Time-global solvability of the Vlasov–Poisson system around the traveling waves

We consider the 1D Vlasov–Poisson system on the real line, and establish the time-global solvability for arbitrarily large initial data $ f_{\pm}^{0} $. We assume that the support of the difference between the initial data $ f_{\pm}^{0} $ and $ f_{\pm}^{*, 0} $ is compact in the phase space, where $ f_{\pm}^{*, 0} $ is the initial value of traveling waves of the 1D Vlasov–Poisson system.