New exact optical solutions for the Lakshmanan–Porsezian–Daniel equation with parabolic law nonlinearity using the \(\phi ^{6}\)-expansion technique

In this study, we investigate new optical soliton solutions for the Lakshmanan–Porsezian–Daniel (LPD) equation incorporating group velocity dispersion (GVD), spatio-temporal dispersion (STD), and the parabolic law of nonlinearity. The considered LPD equation is a generalization of the nonlinear Schrödinger equation which arises in nonlinear optics. It describes the dynamical behavior of optical solitons as they propagate through nonlinear media under the influence of parabolic law nonlinearity. By employing the \(\phi ^{6}\) ${\varphi }^6$ -model expansion technique, we derive several optical soliton waves in terms of the Jacobi elliptic functions \(\Omega (\zeta ,{\textbf {k}})\) $\mathrm{\Omega }(\zeta ,\mathbf{k})$ which may degenerate to either hyperbolic function solutions or trigonometric function solutions depending on the limiting values of \({\textbf {k}}\) $\mathbf{k}$ . This allows us to extract various types of solutions with dark, periodic, bright, dark–bright, singular, and mixed soliton wave structures. In highlighting the reliability, practicality, and effectiveness of the applied computational technique, some of the obtained soliton solutions are captured in 3D, 2D, and contour representations for appropriate values of the parameters consistent with the restrictive conditions arising from the \(\phi ^{6}\) ${\varphi }^6$ -model expansion method. This demonstrated its applicability to a wide range of complex problems and further underscored the relevance of this work in the field.