# Ngô Quốc Anh

## February 19, 2011

### Derivation of the relation between the nonlinear Schrodinger equation and its associated nonlinear elliptic equation

Filed under: Uncategorized — Ngô Quốc Anh @ 19:23

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, $\psi$, for the following nonlinear Schrodinger equation in $\mathbb R^n$

$\displaystyle -i\frac{\partial\psi}{\partial t}=\Delta \psi - \widetilde V(y)\psi + |\psi|^{p-1}\psi$

where $p>1$.

By means of standing wave we look for $\psi$ to be of the form

$\psi(t,y)=e^{i\lambda t}u(y)$.

A simple calculation shows that

$\displaystyle - i\frac{{\partial \psi }}{{\partial t}} = \lambda {e^{i\lambda t}}u(y)$.

Thus the NSL can be rewritten as follows

$\displaystyle\lambda {e^{i\lambda t}}u(y) = {e^{i\lambda t}}\Delta u(y) - \widetilde V(y){e^{i\lambda t}}u(y) + |u(y){|^{p - 1}}u(y)$

which is nothing but the following nonlinear elliptic equation

$\displaystyle - \Delta u + V(y)u = |u{|^{p - 1}}u$

where $V(y)=\widetilde V(y)+\lambda$.

In the literature, function $u$ 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

$u>0, \quad \mathop {\lim }\limits_{|y| \to + \infty } u(y) = 0.$

In other words, our nonlinear elliptic equations reads as the following

$\displaystyle - \Delta u + V(y)u = u^{p}, \quad y \in \mathbb R^n.$

This kind of PDE has been extensively studied over years.