Bellazzini, Jacopo and Georgiev, Vladimir and Lenzmann, Enno and Visciglia, Nicola.
(2019)
* On traveling solitary waves and absence of small data scattering for nonlinear half-wave equations.*
Communications in Mathematical Physics, 372 (2).
pp. 713-732.

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Official URL: https://edoc.unibas.ch/75290/

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## Abstract

We consider nonlinear half-wave equations with focusing power-type nonlinearity

i∂tu=−Δ−−−√u−|u|p−1u,with(t,x)∈R×Rd

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)=eiωtQv(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.

i∂tu=−Δ−−−√u−|u|p−1u,with(t,x)∈R×Rd

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)=eiωtQv(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.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann) |
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UniBasel Contributors: | Lenzmann, Enno |

Item Type: | Article, refereed |

Article Subtype: | Research Article |

Publisher: | Springer Nature |

ISSN: | 0010-3616 |

Note: | Publication type according to Uni Basel Research Database: Journal article |

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Last Modified: | 30 Jun 2020 15:31 |

Deposited On: | 24 Jun 2020 13:19 |

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