Evolution and interactions of solitary waves (solitons) in nonlinear dissipative systems

A generalized wave equation containing production and dissipation of energy is derived in a heuristic fashion so as to have in the dissipationless limit the (two-way wave) Boussinesq equation, while for the slowly evolving in a moving frame (one-way) wave system, it reduces to the dissipation modified KDV equation (KDV-KSV) with the same energy-balance law. The new equation allows investigating the head-on collision of dissipative localized structures. A special difference scheme is devised which faithfully represents the balance law for energy. The numerical simulations show that if the production-dissipation rate is of order of a small parameter, the coherent structures upon collisions preserve their localized character and within a time interval proportional to the inverse of the small parameter they behave like (imperfect) solitons. The collisions are almost ideal without phase shift. The only difference from the strictly soliton collision is that during the time of interaction the dissipative structures are "aging" and changing their shapes.

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