Uniqueness for the Nonlocal Liouville Equation in $\mathbb{R}$

TL;DR

This paper proves the uniqueness of solutions to a nonlocal Liouville equation on the real line with finite Q-curvature, using a novel connection to integrable PDEs like the Calogero--Moser system.

Contribution

It establishes the first uniqueness results for the nonlocal Liouville equation with symmetric, decaying curvature functions, linking it to integrable soliton equations.

Findings

Uniqueness of solutions for the nonlocal Liouville equation with symmetric curvature functions.

Special case of Gaussian curvature function $K(x) = e^{-x^2}$ also admits unique solutions.

Connection established between the nonlocal Liouville equation and ground state solitons of Calogero--Moser derivative NLS.

Abstract

We prove uniqueness of solutions for the nonlocal Liouville equation $$ (-\Delta)^{1/2} w = K e^w \quad \mbox{in $\mathbb{R}$} $$ with finite total $Q$-curvature $\int_{\mathbb{R}} K e^w \, dx< +\infty$. Here the prescribed $Q$-curvature function $K=K(|x|) > 0$ is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian case with $K(x) = \exp(-x^2)$. Our uniqueness proof exploits a connection of the nonlocal Liouville equation to ground state solitons for Calogero--Moser derivative NLS, which is a completely integrable PDE recently studied by P. G\'erard and the second author.