For the mass critical generalized KdV equation $$\partial_t u + \partial_x (\partial_x^2 u + u^5)=0$$ on $\mathbb R$, we construct a full family of flattening solitary wave solutions.

Let $Q$ be the unique even positive solution of $Q''+Q^5=Q$. For any $\nu\in (0,\frac 13)$, there exist global (for $t\geq 0$) solutions of the equation with the asymptotic behavior

\[u(t,x)= t^{-\frac{\nu}2} Q\left(t^{-\nu} (x-x(t))\right)+w(t,x)\]

where, for some $c>0$,

\[x(t)\sim c t^{1-2\nu} \quad \mbox{and}\quad \|w(t)\|_{H^1(x>\frac 12 x(t))} \to 0\quad \mbox{as $t\to +\infty$.}\]

Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data.

This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation.

This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.

This talk is based on a joint work with Yvan Martel (Ecole Polytechnique).