Inelasticity of soliton collisions for the 5D energy critical wave equation

摘要

For the focusing energy critical wave equation in 5D, we construct a solution showing the inelastic nature of the collision of two solitons for any choice of sign, speed, scaling and translation parameters, except the special case of two solitons of same scaling and opposite signs. Beyond its own interest as one of the first rigorous studies of the collision of solitons for a non-integrable model, the case of the quartic gKdV equation being partially treated in Martel and Merle (Ann Math 174(2):757-857, 2011; Invent Math 183(3):563-648, 2011; Int Math Res Notices 2015(3):688-739, 2015), this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation. This conjecture has already been proved in the 3D radial case in Duyckaerts et al. (Camb J Math 1:75-144, 2013) and in the general case in3,4 and5D along a sequence of times in Duyckaerts et al. (Geom Funct Anal 27(4):798-862, 2017). Compared with the construction of an asymptotic two-soliton in Martel and Merle (Arch Ration Mech Anal 222(3):1113-1160, 2016), the study of the nature of the collision requires a more refined approximate solution of the two-soliton problem and a precise determination of its space asymptotics. To prove inelasticity, these asymptotics are combined with the method of channels of energy from Duyckaerts et al. (Camb J Math 1:75-144, 2013), Kenig et al. (Geom Funct Anal 24:610-647, 2014).