The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the $\left(\frac{{\mathsf{\Omega }}^{\prime }}{\mathsf{\Omega }}, \frac{1}{\mathsf{\Omega }}\right)$ method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields.

Solitary waves, inherent in nonlinear wave equations, manifest across various physical systems like water waves, optical fibers, and plasma waves. In this study, we present this type of wave solution within the integrable Mikhailov–Novikov–Wang (MNW) equation, an integrable system known for representing localized disturbances that persist without dispersing, retaining their form and coherence over extended distances, thereby playing a pivotal role in understanding nonlinear dynamics and wave phenomena. Beyond this innovative work, we examine the stability and modulation instability of its gained solutions. These new solitary wave solutions have potential applications in telecommunications, spectroscopy, imaging, signal processing, and pulse modeling, as well as in economic systems and markets. To derive these solitary wave solutions, we employ two effective methods: the improved Sardar subequation method and the (℧′/℧, 1/℧) method. Through these methods, we develop a diverse array of waveforms, including hyperbolic, trigonometric, and rational functions. We thoroughly validated our results using Mathematica software to ensure their accuracy. Vigorous graphical representations showcase a variety of soliton patterns, including dark, singular, kink, anti-kink, and hyperbolic-shaped patterns. These findings highlight the effectiveness of these methods in showing novel solutions. The utilization of these methods significantly contributes to the derivation of novel soliton solutions for the MNW equation, holding promise for diverse applications throughout different scientific domains.

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, suitability for optical signal processing in optical physics, etc. Through the powerful ($G^{\prime }/G$, $1/G$)-expansion analytical method, a variety of soliton solutions are expressed in three distinct forms: trigonometric, hyperbolic, and rational expressions. Rigorous validation using Mathematica software ensures precision, while dynamic visual representations vividly portray various soliton patterns such as kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton solutions. Indeed, a sensitivity analysis was conducted to assess how changes in parameters affect the exact solutions, aiding in the understanding of system behavior and informing decision-making, especially in accurately designing or analyzing real-world optical phenomena. This investigation reveals the significant influence of parameters $\lambda$, $\tau$, $c$, $\mathcal{B}$, and $\mathsf{{\rm K}}$ on the precise solutions in Kerr and power law nonlinearities within the BME. Notably, parameter $\lambda$ exhibits consistently high sensitivity across all scenarios, while parameters $\tau$ and $c$ demonstrate pronounced sensitivity in scenario III. The outcomes derived from this method are distinctive and carry significant implications for the dynamics of optical fibers and wave phenomena across various optical systems.

Pollutant dispersion in urban areas has been a key concern in densely populated cities because pollutants can negatively impact human health and the environment. Topography and urban obstructions, such as buildings, greatly affect the atmospheric fluid flow, leading to the dispersion process. Street canyons are usually formed in highly populous cities due to the close proximity of buildings to streets to serve commercial purposes. It can also be termed as pollutant traps that exact adverse effects on urban life. Street canyons cause changes in pollutant dispersion, particularly in the case of vehicle exhaust air pollutants, which cannot be transported by the wind owing to the presence of buildings acting as impediments regardless of the wind flow. Hence, it is imperative to completely understand the behavior of pollutants within the confined urban surroundings for further improving the urban air quality. This study explores the effects of street canyons on the busiest street of Dhaka city in Bangladesh. The Reynolds-averaged Navier–Stokes based dispersion model is used to investigate the air quality phenomenon. Building height, width, building distance, and road width have been taken into consideration in the simulation. It is expected that the findings of the current study would be helpful for urban planners and designers of Dhaka city.

This study, highlights the exact optical soliton solutions in the context of optical physics, centering on the intricate Hamiltonian amplitude equation with bifurcation and sensitivity analysis. This equation is pivotal in optics which underpins the understanding of optical manifestations, encompassing solitons, nonlinear consequences, and wave interactions. Applying an analytical expansion approach, we extract diverse optical solutions, having trigonometric, hyperbolic, and rational functions. Next, we utilize concepts from the principle of planar dynamical systems to investigate the bifurcation processes and chaotic behaviors present in this derived system. Additionally, we use the Runge–Kutta scheme to carry out a thorough sensitivity analysis of the dynamical system. It has been verified through this analytical process that small variations in beginning conditions have negligible effects on the stability of the solution using bifurcation analysis. Validation via Mathematica software ensures the accuracy of these findings. Furthermore, we employ dynamic visualizations, such as 2D, 3D, and contour plots, to illustrate various soliton patterns, including kink, multi-kink, single periodic, multi-periodic, singular, and semi-bell-shaped configurations. These visual representations provide a glimpse into the fascinating behavior of optical phenomena. The solutions obtained via this proposed method showcase its efficacy, dependability, and simplicity in comparison to various alternative approaches.