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Lagrangian Density, Non Linear Schrodinger eq

  1. Apr 19, 2012 #1
    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
     
  2. jcsd
  3. Apr 21, 2012 #2
    Nwm i made an error the solution should be:
    [itex] |u|^2u + i\partial_t u + \partial_{uxx} = 0 [/itex]
     
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