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 ∂ t u = - Δ u - | u | p - 1 u , with ( t , x ) ∈ R × R d 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 ) = e i ω t Q v ( x - v t ) 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 + 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 ≥ 2.