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Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 127, Number 5, October 2005
- pp. 1103-1140
- 10.1353/ajm.2005.0033
- Article
- Additional Information
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We consider the generalized Korteweg-de Vries equations
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≤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.