Inelastic interaction of nearly equal solitons for the quartic gKdV equation

摘要

This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation, We call soliton a solution of (0.1) of the form u(t,x)=Q_c(x-ct-y_0), where c>0, y_0∈? and Q″_c+Q~4_c=cQ_c. Since (0.1) is not an integrable model, the general question of the collision of two given solinfitons Qc_1(x-c_t),Qc_2(x-c_2t) with c_1≠c_2 is an open problem. We focus on the special case where the two solitons have nearly equal speeds: let U(t) be the solution of (0. 1) satisfying for small. By constructing an approximate solution of (0.1), we prove that, for all time t∈?,U(t)=Q_(c1)(t)(x-y_1(t))+Q_(c2)(t)(x-y2(t))+w(t),with y_1(t)-y_2(t)>2{pipe}ln μ_0{pipe}+C, for some C∈?. These estimates mean that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable KdV equation in this regime. However, unlike in the integrable case, we prove that the collision is not perfectly elastic, in the following sense, for some C>0, and in H1 as t→+∞.