This study investigates the 1+1-dimensional Sharma–Tasso–Olver–Burgers (STOB) equation, a nonlinear model arising in mathematical physics. The modified F-expansion method is employed to derive a variety of exact soliton solutions, including trigonometric, hyperbolic, singular, periodic, and exponential forms. In addition to analytical solution construction, the dynamical behavior of the model is examined through bifurcation and chaos analysis. Quantitative and qualitative tools such as return maps, power spectra, and time series are utilized to characterize the system dynamics and identify transitions to chaotic regimes. The results demonstrate that the proposed approach is efficient and reliable, providing new insights into nonlinear wave structures and complex dynamical phenomena in nonlinear systems.
This study numerically investigates the unsteady magnetohydrodynamic (MHD) flow of a blood-based Casson hybrid nanofluid containing gold (Au) and iron oxide (Fe3O4) nanoparticles, flowing through a stenosed artery. The viscous dissipation, Joule heating, thermal radiation and porosity effects, along with a first-order homogeneous chemical reaction, are included in the mathematical model. A semi-implicit Crank–Nicolson finite-difference scheme is applied to solve the governing dimensionless equations in a MATLAB package, and the numerical method is tested for grid independence and benchmarking with other available solutions. The results indicate that the axial velocity is reduced with increasing Hartmann number; however, the increasing Reynolds number, Darcy number and Casson parameters increase the flow. The temperature is increased by the viscous dissipation and Joule heating as the Eckert number increases, and the thermal boundary layer is decreased as the Prandtl and radiation parameters increase. Increasing the volume fraction of nanoparticles has increased the skin friction and heat transfer, while the chemical reaction and Schmidt numbers have reduced the concentration field. Greater stenosis severity significantly suppresses momentum, heat and mass transport. The proposed model provides useful insight into the combined effects of magnetic fields, hybrid nanoparticles and arterial stenosis on blood flow, heat transfer and reactive species transport, with potential applications in magnetically guided drug delivery and cardiovascular thermal therapies.
In this study, we work on a Holling–Tanner prey-predator (HTPP) model by incorporating the impact of predator cannibalism and later introducing a delay in prey growth. We first examine its behavior without delay: we identify equilibrium points, describe their stability, and explain when the system remains stable or goes into oscillation. Then we add a time lag in the prey growth, which captures the natural lag in biological systems. We establish positivity, and by applying the Routh–Hurwitz criterion, we obtain explicit criteria for local stability and also prove global stability (GS) by applying a Lyapunov approach. Numerical simulations indicate that populations can converge in the ordinary model but oscillate indefinitely as soon as the delay passes a threshold. Aside from analysis, we also plot a detailed graph to show the findings. The results validate that delay significantly destabilizes the system, and cannibalism determines the long-term dynamics.
A numerical collocation approach employing independence polynomials of paths is formulated to approximate the solutions of high-order linear Fredholm–Volterra integro-differential problems under mixed conditions. The proposed approach transforms the given equation and conditions into a matrix form, leading to a linear system whose unknowns are the independence coefficients. To demonstrate the method’s validity and efficiency, several numerical examples are presented, with comparisons to existing results made in selected cases. The computations were performed in MATLAB.
The propagation of ultrashort optical pulses in nonlinear dissipative media is investigated through a generalized higher-order cubic–quintic complex Ginzburg–Landau equation (HG CQ-CGLE) incorporating third-order dispersion (TOD), fourth-order dispersion (FOD), self-steepening (SS), and intrapulse Raman scattering (IRS) effects. A consistent traveling-wave reduction is established, yielding a fourth-order nonlinear ordinary differential equation (NODE) subject to explicit compatibility constraints. Three complementary analytical approaches, the unified Riccati equation expansion method (UREEM), the F-expansion method, and a modified trial equation technique, are employed to construct several classes of exact nonlinear wave structures, including bright dissipative solitons, kink waves, singular solutions, rational states, and elliptic periodic patterns. The resulting solution families demonstrate how higher-order dispersive and dissipative mechanisms control wave localization, asymmetry, and nonlinear stabilization. To assess the physical relevance of the obtained structures, a modulational instability analysis of the continuous-wave (CW) background is performed. The results show that FOD can induce instability in normal-dispersion regimes, quintic nonlinearity regulates gain saturation, and the interplay between SS and IRS suppresses instability growth. Furthermore, spectral filtering significantly modifies the instability bandwidth and gain spectrum. A direct connection between modulation instability (MI) and nonlinear structure formation is established, providing a physical interpretation of the emergence and persistence of the analytical waveforms. The present study therefore offers a unified analytical, physical, and stability-oriented framework for understanding dissipative soliton formation, nonlinear wave transitions, and ultrashort pulse dynamics in generalized higher-order optical systems.
