Multi-soliton solutions for the supercritical gKdV equations

Vianney Combet (1)

1 LMV - Laboratoire de Mathématiques de Versailles

For the L^2 subcritical and critical (gKdV) equations, Martel proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N solitons asymptotically as time goes to infinity. More recently, for the L^2 supercritical case, Cote, Martel and Merle proved the existence of at least one multi-soliton. In the present paper, as suggested by a previous work concerning the one soliton case, we first construct an N-parameter family of multi-solitons for the supercritical (gKdV) equation, for N arbitrarily given solitons, and then prove that any multi-soliton belongs to this family. In other words, we obtain a complete classification of multi-solitons for (gKdV).

Domaines: Equations aux dérivées partielles [math.AP]

Vianney Combet. Multi-soliton solutions for the supercritical gKdV equations. Communications in Partial Differential Equations, 2010, 36 (3), pp.380-419. ⟨10.1080/03605302.2010.503770⟩. ⟨hal-00456077⟩