## 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

### Citations:

Zbl 1126.35055; Zbl 0963.37058; Zbl 0981.35073; Zbl 1064.35171
Full Text:

### References:

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