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.

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