Nondispersive solutions to the L2-critical half-wave equation

Abstract

We consider the focusing $L^2$-critical half-wave equation in one space dimension $$i{\partial }_{t}u=Du-|u{|}^{2}u,$$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_{\ast }>0$ such that all $H^{1/2}$ solutions with $\Vert u{\Vert }_{L^2}<M_{\ast }$ extend globally in time, while solutions with $\Vert u{\Vert }_{L^2}\ge M_{\ast }$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass $\Vert u_0{\Vert }_{L^2}=M_{\ast }$. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy $E_0>0$ and the linear momentum $P_0\in \mathbb{R}$. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for $L^2$-critical nonlinear PDE with nonlocal dispersion.

51 pages

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