Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation

We consider the following class of equations of (gKdV) type partial derivative(t)u + partial derivative(x) (partial derivative(2)(x) u + vertical bar u vertical bar p(-1) u) = 0, p integer, t, x is an element of R with mass sub-critical (2 < p < 5) and mass super-critical nonlinearities (p > 5). We prove the existence of two-solitary wave solutions with logarithmic relative distance, i.e. solutions u(t) satisfying parallel to u(t) - (Q(. - t - log(ct)) + sigma Q(. - t + log(ct))) parallel to H-1 -> 0 as t -> +infinity, where c = c(p) > 0 is a fixed constant, sigma = - 1 in sub-critical cases and sigma = 1 in super-critical cases. This regime corresponds to strong attractive interactions. For sub-critical p, it was known that opposite-sign traveling waves are attractive. For super-critical p, we derive from our computations that same-sign traveling waves are attractive.