Advertisement

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
Article

Abstract

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).

Notes

Acknowledgements

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.

References

  1. 1.
    Bona, J.L., Pritchard, W.G., Scott, L.R.: Solitary-wave interaction. Phys. Fluids 23, 438 (1980)CrossRefGoogle Scholar
  2. 2.
    Côte, R., Kenig, C.E., Schlag, W.: Energy partition for the linear radial wave equation. Math. Ann. 358, 573–607 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Côte, R., Martel, Y., Merle, F.: Construction of multi-soliton solutions for the \(L^2\)-supercritical gKdV and NLS equations. Rev. Mat. Iberoam. 27, 273–302 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Côte, R., Muñoz, C.: Multi-solitons for nonlinear Klein–Gordon equations. In: Forum of Mathematics, Sigma vol. 2 (2014)Google Scholar
  5. 5.
    Craig, W., Guyenne, P., Hammack, J., Henderson, D., Sulem, C.: Solitary water wave interactions. Phys. Fluids 18, 057106 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Deift, P.A., Zhou, X.: Long-time asymptotics for integrable nonlinear wave equations. In: Important Developments in Soliton Theory, Springer Series in Nonlinear Dynamics, pp. 181–204. Springer, Berlin (1993)zbMATHGoogle Scholar
  7. 7.
    del Pino, M., Musso, M., Pacard, F., Pistoia, A.: Large energy entire solutions for the Yamabe equation. J. Differ. Equ. 251, 2568–2597 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ding, W.Y.: On a conformally invariant elliptic equation on \({\mathbb{R}}^n\). Commun. Math. Phys. 107, 331–335 (1986)CrossRefGoogle Scholar
  9. 9.
    Duyckaerts, T., Merle, F.: Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. 2008, Art ID rpn002 (2008)Google Scholar
  10. 10.
    Duyckaerts, T., Kenig, C.E., Merle, F.: Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation. J. Eur. Math. Soc. 13, 533–599 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Duyckaerts, T., Kenig, C.E., Merle, F.: Classification of radial solutions of the focusing, energy-critical wave equation. Camb. J. Math. 1, 75–144 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Duyckaerts, T., Kenig, C .E., Merle, F.: Solutions of the focusing nonradial critical wave equation with the compactness property. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15, 731–808 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Duyckaerts, T., Kenig, C.E., Merle, F.: Scattering profile for global solutions of the energy-critical wave equation. J. Eur. Math. Soc. (to appear). arXiv:1601.02107
  14. 14.
    Duyckaerts, T., Jia, H., Kenig, C.E., Merle, F.: Soliton resolution along a sequence of times for the focusing energy critical wave equation. Geom. Funct. Anal. 27(4), 798–862 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19. AMS (2010)Google Scholar
  16. 16.
    Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems, I. In: Newell, A.C. (eds) Los Alamos Report LA1940 (1955); Reproduced in Nonlinear Wave Motion, pp. 143–156. AMS, Providence, RI (1974)Google Scholar
  17. 17.
    Gérard, P., Lenzmann, E., Pocovnicu, O., Raphaël, P.: A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line. Annals of PDE. (to appear). arXiv:1611.08482
  18. 18.
    Hammack, J., Henderson, D., Guyenne, P., Yi, M.: Solitary-wave collisions. In: Proceedings of the 23rd ASME Offshore Mechanics and Artic Engineering (A Symposium to Honor Theodore Yao-Tsu Wu), Vancouver, Canada, June 2004. Word Scientific, Singapore (2004)Google Scholar
  19. 19.
    Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)CrossRefGoogle Scholar
  20. 20.
    Jendrej, J.: Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5. J. Funct. Anal. 272(3), 866–917 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jendrej, J.: Construction of two-bubble solutions for energy-critical wave equations. Am. J. Math. (to appear). arXiv:1602.06524
  22. 22.
    Jendrej, J., Lawrie, A.: Two-bubble dynamics for threshold solutions to the wave maps equation. Invent. Math. 213(3), 1249–1325 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kenig, C.E., Lawrie, A., Liu, B., Schlag, W.: Channels of energy for the linear radial wave equation. Adv. Math. 285, 877–936 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kenig, C.E., Lawrie, A., Schlag, W.: Relaxation of wave maps exterior to a ball to harmonic maps for all data. Geom. Funct. Anal. 24, 610–647 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201, 147–212 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Krieger, J., Nakanishi, K., Schlag, W.: Threshold phenomenon for the quintic wave equation in three dimensions. Commun. Math. Phys. 327(1), 309–332 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics away from the ground state for the energy-critical nonlinear wave equation. Am. J. Math. 135(4), 935–965 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Krieger, J., Schlag, W., Tataru, D.: Slow blow-up solutions for the \(H^1({\mathbb{R}}^3)\) critical focusing semilinear wave equation. Duke Math. J. 147(1), 1–53 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Li, Yi, Sattinger, D.H.: Soliton collisions in the ion acoustic plasma equations. J. Math. Fluid Mech. 1, 117–130 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Martel, Y.: Asymptotic \(N\)-soliton-like solutions of the subcritical and critical generalized Korteweg–de Vries equations. Am. J. Math. 127, 1103–1140 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Martel, Y., Merle, F.: Multi-solitary waves for nonlinear Schrödinger equations. Ann. lnst. H. Poincaré Non-linear Anal. 23, 849–864 (2006)CrossRefGoogle Scholar
  33. 33.
    Martel, Y., Merle, F.: Description of two soliton collision for the quartic gKdV equation. Ann. Math. (2) 174(2), 757–857 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Martel, Y., Merle, F.: Inelastic interaction of nearly equal solitons for the quartic gKdV equation. Invent. Math. 183(3), 563–648 (2011)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Martel, Y., Merle, F.: On the nonexistence of pure multi-solitons for the quartic gKdV equation. Int. Math. Res. Notices 3, 688–739 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Martel, Y., Merle, F.: Construction of multi-solitons for the energy-critical wave equation in dimension 5. Arch. Ration. Mech. Anal. 222(3), 1113–1160 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Martel, Y., Merle, F., Tsai, T.-P.: Stability and asymptotic stability in the energy space of the sum of \(N\) solitons for subcritical gKdV equations. Commun. Math. Phys. 231, 347–373 (2002)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Martel, Y., Raphaël, P.: Strongly interacting blow up bubbles for the mass critical NLS. Ann. Sci. Éc. Norm. Supér. (to appear). arXiv:1512.00900
  39. 39.
    Merle, F.: Construction of solutions with exactly \(k\) blow-up points for the Schrödinger equation with critical nonlinearity. Commun. Math. Phys. 129(2), 223–240 (1990)CrossRefGoogle Scholar
  40. 40.
    Merle, F., Raphaël, P.: On universality of blow-up profile for \(L^2\) critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565–672 (2004)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Miura, R.M.: The Korteweg–de Vries equation, a survey of results. SIAM Rev. 18, 412–459 (1976)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Mizumachi, T.: Weak interaction between solitary waves of the generalized KdV equations. SIAM J. Math. Anal. 35(4), 1042–1080 (2003)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Muñoz, C.: On the inelastic two-soliton collision for gKdV equations with general nonlinearity. Int. Math. Res. Notices 9, 1624–1719 (2010)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Nakanishi, K., Schlag, W.: Invariant manifolds and dispersive Hamiltonian evolution equations. In: Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS). Zürich (2011)Google Scholar
  45. 45.
    Perelman, G.: Two soliton collision for nonlinear Schrödinger equations in dimension 1. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(3), 357–384 (2011)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Raphaël, P., Rodnianski, I.: Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems. Publ. Math. Inst. Hautes Études Sci. 115, 1–122 (2012)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Rey, O.: The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 130, 357–426 (1995)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Shih, L.Y.: Soliton-like interaction governed by the generalized Korteweg–de Vries equation. Wave Motion 2, 197–206 (1980)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Wadati, M., Toda, M.: The exact \(N\)-soliton solution of the Korteweg–de Vries equation. J. Phys. Soc. Jpn. 32, 1403–1411 (1972)CrossRefGoogle Scholar
  50. 50.
    Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure. Appl. Math. 39, 51–68 (1986)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Zabusky, N.J., Kruskal, M.D.: Interaction of “solitons” in a collisionless plasma and recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)CrossRefGoogle Scholar
  52. 52.
    Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)MathSciNetGoogle Scholar

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

Personalised recommendations