We investigate the dynamics of dark solitons in one-dimensional Bose-Einstein condensates. In the large particle limit, by introducing the lens-type transformation, we find that the macroscopic wave function evolves self-similarly when its initial profile strays from that of the equilibrium state, which provides a time-varying background for the propagation of dark solitons. The interaction of dark solitons with this kind of background is studied both analytically and numerically. We find that the center-of-mass motion of the dark soliton is deeply affected by the time-varying background, and the beating phenomena of dark soliton emerge when the intrinsic frequency of the dark soliton approaches that of the background. Lastly, we investigate the propagation of dark solitons in the freely expanding background.
We investigate how the junction law for multipartite entanglement is realized in confining holographic backgrounds, using genuine multi-entropy (GM) as our main diagnostic. We first study an AdS
3 hard-wall toy model as an analytic benchmark, where multi-way cuts and junction geometries can be analyzed explicitly. In this setup, we classify the relevant saddles, determine the dominant phases, and show that the genuinely multipartite contribution diagnosed by GM is localized near the junction. We also examine how this structure depends on subsystem sizes, asymmetry, and the confinement scale, including phase transitions between competing saddles. We then move beyond the hard-wall benchmark to smooth confining geometries, focusing on the D4-soliton and D3-soliton backgrounds and formulating the corresponding framework also for the Klebanov--Strassler background. In the smooth-cap examples, we find that the junction picture persists, while the detailed phase structure differs from the hard-wall case: in particular, the hard-wall plateau does not survive, and GM instead decreases monotonically and vanishes at a finite critical scale. We also find that the short-distance behavior is background-dependent, with
GM(3)∼L−4 in the D4-soliton background,
GM(3)∼L−2 in the D3-soliton background, and
GM(3)∼L−2⋅(logL)2 in the Klebanov--Strassler background. These results clarify which features of the junction-law picture are robust in confining holography and which features of the phase structure and short-distance scaling are background-dependent.