Nondispersive solutions to the mass critical half-wave equation in two dimensions

TL;DR

This paper studies the two-dimensional mass critical half-wave equation, proving the existence of traveling solitary waves and finite-time blowup solutions with minimal mass, advancing understanding of its nonlinear dynamics.

Contribution

It establishes the existence of traveling solitary waves and minimal mass blowup solutions for the 2D mass critical half-wave equation, which were previously unknown.

Findings

Existence of a family of traveling solitary waves.

Existence of finite-time blowup solutions with minimal mass.

Characterization of ground state solutions Q.

Abstract

We consider the half-wave equation with mass critical in two dimension \begin{eqnarray*} \begin{cases} iu_t=Du-|u|u,\,\,\, \\ u(0,x)=u_0(x), \end{cases} \end{eqnarray*} First, we prove the existence of a family of traveling solitary waves. We then show the existence of finite-time blowup solutions with minimal mass $\|u_0\|_2=\|Q\|_2$, where $Q$ is the ground state solution of equation $DQ+Q=Q^2$.