Inelastic interaction of nearly equal solitons for the quartic gKdV equation

Abstract

This paper describes the interaction of two solitons with nearly equal speeds for the quartic (gKdV) equation $$\partial_tu+\partial_x(\partial_x^2u+u^4)=0,\quad t,x\in \mathbb{R}.$$ (0.1) We call soliton a solution of (0.1) of the form u(t,x)=Qc(x−ct−y0), where c>0, y0∈ℝ and \(Q_{c}''+Q_{c}^{4}=cQ_{c}\). Since (0.1) is not an integrable model, the general question of the collision of two given solitons \(Q_{c_{1}}(x-c_{1}t)\), \(Q_{c_{2}}(x-c_{2}t)\) with c1≠c2 is an open problem.