Interplay of solitons and radiation in one-dimensional Bose gases

We study relaxation dynamics in one-dimensional Bose gases, formulated as an initial value problem for the classical nonlinear Schrödinger equation. We propose an analytic technique which takes into account the exact spectrum of nonlinear modes that is both soliton excitations and dispersive continuum of radiation modes. Our method relies on the exact large-time asymptotics and uses the so-called dressing transformation to account for the solitons. The obtained results are quantitatively compared with the predictions of the linearized approach in the framework of the Bogoliubov theory. In the attractive regime, the interplay between solitons and radiation yields a damped oscillatory motion of the profile which resembles breathing. For the repulsive interaction, the solitons are confined in the sound cone region separated from the supersonic radiation.

Figure 1

Time evolution of quenched profile (7) with $\varkappa =-1$ and $\eta =1/3$. The dashed lines denote the half-width of the profile, indicating a ballistic (i.e., linear in time) expansion of the half-width.

Figure 2

Time evolved profile (7) for $\varkappa =-1$ and $\eta =1/3$ at $t=5$, obtained by numerical integration (solid line) and from the asymptotic solution ${\psi }_a$ given by Eq. (24) (dashed line). The inset plot shows the relative difference between the two integration methods $\mathrm{\Delta }=|\psi (x,t)-{\psi }_a(x,t)|/|\psi (x,t)|$.

Figure 3

Time evolution of the intensity $\left|\psi \right(x,t\left)\right|$ of the quenched profile (7) with $\varkappa =-1$ and $\eta =\sqrt{2}$, showing persistent oscillations referred to as the soliton breathing. The red lines denote the half-width of the profile.

Figure 4

Time dependence of the profile's half-width $w\left(t\right)$ obtained from the linearized (BdG) equation with parameter $\varkappa =-1$ and $\eta =\sqrt{2}$, with (diamonds) and without (dashed line) the zero-mode components, compared to the exact numerical integration.

Figure 5

Solution to BdG equation (30) (diamonds) versus the solution constructed from the asymptotics profile dressed by a soliton (24) (dashed), shown for $\eta =\sqrt{2}$ ($\varkappa =-1$) at times $t=5.5$, 8, 10.5, and 13, respectively. The results are benchmarked against the numerical integration (solid line).

Figure 6

The asymptotic solution of Eq. (76) at $t=10$ (dotted line), compared with the numerical solution of the exact equation of motion (57) (solid line), for parameter $\varkappa =1$ and $\eta =\frac{1}{2}$.

Figure 7

Time evolution of $\left|\psi \right(x,t\left)\right|-1$ of the quenched profile (60) with coupling $\varkappa =1$ and quench parameter $\eta =1/2$. The radiation modes propagate faster than any dark (gray) solitons, with lower velocity bound $\zeta$. All the radiation modes are confined outside of the sound cone, marked with dashed lines.

Figure 8

Solution of BdG equation (87) (diamonds) compared to the exact numerical integration (solid line) for the 1D NLSE with finite asymptotic density (57), with parameters $\eta =1/2$ at $t=5$. The imaginary part of the solution (${\phi }^{-}$ and ${\varphi }^{-}$ modes) is responsible for the observed excess in the asymptotic density (as shown by the dotted line).