A Generalized Schrödinger Equation Involving the Parabolic Law of Nonlinear Refractive Index: Its Sideband Instability, Bifurcation Analysis, and Solitary Waves

Throughout the past few decades, the exploration of mathematical models associated with Earth systems has received substantial scholarly focus. The current paper aims to comprehensively explore a generalized Schrödinger equation involving the parabolic law of nonlinear refractive index under an external potential. Specifically, the governing equation for the sideband instability (SI) is examined in detail to distinguish the effects of cubic and quantic nonlinearities on stable and unstable zones. Further, the bifurcation analysis is theoretically and numerically conducted to locate equilibrium points of the generalized Schrödinger equation and identify its solitary and periodic waves. In the end, the impact of nonlinear effects on the propagation of shock and dark waves modeled by the generalized Schrödinger equation is thoroughly analyzed. In light of the results given in the present paper, an increase in the coefficient of the cubic (quantic) nonlinearity leads to an increase (decrease) in the amplitude of shock and dark waves.