Uniqueness of non-linear ground states for fractional Laplacians in $BbbR

We prove uniqueness of ground state solutions Q = Q(vertical bar x vertical bar) >= 0 of the non-linear equation (-Delta)(s)Q+Q-Q(alpha+1)=0 in R, (-Delta) s Q + Q - Q alpha + 1 = 0 in R, where 0 < s < 1 and 0 < alpha < 4s/(1-2s) for s<1/2 s<1 2 and 0 < alpha < infinity for s >= 1/2 s >= 1 2. Here (-Delta)(s) denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig-Martel-Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s=1/2 s=1 2 and alpha = 1 in [5] for the Benjamin-Ono equation.As a technical key result in this paper, we show that the associated linearized operator L+ = (-Delta)(s) +1-(alpha+1)Q(alpha) is non-degenerate; i.e., its kernel satisfies ker L+=span{Q'}. This result about L+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.