Lagrangian Density, Non Linear Schrodinger eq

[dikmikkel]

Homework Statement

Derive the Non-Linear Schrödinger from calculus of variations

Homework Equations

Lagrangian Density $\mathcal{L} = \text{Im}(u^*\partial_t u)+|\partial_x u|^2 -1/2|u|^4$
The functional to be extreme: $J = \int\limits_{t_1}^{t_2}\int\limits_{-\infty}^{\infty}\! \mathcal{L}\,\text{d}x\,\text{d}t$

The Attempt at a Solution

I integrate by parts make a variation function which is demanded differentiable in x,t and get the following Euler Equation(2d) so i consider the Lagrangian density:
$\dfrac{\partial \mathcal{L} }{\partial u} = \dfrac{d}{dt}\left(\dfrac{\partial \mathcal{L} }{\partial u_t} \right) + \dfrac{d}{dx}\left(\dfrac{\partial \mathcal{L}}{\partial u_x}\right)$
Inserting into the above i arrive at a equation not entierly similar to the Non-linear Schrödinger equation:
$-|u|^2u-\dfrac{i}{2}\dfrac{\partial u}{\partial t} -\dfrac{\partial^2u}{\partial x^2} = 0$
My question is: Is this wrong? it looks a lot like the NSE but it is not entierly equal to

 

[dikmikkel]

Nwm i made an error the solution should be:
$|u|^2u + i\partial_t u + \partial_{uxx} = 0$