$\sqrt \gamma z_j$ for $j=1, 2, ..., N$ and then take the limit, producing the dark $N$-soliton solution of the NLS equation. Note, in this limit, that the constraints (3.44) reduce to $p_jp_j^*=-\mu\rho^2/2$ and hence the dark soliton solutions exist only if the condition $\mu<0$ is satisfied.

[*3.6. Stability of the plane wave*]{}

We have considered the dark solitons on the background of a plane wave $\rho\,{\rm e}^{{\rm i}(\kappa x-\omega t)}$ with $\omega =1/\kappa +2\rho^2$. It is important to see whether the background field is stable or not against perturbations. If unstable, then dark solitons would not exist, as will be demonstrated in the next section. To this end, we perform the linear stability analysis of the plane wave.

Following the standard procedure, we seek a solution of the form $$u=(\rho+\Delta\rho)\,{\rm e}^{{\rm i}(\kappa x-\omega t+\Delta\phi)}, \eqno(3.45)$$ where $\Delta\rho=\Delta\rho(x,t)$ and $\Delta\phi=\Delta\phi(x,t)$ are small perturbations. Substituting (3.45) into the FL equation (1.1) and linearizing about the plane wave, we obtain the system of linear PDEs for $\Delta\rho$ and $\Delta\phi$ $$\Delta\rho_{xt}+\rho(\omega-2\rho^2)\Delta\phi_x-\kappa\rho\Delta\phi_t-4\kappa\rho^2\Delta\rho=0, \eqno(3.46a)$$ $$\rho\Delta\phi_{xt}-(\omega-2\rho^2)\Delta\rho_x+\kappa\Delta\rho_t=0. \eqno(3.46b)$$ Assume the perturbations of the form ${\rm e}^{{\rm i}(\lambda x-\nu t)}$ with $\lambda$ real and $\nu$ possibly complex and substitute them into (3.46) to obtain a homogeneous linear system 