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 ∈ R inline-graphic xmlns: xlink="http: //www. w3. org/1999/xlink" xlink: href="01i" / 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 ∈ 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 inline-graphic xmlns: xlink="http: //www. w3. org/1999/xlink" xlink: href="02i" / This solution behaves asymptotically as t → +∞ as the sum of N solitons without loss of mass by dispersion. This is an exceptional behavior, indeed, being given the parameters c j 1≤ j ≤ N, x j 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.