Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations

We consider the generalized Korteweg-de Vries equations u(t) + (u(xx) + u(P))(x) = 0, t,x is an element of R, in the subcritical and critical cases p = 2,3,4 or 5. Let R-j(t,x) = Qc(j)(x - c(j)t - x(j)), where j is an element of {1,..., N}, be N soliton solutions of this equation, with corresponding speeds 0 < c(1) < c(2) <...< c(N). In this paper, we construct a solution u(t) of the generalized Korteweg-de Vries equation such that N [GRAPHIC] This solution behaves asymptotically as t -> +infinity as the sum of N solitons without loss of mass by dispersion. This is an exceptional behavior, indeed, being given the parameters {cj}1 <= j <= N, {xj} 1 <= j <= N, we prove uniqueness of such a solution. In the integrable cases p = 2 and 3, such solutions are explicitly known and their properties were extensively studied in the literature (they are called N-soliton solutions). Therefore, the existence result is new only for the nonintegrable cases. The uniqueness result is new for all cases.