Semi analytical technique implementation upon 4th-order Schrödinger equations with cubic–quintic nonlinearity

Higher-order nonlinear Schrödinger equations are frequently analyzed as a result of research into nonlinear wave mechanics in intricate physical systems. In this work, the $4^{\mathrm{th}}$-order Schrödinger equation with cubic-quintic nonlinearity is solved using semi-analytical method named Shehu HPM. Higher-order dispersive effects are considered by $4^{\mathrm{th}}$-order components, and equilibrium between self-focusing and saturation events in nonlinear media is modeled by the cubic–quintic nonlinearity. The intricate interaction of higher-order components with nonlinearity is frequently too complex for traditional numerical methods to handle, requiring reliable and precise semi-analytical techniques. The fetched results demonstrate exceptional agreement between exact and approximated solutions, validated through rigorous graphical compatibility analysis. The success of this approach underscores its effectiveness in handling higher-order dispersive and nonlinear terms, offering a reliable alternative to purely numerical techniques.