Abstract

We consider the generalized Korteweg-de Vries equations

ut+uxx+upx=0t, 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 Rj(t, x) = Qcj(x - cjt - xj), where j ∈ {1, . . . , N}, be N soliton solutions of this equation, with corresponding speeds 0 < c1 < c2 < ... < cN. 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 {cj}1≤jN, {xj}1≤jN, 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.

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