Inventiones mathematicae

, Volume 214, Issue 3, pp 1267–1363 | Cite as

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

  • Yvan Martel
  • Frank Merle


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 in 3, 4 and 5D 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).



The authors thank the anonymous referees for their useful comments. The authors are also grateful to Xu Yuan for his careful reading of the article and his remarks. This work was partially supported by ERC 291214 BLOWDISOL. This material is partly based upon work supported by the National Science Foundation under Grant No. 0932078 000 while Y.M. was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2015 semester.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CMLSÉcole polytechnique, CNRS, Université Paris-SaclayPalaiseau CedexFrance
  2. 2.AGMUniversité de Cergy-Pontoise and Institut des Hautes Études Scientifiques, CNRSCergy-PontoiseFrance

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