Badgerific

The nonlinear Schrodinger equation may be written as
$j{\frac {\partial u(x,z)}{\partial z}}+{\frac {\partial ^{2}u(x,z)}{\partial z^{2}}}+|u(x,z)|^{2}u(x,z)$

and I write $u(x,z)=u_{i}^{n}$ such that $u(x+\Delta x,z)=u_{i+1}^{n}$ and $u(x,z+\Delta z)=u_{i}^{n+1}$
(To avoid confusion j is the complex number)
Crank Nicholson method for this woudl look something like;

$j{\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta z}}=-{\frac {1}{2}}\left({\frac {u_{i+1}^{n+1}-2u_{i}^{n+1}+u_{i-1}^{n+1}}{\Delta x^{2}}}+|u_{i}^{n+1}|^{2}u_{i}^{n+1}+{\frac {u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}}{\Delta x^{2}}}+|u_{i}^{n}|^{2}u_{i}^{n}\right)$
I can't see how to evaluate forwards one step in n from this scheme... Badgerific (talk) 17:56, 29 March 2010 (UTC)