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∂tu=−Δ−−−√u−|u|p−1u,with(t,x)∈R×Rd
with exponents 1<p<∞ for d = 1 and 1<p<(d+1)/(d−1) for d ≥ 2. We study traveling solitary waves of the form
u(t,x)=eiωtQv(x−vt)
with frequency ω∈R, velocity v∈Rd, and some finite-energy profile Qv∈H1/2(Rd), Qv≢0. We prove that traveling solitary waves for speeds |v|≥1 do not exist. Furthermore, we generalize the non-existence result to the square root Klein–Gordon operator −Δ+m2−−−−−−−−√ 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 ≥ 2.