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

Abstract

We construct infinite-time Sobolev cascades for the cubic nonlinear half-wave equation on the real line. For $\frac{1}{2}<s\le 1$ and $0\le \kappa <2s(2s-1)/(4s-1)$, spatial gluing of permanently saturated two-soliton packets gives a dense set of strictly mass-subthreshold data for which $\Vert u(t){\Vert }_{H^s}/(1+t{)}^{\kappa }\to \infty$ in the focusing case. A second construction replicates finite-burst Szegő packets without rescaling. It yields a dense $G_{\delta }$ subset of $H^{\infty }(\mathbb{R})$ whose solutions perform logarithmic cascades in every $H^s$ with $s>\frac{1}{2}$ in the focusing case and in the defocusing case.