## Refined asymptotics around solitons for gKdV equations.(English)Zbl 1137.35062

Summary: We consider the generalized Korteweg-de Vries equation
$\partial_tu+ \partial_x(\partial_x^2u+ f(u))=0, \quad (t,x)\in\mathbb R\times\mathbb R,\tag{1}$
with $$C^3$$ nonlinearity $$f$$. Under some explicit condition on $$f$$ and $$c>0$$, there exists a solution of (1) in the energy space $$H^1$$ of the type $$u(t,x)=Q_c(x -x_0-ct)$$, called soliton.
In [the authors, J. Am. Math. Soc. 15, No. 3, 617–664 (2002; Zbl 0996.35064), Nonlinearity 18, No. 1, 55–80 (2005; Zbl 1064.35171)], it was proved that for $$f(u)=u^p$$, $$p=2,3,4$$, the family of solitons is asymptotically stable in some local sense in $$H^1$$, i.e. if $$u(t)$$ is close to $$Q_c$$, then $$u(t,.+\rho(t))$$ locally converges in the energy space to some $$Q_{c_+}$$ as $$t\to+\infty$$, for some $$c^+\sim c$$ and some function $$\rho(t)$$ such that $$\rho'(t)\sim c^+$$. Then, in [the the author, SIAM J. Math. Anal. 38, No. 3, 759–781 (2006; Zbl 1126.35055) and the authors, Math. Ann. 341, No. 2, 391–427 (2008; Zbl 1153.35068)], these results were extended with shorter proofs under general assumptions on $$f$$.
The first objective of this paper is to give more information about the function $$\rho(t)$$. In the case $$f(u)= u^p$$, $$p=2,3,4$$ and under the additional assumption $$x_+u\in L^2(\mathbb R)$$, we prove that the function $$\rho(t)-c^+t$$ has a finite limit as $$t\to+\infty$$.
Second, we prove stability and asymptotic stability results for two solitons for a general nonlinearity $$f(u)$$, in the case where the ratio of the speeds of the two solitons is small.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs

### Keywords:

generalized KdV equation soliton; asymptotic stability

### Citations:

Zbl 0996.35064; Zbl 1064.35171; Zbl 1126.35055; Zbl 1153.35068
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