The main point of this entry is to derive the relation between the nonlinear Schrodinger equation and its associated nonlinear elliptic equation.
Let us start with the standing waves, , for the following nonlinear Schrodinger equation in
By means of standing wave we look for to be of the form
A simple calculation shows that
Thus the NSL can be rewritten as follows
which is nothing but the following nonlinear elliptic equation
In the literature, function is usually called the amplitude. It is well-known that the amplitude is often assumed to be positive and to vanish at infinity, thus, it is reasonable to assume
In other words, our nonlinear elliptic equations reads as the following
This kind of PDE has been extensively studied over years.