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

We consider the generalized Korteweg-de Vries equations $$u_t+u_{\mathrm{xx}}+u_p{x}_{}=0_{}\text{t, x ∈ R}$$ [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_jt - x_j), where j ∈ {1, . . . , N}, be N soliton solutions of this equation, with corresponding speeds 0 < c₁ < c₂ < ... < 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.