The system is

$i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \phi_x,\,$

$\phi_{xx} + c_3 \phi_{yy} = ( |u|^2 )_x.\,$

This is like the NLS but with the extra y-dimension. The NLS has the lagrangian formulation

$\mathcal{L}=\frac{i}{2}(\psi_{t}\bar{\psi}-\psi\bar{\psi}_{t})-|\bigtriangledown\psi|^{2}+|\psi|^{2(\sigma+1)}.$

For DS since we have two eqns and unknowns, we will have two Euler-Lagrangian equations. But I was wondering if there is a way to express them as one Lagrangian (to make applying Noether's theorem easier).