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Evolution equation for soft physics at high energy

Based on the non-linear logistic equation we study, in a qualitative and semi-quantitative way, the evolution with energy and saturation of the elastic differential cross-section in $pp(\bar{p}p)$ collisions at high energy. Geometrical scaling occurs at the black disk limit, and scaling develops first for small values of the scaling variable $|t|σ_{tot.}$. Our prediction for $d σ/ \ d t$ at LHC, with two zeros and a minimum at large $|t|$ differs, as far as we know, from all existing ones.