On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations
| Autor: | Bellazzini, Jacopo, Georgiev, Vladimir, Lenzmann, Enno, Visciglia, Nicola |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00220-019-03374-y |
| Popis: | We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i \pt_t u = \sqrt{-\Delta} \, u - |u|^{p-1} u, \quad \mbox{with $(t,x) \in \R \times \R^d$} $$ with exponents $1 < p < \infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d \geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{i\omega t} Q_v(x-vt) $$ with frequency $\omega \in \R$, velocity $v \in \R^d$, and some finite-energy profile $Q_v \in H^{1/2}(\R^d)$, $Q_v \not \equiv 0$. 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 $\sqrt{-\DD+m^2}$ 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 $d \geq 2$. Comment: 17 pages |
| Databáze: | arXiv |
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