Non-Markovian dynamics of the sine-Gordon solitons

The sine-Gordon equation exhibits a gap in its linear spectrum. That gives rise to memory, or non-Markovian, effects in the soliton formation processes. The generalized variational approach is suggested to derive the model equation that governs the solitary wave evolution. The detailed analytical and numerical studies show that the soliton relaxation dynamics exhibits the main specific features of quantum emitters decay processes in photonic band gap materials. In particular, the non-Markovian effects lead to the extremely long-lived oscillations of the sine-Gordon solitons. That results in the bound state of the soliton with the radiated linear waves.