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We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for thenonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\mathbb{R}$, where$0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha <\infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ denotes the fractional Laplacianin one dimension. In particular, we generalize (by completely differenttechniques) the specific uniqueness result obtained by Amick and Toland for$s=1/2$ and $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As atechnical key result in this paper, we show that the associated linearizedoperator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate;i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$.This result about $L_+$ proves a spectral assumption, which plays a centralrole for the stability of solitary waves and blowup analysis for nonlineardispersive PDEs with fractional Laplacians, such as the generalizedBenjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.

Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q + Q - Q^{\alpha+1} = 0$ in $\mathbb{R}$