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Description of two soliton collision for the quartic gKdV equation. (English) Zbl 1300.37045
Summary: 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 $$\|\eta(t)\|_{H^1} \ll \|Q_{c_2}\|_{H^1}$$ 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. $$\mathrm{lim}_{t\to -\infty} |\eta(t)|_{H^1}=0)$$, we obtain $$c_1^{+}>c_1$$, $$c_2^{+}< c_2$$ and for the residual part, $$\mathrm{lim}_{t\to +\infty} |\eta(t)|_{H^1}> 0$$. 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.

##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations
##### Keywords:
collision; gKdV equations; multi-soliton; soliton
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