Nondispersive solutions to the L 2-critical Half-Wave Equation

We consider the focusing L ²-critical half-wave equation in one space dimension, $$i partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H ^1/2 solutions with ${|u|_{L^2} < M_*}$ extend globally in time, while solutions with ${|u|_{L^2} geq M_*}$ 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 ${|u_0|_{L^2} = M_*}$ . 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 PDEs with nonlocal dispersion.