Communications in Mathematical Physics

, Volume 372, Issue 2, pp 713–732 | Cite as

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

  • Jacopo BellazziniEmail author
  • Vladimir Georgiev
  • Enno Lenzmann
  • Nicola Visciglia


We consider nonlinear half-wave equations with focusing power-type nonlinearity
$$i \partial_t u = \sqrt{-\Delta} \, u - |u|^{p-1} u, \quad \mbox{with}\, (t,x) \in \mathbb{R} \times \mathbb{R}^d$$
with exponents \({1 < p < \infty}\) for d = 1 and \({1 < p < (d+1)/(d-1)}\) for d ≥  2. We study traveling solitary waves of the form
$$u(t, x) = e^{i\omega t} Q_v(x-vt)$$
with frequency \({\omega \in \mathbb{R}}\), velocity \({v \in \mathbb{R}^d}\), and some finite-energy profile \({Q_v \in H^{1/2}(\mathbb{R}^d)}\), \({Q_v \not \equiv 0}\). We prove that traveling solitary waves for speeds \({|v| \geq 1}\) do not exist. Furthermore, we generalize the non-existence result to the square root Klein–Gordon operator \({\sqrt{-\Delta+m^2}}\) 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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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].


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Università di SassariSassariItaly
  2. 2.Dipartimento di MatematicaUniversità di PisaPisaItaly
  3. 3.Faculty of Science and EngineeringWaseda UniversityTokyoJapan
  4. 4.IMI–BASSofiaBulgaria
  5. 5.Departement Mathematik und InformatikUniversität BaselBaselSwitzerland

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