  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.
