(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Derive the Non-Linear Schrödinger from calculus of variations

2. Relevant equations

Lagrangian Density [itex] \mathcal{L} = \text{Im}(u^*\partial_t u)+|\partial_x u|^2 -1/2|u|^4[/itex]

The functional to be extreme: [itex] J = \int\limits_{t_1}^{t_2}\int\limits_{-\infty}^{\infty}\! \mathcal{L}\,\text{d}x\,\text{d}t[/itex]

3. 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:

[itex] \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)[/itex]

Inserting into the above i arrive at a equation not entierly similar to the Non-linear Schrodinger equation:

[itex] -|u|^2u-\dfrac{i}{2}\dfrac{\partial u}{\partial t} -\dfrac{\partial^2u}{\partial x^2} = 0[/itex]

My question is: Is this wrong? it looks a lot like the NSE but it is not entierly equal to

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Lagrangian Density, Non Linear Schrodinger eq

**Physics Forums | Science Articles, Homework Help, Discussion**