### 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) = 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.

Original language | English |
---|---|

Journal | Communications in Mathematical Physics |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*. https://doi.org/10.1007/s00220-019-03374-y

**On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations.** / Bellazzini, Jacopo; Gueorguiev, Vladimir Simeonov; Lenzmann, Enno; Visciglia, Nicola.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*. https://doi.org/10.1007/s00220-019-03374-y

}

TY - JOUR

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

AU - Bellazzini, Jacopo

AU - Gueorguiev, Vladimir Simeonov

AU - Lenzmann, Enno

AU - Visciglia, Nicola

PY - 2019/1/1

Y1 - 2019/1/1

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) = 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.

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) = 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.

UR - http://www.scopus.com/inward/record.url?scp=85062012758&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062012758&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03374-y

DO - 10.1007/s00220-019-03374-y

M3 - Article

AN - SCOPUS:85062012758

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -