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Description and Classification of 2-Solitary Waves for Nonlinear Damped Klein–Gordon Equations

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Abstract

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein–Gordon equation

$$\begin{aligned} \partial _{tt}u+2\alpha \partial _{t}u-\Delta u+u-|u|^{p-1}u=0 \end{aligned}$$

on \(\mathbb {R}^N\), for \(1\leqslant N\leqslant 5\) and energy subcritical exponents \(p>2\). The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solitary waves in the case of opposite signs. Close to the sum of two remote solitary waves, it turns out that only the components of the initial data in the unstable direction of each ground state are relevant in the large time asymptotic behavior of the solution. In particular, we show that 2-solitary waves have a universal behavior: the distance between the solitary waves is asymptotic to \(\log t\) as \(t\rightarrow \infty \). This behavior is due to damping of the initial data combined with strong interactions between the solitary waves.

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References

  1. Bates, P.W., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. Dynamics reported, Vol. 2, 1–38, Dyn. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester (1989)

  2. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313–345 (1983)

  3. Burq, N., Raugel, G., Schlag, W.: Long time dynamics for damped Klein-Gordon equations. Ann. Sci. Éc. Norm. Supér. (4) 50(6), 1447–1498 (2017)

  4. Combet, V.: Multi-soliton solutions for the supercritical gKdV equations. Commun. Partial Differ. Equ. 36(3), 380–419 (2011)

    Article  MathSciNet  Google Scholar 

  5. Combet, V., Martel, Y.: Construction of multi-bubble solutions for the critical gKdV equation. SIAM J. Math. Anal. 50(4), 3715–3790 (2018)

    Article  MathSciNet  Google Scholar 

  6. Côte, R.: On the soliton resolution for equivariant wave maps to the sphere. Commun. Pure Appl. Math. 68, 1946–2004 (2015)

    Article  MathSciNet  Google Scholar 

  7. Côte, R., Martel, Y., Merle, F.: Construction of multi-soliton solutions for the \(L^2\)-supercritical gKdV and NLS equations. Rev. Mat. Iberoamericana 27, 273–302 (2011)

    Article  MathSciNet  Google Scholar 

  8. Côte, R., Martel, Y., Yuan, X.: Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation. Arch. Rational Mech. Anal. 239, 1837–1874 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  9. Côte, R., Muñoz, C.: Multi-solitons for nonlinear Klein–Gordon equations. Forum of Mathematics, Sigma 2 (2014)

  10. Côte, R., Yuan, X.: Asymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation. Preprint arXiv:2102.11178

  11. Duyckaerts, T., Kenig, C.E., Merle, F.: Classification of radial solutions of the focusing, energy-critical wave equation. Cambridge J. Math. 1, 75–144 (2013)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. Feireisl, E.: Convergence to an equilibrium of semilinear wave equations on unbounded intervals. Dyn. Syst. Appl. 3, 423–434 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Feireisl, E.: Finite energy travelling waves for nonlinear damped wave equations. Quart. Appl. Math. 56(1), 55–70 (1998)

    Article  MathSciNet  Google Scholar 

  15. Jendoubi, M.A.: A remark on the convergence of global and bounded solutions for a semilinear wave equation on \(\mathbb{R}^N\). J. Dyn. Differ. Equ. 14(3), 589–596 (2002)

    Article  Google Scholar 

  16. Jendrej, J.: Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XVIII (2018), 1–44

  17. Jendrej, J.: Construction of two-bubble solutions for energy-critical wave equations. Amer. J. Math. 141(1), 55–118 (2019)

    Article  MathSciNet  Google Scholar 

  18. Jendrej, J., Lawrie, A.: Two-bubble dynamics for threshold solutions to the wave maps equation. Invent. Math. 213(3), 1249–1325 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  19. Jendrej, J.: Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations. Preprint arXiv:1802.06294

  20. Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u-u+u^p=0\) in \(\mathbb{R}^n\). Arch. Rational Mech. Anal. 105(3), 243–266 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  21. Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Am. Math. Soc. 19, 815–920 (2006)

    Article  Google Scholar 

  22. Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics above the ground state energy for the one-dimensional NLKG equation. Math. Z. 272(1–2), 297–316 (2012)

    Article  MathSciNet  Google Scholar 

  23. Li, Z., Zhao, L.: Asymptotic decomposition for nonlinear damped Klein-Gordon equations. J. Math. Study 53(3), 329–352 (2020)

    Article  MathSciNet  Google Scholar 

  24. 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)

    Article  MathSciNet  Google Scholar 

  25. Martel, Y., Merle, F., Nakanishi, K., Raphaël, P.: Codimension one threshold manifold for the critical gKdV equation. Commun. Math. Phys. 342, 1075–1106 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  26. Martel, Y., Merle, F., Tsai, T.-P.: Stability and asymptotic stability for subcritical gKdV equations. Commun. Math. Phys. 231, 347–373 (2002)

    Article  ADS  Google Scholar 

  27. Martel, Y., Nguyen, T.V.: Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system. Discrete Contin. Dyn. Syst. 40, 1595–1620 (2020)

    Article  MathSciNet  Google Scholar 

  28. Martel, Y., Raphaël, P.: Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation. Ann. Sci. Éc. Norm. Supér. 51(3), 701–737 (2018)

    Article  MathSciNet  Google Scholar 

  29. Miura, R.M.: The Korteweg-de Vries equation, a survey of results. SIAM Rev. 18, 412–459 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  30. Nakanishi, K., Schlag, W.: Invariant manifolds and dispersive Hamiltonian evolution equations. Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2011)

  31. Nakanishi, K., Schlag, W.: Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation. SIAM J. Math. Anal. 44, 1175–1210 (2012)

    Article  MathSciNet  Google Scholar 

  32. Nguyen, T.V.: Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation. Nonlinearity 30(12), 4614–4648 (2017)

    Article  MathSciNet  Google Scholar 

  33. Nguyen, T.V.: Existence of multi-solitary waves with logarithmic relative distances for the NLS equation. C. R. Math. Acad. Sci. Paris 357(1), 13–58 (2019)

    Article  MathSciNet  Google Scholar 

  34. Zakharov, T., 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)

    ADS  MathSciNet  Google Scholar 

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Acknowledgements

We thank Kenji Nakanishi for pointing out to us an error in an early version of this article, regarding the admissible range of exponents p in Theorem 1.5. We are also grateful to the anonymous referees for their useful comments and suggestions.

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Author notes

  1. Xu Yuan

    Present address: Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

Authors and Affiliations

  1. IRMA UMR 7501, Université de Strasbourg, CNRS, 67000, Strasbourg, France

    Raphaël Côte

  2. CMLS, École polytechnique, CNRS, Institut Polytechnique de Paris, 91128, Palaiseau Cedex, France

    Yvan Martel & Xu Yuan

  3. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

    Lifeng Zhao

Authors
  1. Raphaël Côte
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  2. Yvan Martel
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Correspondence to Yvan Martel.

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Communicated by K. Nakanishi.

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R. C. was partially supported by the French ANR contract MAToS ANR-14-CE25-0009-01. Y. M. and X. Y. thank IRMA, Université de Strasbourg, for its hospitality. L. Z. thanks CMLS, École Polytechnique, for its hospitality. L. Z. was partially supported by the NSFC Grant of China (11771415).

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Côte, R., Martel, Y., Yuan, X. et al. Description and Classification of 2-Solitary Waves for Nonlinear Damped Klein–Gordon Equations. Commun. Math. Phys. 388, 1557–1601 (2021). https://doi.org/10.1007/s00220-021-04241-5

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  • DOI: https://doi.org/10.1007/s00220-021-04241-5

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