In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.

The introduction of hybrid nanofluids is an important concept in various engineering and industrial applications. It is used prominently in various engineering applications, such as wider absorption range, low-pressure drop, generator cooling, nuclear system cooling, good thermal conductivity, heat exchangers, etc. In this article, the impact of variable magnetic field on the flow field of hybrid nano-fluid for the improvement of heat and mass transmission is investigated. The main objective of this study is to see the impact of hybrid nano-fluid (ferrous oxide water and carbon nanotubes) CNTs-$F{e}_3{O}_4$, $H_{2}O$ between two parallel plates with variable magnetic field. The governing momentum equation, energy equation, and the magnetic field equation have been reduced into a system of highly nonlinear ODEs by using similarity transformations. The parametric continuation method (PCM) has been utilized for the solution of the derived system of equations. For the validity of the model by PCM, the proposed model has also been solved via the shooting method. The numerical outcomes of the important flow properties such as velocity profile, temperature profile and variable magnetic field for the hybrid nanofluid are displayed quantitatively through various graphs and tables. It has been noticed that the increase in the volume friction of the nano-material significantly fluctuates the velocity profile near the channel wall due to an increase in the fluid density. In addition, single-wall nanotubes have a greater effect on temperature than multi-wall carbon nanotubes. Statistical analysis shows that the thermal flow rate of ($F{e}_3{O}_4$-SWCNTs-water) and ($F{e}_3{O}_4$-MWCNTs-water) rises from 1.6336 percent to 6.9519 percent, and 1.7614 percent to 7.4413 percent, respectively when the volume fraction of nanomaterial increases from 0.01 to 0.04. Furthermore, the body force accelerates near the wall of boundary layer because Lorentz force is small near the squeezing plate, as the current being almost parallel to the magnetic field.

In this article, the behavior of transient electroviscous fluid flow is investigated through squeezing plates containing hybrid nanoparticles. A hybrid nanofluid ${\mathrm{MoS}}_2+\mathrm{Au}$/${\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}$ was formulated by dissolving the components of an inorganic substance such as molybdenum disulfide $\left({\mathrm{MoS}}_2\right)$ and gold (Au) in a base fluid of ethylene glycol/water. This hybrid non-liquid flow was modeled by various nonlinear mathematical fluid flow models and subsequently solved by numerical as well as analytical methods. For the numerical solution of nonlinear ODEs, a built-in function BVP4C was used in MATLAB, and the same problem was solved in MATHEMATICA by HAM. The result of the present problem related to the results obtained from the existing literature under certain conditions. The outcomes revealed that the concentration profiles were more sensitive to homogeneity diversity parameters. The simulation of the various physical parameters of the model indicated that the heat transfer through a mixture of hybrid nanofluids was greater than a simple nanofluid. In addition, the phenomenon of mixed convection was considered to improve the velocity of simple nanofluids and hybrid nanofluids, when both cases have low permeability. A rise in the volume fraction of the nanomaterials, $\Phi$, was associated with an increase in the heat transfer rate. It was observed that the heat transfer rate of the hybrid nanofluids ${\mathrm{MoS}}_2+{\mathrm{Au}/\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}$ was higher than that of the single nanofluids ${\mathrm{MoS}}_2/{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2-{\mathrm{H}}_2\mathrm{O}$.