On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations

We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i pt_t u = sqrt{-Delta} , u - |u|^{p-1} u, quad mbox{with $(t,x) in R times R^d$} $$ with exponents $1 < p < infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{iomega t} Q_v(x-vt) $$ with frequency $omega in R$, velocity $v in R^d$, and some finite-energy profile $Q_v in H^{1/2}(R^d)$, $Q_v ot equiv 0$. We prove that traveling solitary waves for speeds $|v| geq 1$ do not exist. Furthermore, we generalize the non-existence result to the square root Klein--Gordon operator $sqrt{-DD+m^2}$ and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds $|v| < 1$. Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions $d geq 2$.