Maxwell's Equations, when converted to cylindrical coordinates, and with the boundary conditions for an optical fiber while including birefringence as an effect taken into account, will yield the coupled nonlinear Schrödinger equations.
represent the envelope of the electromagnetic field as an initial condition.
For theoretical purposes, the integral equation version is often very useful.
It is as follows: One may make further substitutions and simplifications, depending on the limits used and the assumptions about boundary or initial conditions.
If a non-zero solution is desired, the imaginary part of the eigenvalue cannot change sign; accordingly, most researchers take the imaginary part to be positive.