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 $\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$ $(x\mathrm{/}t\sim \mathcal{O}(1))$ 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$,withfinite-densityinitialdata$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.