Refined asymptotics around solitons for gKdV equations

We consider the generalized Korteweg-de Vries equation $$\partial_t u + \partial_x (\partial_x^2 u + f(u))=0, \quad (t,x)\in [0,T)\times \mathbb{R}$$ with general $C^2$ 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. Stability theory for $Q_c$ is well-known. In previous works, we have 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 $Q_{c}$ (for all $t\geq 0$), then $u(t,.+\rho(t))$ locally converges in the energy space to some $Q_{c_+}$ as $t\to +\infty$, for some $c^+\sim c$. Then, the asymptotic stability result could be extended to the case of general assumptions on $f$ and $Q_c$. The objective of this paper is twofold. The main objective is to prove that in the case $f(u)=u^p$, $p=2,3,4$, $\rho(t)-c_+ t$ has limit as $t\to +\infty$ under the additional assumption $x_+ u\in L^2$. The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity $f(u)$, when the ratio of the speeds of the solitons is small. The motivation is to accompany forthcoming works devoted to the collision of two solitons in the nonintegrable case. The arguments are refinements of previous works specialized to the case $u(t)\sim Q_{c_1}+Q_{c_2}$, for $0< c_2 \ll c_1$.