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

Jacopo Bellazzini*, Vladimir Georgiev, Enno Lenzmann, Nicola Visciglia

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)713-732
Number of pages20
JournalCommunications in Mathematical Physics
Volume372
Issue number2
DOIs
Publication statusPublished - 2019 Dec 1

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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