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 ∂tu+∂x(∂2xu+u4)=0, which behave as t→−∞ like
u(t,x)=Qc1(x−c1t)+Qc2(x−c2t)+η(t,x),
where Qc(x−ct) is a soliton and ∥η(t)∥H1≪∥Qc2∥H1≪∥Qc1∥H1.
The global behavior of u(t) is given by the following stability result: for all t∈ℝ, u(t,x)=Qc1(t)(x−y1(t))+Qc2(t)(x−y2(t))+η(t,x), where |η(t)∥H1≪∥Qc2|H1 and limt→+∞c1(t)=c+1, limt→+∞c2(t)=c+2.
In the case where u(t) is a pure 2-soliton solution as t→−∞ (i.e. limt→−∞|η(t)|H1=0), we obtain c+1>c1,c+20. 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.

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