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

Abstract

We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i \pt_t u = \sqrt{-Δ} \, u - |u|^{p-1} u, \quad \mbox{with $(t,x) \in \R \times \R^d$} $$ with exponents $1<p<\infty$ for $d=1$ and $1<p<(d+1)/(d-1)$ for $d\ge 2$. 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 $|v|\ge 1$ 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 $d\ge 2$.

17 pages

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• Nondispersive solutions to the mass critical half-wave equation in two dimensions
• Blowup dynamics for mass critical Half-wave equation in 2D