American Institute of Mathematical Sciences

$\partial_t u + \partial_x (\partial_x^2 u + f(u))=0, (t,x)\in \mathbb{R}\times \mathbb{R}$(1)

with $C^3$ nonlinearity $f$. Under an 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 [11], [13], 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 $c^+$~$c$ and some function $\rho(t)$ such that $\rho'(t)$~$c^+$. Then, in [9] and [14], 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.