Nondispersive Solutions to the L-critical Half-wave Equation

We consider the focusing L2-critical half-wave equation in one space dimension i∂tu = Du− |u|u, where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold M∗ > 0 such that all H1/2 solutions with ‖u‖L2 < M∗ extend globally in time, while solutions with ‖u‖L2 > 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 ‖u0‖L2 = M∗. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E0 > 0 and the linear momentum P0 ∈ R. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L2-critical nonlinear PDE with nonlocal dispersion.