Description of two soliton collision for the quartic gKdV equation

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 \[ u(t,x)=Q_{c_1}(x -c_1 t) + Q_{c_2}(x-c_2 t) + \eta(t,x), \] where $Q_{c}(x-ct)$ 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 $t\in \mathbb{R}$, $u(t,x)=Q_{ c_1(t)}(x-y_1(t)) + Q_{ c_2(t)}(x-y_2(t)) + \eta(t,x)$, 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 2-soliton solution exists and the collision is inelastic. A different notion of global 2-soliton is then proposed.