Nondispersive solutions to the L2-critical half-wave equation

TL;DR

This paper studies the focusing L^2-critical half-wave equation in one dimension, demonstrating the existence of traveling waves with small mass and finite-time blowup solutions at the critical mass, highlighting nonlocal dispersion effects.

Contribution

It constructs minimal mass blowup solutions parametrized by energy and momentum, and provides examples of nondispersive dynamics in a nonlocal dispersive PDE.

Findings

Existence of traveling waves with arbitrarily small mass.

Construction of finite-time blowup solutions at critical mass.

Illustration of minimal mass blowup as a model for nonlocal dispersive PDEs.

Abstract

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 $P_0 \in \R$. 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.