The nonlinear Schr\"odinger equation: A mathematical model with its wide-ranging applications

The nonlinear Schr\"odinger equation (NLSE) models the slowly varying envelope dynamics of a weakly nonlinear quasi-monochromatic wave packet in dispersive media. In the context of Bose-Einstein condensate (BEC), it is often referred to as the Gross-Pitaevskii equation (GPE). The NLSE is one example of integrable systems of a nonlinear partial differential equation (PDE) in $(1 + 1)$D and it possesses an infinite set of conservation laws. This nonlinear evolution equation arises in various physical settings and admits a wide range of applications, including but not limited to, surface gravity waves, superconductivity, nonlinear optics, and BEC. This chapter discusses not only the modeling aspect of the NLSE but also provides an overview of the applications in these four exciting research areas. The former features derivations of the NLSE heuristically and by employing the method of multiple-scale from other mathematical models as governing equations. Depending on how the variables are interpreted physically, the resulting NLSE can model a different dynamics of the wave packet. Furthermore, depending on the adopted assumptions and the chosen governing equations, each approach may provide different values for the corresponding dispersive and nonlinear coefficients.