investigated $n$-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in $3$-dimensional space at the end of the paper.