Coupling by reflection mixed with synchronous coupling is constructed for a class of stochastic differential equations (SDEs) driven by Lévy noises. As an application, we establish the exponential contractivity of the associated semigroups (P_{t})_{t\ge0} with respect to the standard L^{p}-Wasserstein distance for all p\in[1,\infty). In particular, consider the following SDE:
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\[\mathrm{d}X_{t}=\mathrm{d}Z_{t}+b(X_{t})\,\mathrm{d}t,\] where (Z_{t})_{t\ge0} is a symmetric \alpha-stable process on \mathbb{R}^{d} with \alpha\in(1,2). We show that if the drift term b satisfies that for any x,y\in\mathbb{R}^{d},
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\[\langle b(x)-b(y),x-y\rangle\le\cases{K_{1}|x-y|^{2},\quad\phantom{-} |x-y|\le L_{0};\cr-K_{2}|x-y|^{\theta},\quad |x-y|>L_{0}}\] holds with some positive constants K_{1}, K_{2}, L_{0}>0 and \theta\ge2, then there is a constant \lambda:=\lambda(\theta,K_{1},K_{2},L_{0})>0 such that for all p\in[1,\infty), t>0 and x,y\in\mathbb{R}^{d},
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\[W_{p}(\delta_{x}P_{t},\delta_{y}P_{t})\le C(p,\theta,K_{1},K_{2},L_{0})\mathrm{e}^{-\lambda t/p}[\frac{|x-y|^{1/p}\vee|x-y|}{1+|x-y|{\mathbf{1}}_{(1,\infty)\times(2,\infty)}(t,\theta)}].\]