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 - Funding Information:
J.B., V.G., and N.V. are partially supported by Project 2016 “Dinamica di equazioni nonlineari dispersive” of FONDAZIONE DI SARDEGNA. J.B. and V.G. are also partially supported by Project 2017 “Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari” of INDAM, GNAMPA- Gruppo Nazionale per l’AnalisiMatematica, la Probabilita e le loro Applicazioni. V.G. is partially supported by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, by Top Global University Project, Waseda University and the Project PRA 2018 49 of University of Pisa. E.L. was partially supported by the SwissNational Science Foundation (SNSF) throughGrantNo. 200021–149233. In addition, E. L. thanks Rupert Frank for valuable discussions and providing us with reference [3].
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 -