We consider nonlinear half-wave equations with focusing power-type nonlinearityi∂tu=-Δu-|u|p-1u,with(t,x)∈R×Rdwith exponents 1 < p< ∞ for d = 1 and 1 < p< (d+ 1) / (d- 1) for d ≥ 2. We study traveling solitary waves of the formu(t, x) = e i ω t Q v (x- vt) with frequency ω∈ R, velocity v∈ R d , and some finite-energy profile Q v ∈ H 1 / 2 (R d ) , Q v ≢ 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics