Examining the (1 + 1)-dimensional Schrödinger–Hirota equation with Kerr effect under inter-modal dispersion using the invariance theory

In this paper, \((1+1)\)-dimensional Schrödinger–Hirota equation with Kerr law having inter-modal dispersion is considered. This model designates the propagation of pulses in optical fibers, so it has a significant impact on the model and optimization of optical fiber communication systems. Utilizing the Lie symmetry technique, three generators are produced and three sub-algebras are derived by the combination of these generators. Lie symmetry analysis is the basis of the reduction of the presented nonlinear partial differential equation to the nonlinear ordinary differential equation system. We implement the unified Riccati equation expansion method so that we acquire the analytical soliton solutions. Thus, periodic singular, dark, and singular solitons are retrieved; besides, their 3-dimensional portraits are illustrated by selecting appropriate parameter values. Moreover, the influence of the parameters of the presented model is investigated via various 2-dimensional graphical visualizations. This study reports that all produced solutions satisfy the presented model.