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 ∂tu + ∂x(∂2xu + u4) = 0, which behave as t → -∞ like u(t, x) = Qc1t) (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+2 < c2 and for the residual part, limt→+∞ ∥η(t)∥H1 > 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.