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Asymptotic stability of solitons of the gKdV equations with general nonlinearity. (English) Zbl 1153.35068

Summary: We consider the generalized Korteweg-de Vries equation (gKdV)
\[ \partial_t u + \partial_x (\partial_x^2 u + f(u)) = 0, \quad (t, x) \in [0, T)\times {\mathbb{R}}, \]
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 \(t \geq 0\)), then \(u (t)\) locally converges in the energy space to some \(Q_{c +}\) as \(t \rightarrow +\infty\). 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 [Y. Martel, SIAM J. Math. Anal. 38, No. 3, 759–781 (2006; Zbl 1126.35055); Y. Martel and F. Merle, J. Math. Pures Appl. (9) 79, No. 4, 339–425 (2000; Zbl 0963.37058), Arch. Ration. Mech. Anal. 157, No. 3, 219–254 (2001; Zbl 0981.35073), Nonlinearity 18, No. 1, 55–80 (2005; Zbl 1064.35171)], devoted to a pure power case.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
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References:

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