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

Source abstract

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.