First-order perturbed Korteweg-de Vries solitons

We consider the Korteweg-de Vries equation with a perturbation arising naturally in many physical situations. Although being asymptotically integrable, we show that the corresponding perturbed solitons do not have the usual scattering properties. Specifically, we show that there is a solution, correct up to O(ε), where ε is the perturbative parameter, consisting, at t→ -∞ of two superposed deformed solitons characterized by wave numbers k1 and k2 that give rise, for t→ +∞, to the same but phase-shifted superposed solitons plus a coupling term depending on k1, and k2. We also find the condition on the original equation for which this coupling vanishes.