In this research, a \(\phi ^6\) -model approximation method is employed to investigate the localized wave solutions for the Lakshmanan-Porsezian-Daniel equation with beta-derivative. This equation integrates the fundamental phenomena such as Space-Time Dispersion (STD), Group Velocity Dispersion (GVD) and parabolic-law-governed by nonlinear behavior. A variety of optical soliton waves are obtained by applying the \(\phi ^6\) -model expansion approach. These waves are expressed as Jacobi elliptic functions \(F({\mathcal {L}},A)\) that based on the particular values of the parameter A, that can be converted into solutions of trigonometric or hyperbolic functions. This technique provides variety of solutions, including dark soliton solutions, hyperbolic solutions, periodic waves solutions, bright solitons, singular soliton solution and singular periodic waves solutions. To further explore the system’s behavior, bifurcation analysis is done. For this analysis planar dynamical system is obtained by using Galilean transformation. This analysis offers deep understanding of the phase portraits, time series, chaotic behavior and sensitivity analysis of the equation to external perturbations. The sensitivity and dynamics of optical solitons are thoroughly investigated that offers significant insights into their behavior within fractional models.
