Abstract
In this paper, we give the first description of the collision of two solitons for a nonintegrable equation in a special regime. We consider solutions of the quartic gKdV equation $\partial_t u + \partial_x (\partial_x^2 u + u^4)=0$, which behave as $t\to \infty$ like \[ u(t,x)=Q_{c_1}(x c_1 t) + Q_{c_2}(xc_2 t) + \eta(t,x), \] where $Q_{c}(xct)$ is a soliton and $\\eta(t)\_{H^1} \ll \Q_{c_2}\_{H^1}\ll \Q_{c_1}\_{H^1}$.
The global behavior of $u(t)$ is given by the following stability result: for all $t\in \mathbb{R}$, $u(t,x)=Q_{ c_1(t)}(xy_1(t)) + Q_{ c_2(t)}(xy_2(t)) + \eta(t,x)$, where $\vert\eta(t)\_{H^1} \ll \Q_{c_2}\vert_{H^1}$ and $\lim_{t\to +\infty} c_1(t)=c_1^{+}$, $\lim_{t\to +\infty} c_2(t)=c_2^{+}$.
In the case where $u(t)$ is a pure $2$soliton solution as $t\to \infty$ (i.e. $\mathrm{lim}_{t\to \infty} \vert\eta(t)\vert_{H^1}=0$), we obtain $ c_1^{+}>c_1, c_2^{+}\lt c_2 $ and for the residual part, $\mathrm{lim}_{t\to +\infty} \vert\eta(t)\vert_{H^1}\gt 0$. Therefore, in contrast with the integrable KdV equation (or mKdV equation), no global pure 2soliton solution exists and the collision is inelastic. A different notion of global 2soliton is then proposed.

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