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

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.