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This manuscript delves into the dynamic behavior of the (3 + 1) -dimensional nonlinear extended quantum Zakharov-Kuznetsov (NLEQZK) equation, exploring soliton solutions, chaotic phenomena, bifurcation, sensitivity, and stability through the lens of planar dynamical system theory. Firstly, the generalized Arnous method is used for securing some soliton solutions. Solutions obtained using the generalized Arnous method include hyperbolic, rational, and logarithmic forms. To visualize these graphically, we select appropriate parameter values and examine the graphical behavior of the obtained solutions. The visual characteristics of the solutions that were created are also evaluated in this procedure through the utilization of 3D surface graphs and line graphs representing various parameter values. The governing equation is derived using the Galilean transformation to facilitate bifurcation analysis. Chaotic behavior in the NLEQZK equation is investigated by introducing a perturbed term in the dynamical system and presenting various analyses, including Poincaré maps, time series, 2-dimensional (2D) phase portraits, and 3-dimensional (3D) phase portraits. Additionally, the Runge–Kutta method is employed for sensitivity analysis, confirming that slight adjustments to the initial conditions and check whether the system is sensitive or not. The outcomes of this study contribute valuable insights to the understanding of nonlinear dynamical systems and soliton theory. [ABSTRACT FROM AUTHOR]

The combined Korteweg-de Vries (KdV)-modified KdV (mKdV) equation widely appears as an integrable model in a wide range of interacting physical phenomena to explore their nonlinear dynamical features with geometrical structures. In this work, we present a comprehensive physical analysis of a bunch of solitary waves associated with a combined KdV-mKdV equation, aiming to elucidate the properties and behavior of these waves in the context of propagation in the background of interactions. The study is specifically concerned with analytically examining of combined KdV-mKdV equation, with a focus on algebraic and geometrical properties of solitary wave solutions with their stability analysis. The nature and stability of solitary waves are studied using mathematical methods such as the Kudrayashov methodology and the exp (- Φ (ζ)) -expansion method. This investigation tries to shed light on the behavior of solitary waves in various beginning situations and parameter regimes. Plotting 3D surface graphs, contour plots, and line graphs for different parameter values enables the observation of the graphical properties of the developed solutions. The obtained solutions exhibit kink, singular periodic, and periodic behaviors. The answers generated using these two methodologies also show the graphical representations of the soliton propagation. [ABSTRACT FROM AUTHOR]

The use of renewable energy sources is leading the charge to solve the world's energy problems, and non-Newtonian nanofluid dynamics play a significant role in applications such as expanding solar sheets, which are examined in this paper, along with the impacts of activation energy and solar radiation. We solve physical flow issues using partial differential equations and models like Casson, Williamson, and Prandtl. To get numerical solutions, we first apply a transformation to make these equations ordinary differential equations, and then we use the MATLAB-integrated bvp4c methodology. Through the examination of dimensionless velocity, concentration, and temperature functions under varied parameters, our work explores the physical properties of nanofluids. In addition to numerical and tabular studies of the skin friction coefficient, Sherwood number, and local Nusselt number, important components of the flow field are graphically shown and analyzed. Consistent with previous research, this work adds important new information to the continuing conversation in this area. Through the examination of dimensionless velocity, concentration, and temperature functions under varied parameters, our work explores the physical properties of nanofluids. Comparing the Casson nanofluid to the Williamson and Prandtl nanofluids, it is found that the former has a lower velocity. Compared to Casson and Williamson nanofluid, Prandtl nanofluid advanced in heat flux more quickly. The transfer of heat rates are 25.87 % , 33.61 % and 40.52 % at R d = 0.5 , R d = 1.0 , and R d = 1.5 , respectively. The heat transfer rate is increased by 6.91 % as the value of Rd rises from 1.0 to 1.5. This study is further strengthened by a comparative analysis with previous research, which is complemented by an extensive table of comparisons for a full evaluation. [ABSTRACT FROM AUTHOR]

In this paper, we study linear and nonlinear mixed convection, activation energy, and heat radiation effects caused by nanoparticles. This study aims to improve the understanding of how nanofluids behave in the presence of rotating disks and develop more efficient and effective cooling technologies. The flow problem consisted of partial differential equations (PDE). It is challenging to calculate these equations as a result of these nonlinear PDEs. Consequently, we use appropriate similarities to transform them into ordinary differential equations (ODEs). The bvp4c Matlab built-in technique is then used to resolve these ODEs. The velocities, temperature, and concentration outcomes with the various factors are examined graphically. Additionally, tables are employed to analyze the skin friction and Nusselt number values. It is analyzed that increasing the linear and linear mixed convection parameters enhances the velocity profiles of nanofluid. Enhancements in heat are analyzed by increasing nonlinear thermal radiation and enhancement in concentration is examined by increasing activation energy. Furthermore, as the variables for thermophoresis and Brownian motion are increased, the Nusselt number falls. The heat transfer rate is 27.16% for R d = 0.6 and 39.28% for R d = 1.4 . Thus, the heat transfer rate is enhanced 12.12%. This study’s practical applications include improving the behavior of fluids and the transfer of heat in rotating frameworks, which may affect energy systems, heat exchangers, and cooling advances in technology. [ABSTRACT FROM AUTHOR]

The purpose of this manuscript is to investigate the precise soliton solution of the (1 + 1) -dimensional Sharma–Tasso–Olver–Burgers equation. This study thoroughly explores the characteristics and solutions of this nonlinear equation. To pinpoint the exact soliton solution, we employ analytical techniques, specifically the exp (- ϕ (ξ)) -expansion method, as well as stability and sensitivity analysis. We begin by introducing the equation and discussing its role in mathematical modeling. It's worth noting that the equation's inherent nonlinearity adds complexity and poses challenges for analysis and solution. Our research objective is to identify solutions through graphical interpretation, allowing us to gain insights into their behavior. To achieve this, we utilize relevant parameter values to emphasize the physical properties of the provided data. We use the Mathematica and Maple software to demonstrate the solutions that have been discovered in 3D, 2D, and contour plots for the purpose of physical expression and graphical representation. The investigated system is shown to be stable since a little change in the initial conditions does not result in an immense shift in solutions. The sensitivity and stability analyses are also provided at various initial conditions. Importantly, these methods have broader applications and are valuable for conveying nonlinear physical models in the field of nonlinear sciences, not limited to the Sharma–Tasso–Olver–Burgers equation alone. [ABSTRACT FROM AUTHOR]