Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Non-Linear Schr\"{o}dinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann-Hilbert problem approach is used to derive the leading-order asymptotics as $| t | \to \infty$ $(x/t \sim \mathcal{O} (1))$ of solutions $(u = u(x,t))$ to the Cauchy problem for the defocusing non-linear Schr\"{o}dinger equation (D${}_{f}$NLSE), $\mi \partial_{t}u + \partial_{x}^{2}u - 2(| u |^{2} - 1) u = 0$, with finite-density initial data $u(x,0) =_{x \to \pm \infty} \exp (\tfrac{\mi (1 \mp 1) \theta}{2})(1 + o(1))$, $\theta \in [0,2\pi)$. The D${}_{f}$NLSE dark soliton position shifts in the presence of the continuum are also obtained.