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In this paper we describe the inelastic character of solitons of some slowlyvarying gKdV equations. We give precise lower bounds, in the energy space, ofthe defect induced by the potential on the solution as time goes to infinite.For the proof, we consider suitable Lyapunov functionals that propagate thedefect along the dynamics in a non trivial fashion, in the spirit of the recentbreakthroughs made by Martel and Merle in the case of the two soliton collisionfor non integrable gKdV equations. In addition, we also consider a special,degenerate case, where the defect is formally of second order.

Inelastic character of solitons of slowly varying gKdV equations