User talk:Badgerific

The nonlinear Schrodinger equation may be written as

$$j\frac{\partial u(x,z)}{\partial z} + \frac{\partial^2 u(x,z)}{\partial z^2} + |u(x,z)|^2u(x,z)$$

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

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