TY - JOUR

T1 - On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations

AU - Bellazzini, Jacopo

AU - Georgiev, Vladimir

AU - Lenzmann, Enno

AU - Visciglia, Nicola

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - 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) = ei ω tQv(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.

AB - 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) = ei ω tQv(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.

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U2 - 10.1007/s00220-019-03374-y

DO - 10.1007/s00220-019-03374-y

M3 - Article

AN - SCOPUS:85062012758

SN - 0010-3616

VL - 372

SP - 713

EP - 732

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -