Infinite cascades for the cubic nonlinear half-wave equation on the line

We construct infinite-time Sobolev cascades for the cubic nonlinear half-wave equation on the real line. For $\frac12<s\leq1$ and $0\leqκ<2s(2s-1)/(4s-1)$, spatial gluing of permanently saturated two-soliton packets gives a dense set of strictly mass-subthreshold data for which $\|u(t)\|_{H^s}/(1+t)