Uniqueness for the nonlocal Liouville equation in R

We prove uniqueness of solutions for the nonlocal Liouville equation(−Δ)1/2w=Kewin R with finite total Q-curvature ∫RKewdx0 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⁡(−x2). 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. Gérard and the second author.