Abstract
We consider the generalized Korteweg-de Vries equation (gKdV)
with general C 3 nonlinearity f. Under an explicit condition on f and c > 0, there exists a solution in the energy space H 1 of the type u(t, x) = Q c (x − x 0 − ct), called soliton. In this paper, under general assumptions on f and Q c , we prove that the family of solitons around Q c is asymptotically stable in some local sense in H 1, i.e. if u(t) is close to Q c (for all t ≥ 0), then u(t) locally converges in the energy space to some Q c+ as t → +∞. Note in particular that we do not assume the stability of Q c . This result is based on a rigidity property of the gKdV equation around Q c in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in Martel (SIAM J. Math. Anal. 38:759–781, 2006); Martel and Merle (J. Math. Pures Appl. 79:339–425, 2000), (Arch. Ration. Mech. Anal. 157:219–254, 2001), (Nonlinearity 1:55–80), devoted to the pure power case.
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This research was supported in part by the Agence Nationale de la Recherche (ANR ONDENONLIN).
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Martel, Y., Merle, F. Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Math. Ann. 341, 391–427 (2008). https://doi.org/10.1007/s00208-007-0194-z
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DOI: https://doi.org/10.1007/s00208-007-0194-z