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Long-time asymptotics are established for finite energy solutions of the scalar Klein–Gordon equation coupled to a relativistic classical particle: any “scattering” solution is asymptotically a sum of a soliton and of a dispersive free wave packet as t→±∞. These asymptotics mean the nonlinear scattering of free wave packets by the soliton.

Scattering of solitons of the Klein–Gordon equation coupled to a classical particle