Euler-Lagrange equations of 1D NLS with periodic external potential

Question

I am trying to find the Euler Lagrange (EL) equations in order to find ODEs for the parameters $a(t), \xi(t), c(t), d(t)$. Consider the following form: $$iu_t+\frac{1}{2}u_{xx}+|u|^2u=V(x)u \tag{1}$$ the ansatz (bright soliton) $$u(x,t)=a(t)sech[a(t)(x-\xi(t))]e^{i(c(t)x+d(t)t)} \tag{2}$$ and the potential $$V(x)=-A\cos(\frac{2\pi }{\lambda}x).\tag{3}$$ To find the EL equations I first consider the Lagrangian density $$ L=\frac{i}{2}(uu_t^*-u^*u_t)+\frac{1}{2}|u_x|^2-\frac{1}{2}|u|^4+V(x)|u|^2.\tag{4}$$ My question is how to proceed to find the EL equations?

Answer 1

OP is essentially asking:

How do you get the Euler-Lagrange (EL) equation (1) wrt. the complex-valued function $u$ from the Lagrangian density (4)?

Answer:

1. The pedestrian/elementary-but-cumbersome way: Vary (4) wrt. ${\rm Re}(u)$ and ${\rm Im}(u)$ independently, thereby obtaining 2 EL eqs. Rearrange to obtain eq. (1).

2. The elegant way: Vary the action $$ S~=~\int \!dt~dx~{\cal L}, \qquad {\cal L}~=~\frac{i}{2}(uu_t^{\ast}-u^{\ast}u_t)+\frac{1}{2}u_xu_x^{\ast}-\frac{1}{2}(uu^{\ast})^2 +V(x)uu^{\ast},\tag{4'} $$ wrt. the complex conjugate function $u^{\ast}$, pretending that $u$ is an independent fixed variable: $$ 0~=~\frac{\delta S}{\delta u^{\ast}}~=~\frac{\partial {\cal L}}{\partial u^{\ast}} - \frac{d}{dt} \frac{\partial {\cal L}}{\partial u^{\ast}_t}-\frac{d}{dx} \frac{\partial {\cal L}}{\partial u^{\ast}_x}. \tag{1'}$$ Eq. (1') gives the correct eq. (1) immediately! The justification for this method (1') is explained in this Phys.SE post.