Abstract
In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to 0 in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations, such as the sine-Gordon equation and \(\phi ^4\) and \(\phi ^6\) models. It also partially answers a question of Soffer and Weinstein (Invent Math 136(1): 9–74, p 19 1999) about nonexistence of breathers for the cubic NLKG in dimension one.
Similar content being viewed by others
References
Alejo, M.A., Muñoz, C.: Nonlinear stability of mKdV breathers. Commun. Math. Phys. 324(1), 233–262 (2013)
Alejo, M.A., Muñoz, C.: Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers. Anal. PDE 8(3), 629–674 (2015)
Alejo, M.A., Muñoz, C., Palacios, J.M.: On the variational structure of breather solutions, preprint. arXiv:1309.0625
Bambusi, D., Cuccagna, S.: On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential. Am. J. Math. 133(5), 1421–1468 (2011)
Béthuel, F., Gravejat, P., Smets, D.: Asymptotic stability in the energy space for dark solitons of the Gross–Pitaevskii equation. Ann. Sci. Éc. Norm. Supér. 48(6),1327–1381 (2015)
Birnir, B., McKean, H., Weinstein, A.: The rigidity of sine-Gordon breathers. Commun. Pure Appl. Math. XLVII, 1043–1051 (1994)
Buslaev, V., Perelman, G.: Scattering for the nonlinear Schrödinger equations: states close to a soliton. St. Petersburgh Math. J. 4(6), 1111–1142 (1993)
Buslaev, V., Perelman, G.: On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear Evol. Equ. 75–98, Amer. Math. Soc. Transl. Ser. vol. 2, No. 164, American Mathematical Society, Providence (1995)
Buslaev, V., Sulem, C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(3), 419–475 (2003)
Coron, J.-M.: Période minimale pour une corde vibrante de longueur infinie. C.R. Acad. Sc. Paris Série 294, 127 (1982)
Cuccagna, S.: On asymptotic stability in 3D of kinks for the \(\phi ^4\) model. Trans. Am. Math. Soc. 360(5), 2581–2614 (2008)
Cuccagna, S.: The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states. Commun. Math. Phys. 305(2), 279–331 (2011)
Cuccagna, S.: On asymptotic stability of moving ground states of the nonlinear Schrödinger equation. Trans. Am. Math. Soc 366, 2827–2888 (2014)
Delort, J.-M.: Existence globale et comportement asymptotique pour l’équation de Klein–Gordon quasi linéaire à données petites en dimension 1. Ann. Sci. École Norm. Sup. 34(4), 1–61 (2001)
Delort, J.-M.: Semiclassical microlocal normal forms and global solutions of modified one-dimensional KG equations. Annales de l’Institut Fourier 66, 1451–1528 (2016)
Denzler, J.: Nonpersistence of breather families for the perturbed sine Gordon equation. Commun. Math. Phys. 158, 397–430 (1993)
Gol’dman, I.I., Krivchenkov, V.D., Geĭlikman, B.T., Marquit, E., Lepa, E.: Problems in Quantum Mechanics, Authorised revised ed. Edited by B. T. Geilikman; translated from the Russian by E. Marquit and E. Lepa. Pergamon Press, London (1961)
Henry, D.B.J., Perez, F., Wreszinski, W.F.: Stability theory for solitary-wave solutions of scalar field equations. Commun. Math. Phys. 85(3), 351–361 (1982)
Kichenassamy, S.: Breather solutions of the nonlinear wave equation. Commun. Pure Appl. Math. XLIV, 789–818 (1991)
Kopylova, E., Komech, A.I.: On asymptotic stability of kink for relativistic Ginzburg–Landau equations. Arch. Ration. Mech. Anal. 202(1), 213–245 (2011)
Kowalczyk, M., Martel, Y., Muñoz, C.: Kink dynamics in the \(\phi ^4\) model: asymptotic stability for odd perturbations in the energy space. J. Am. Math. Soc. doi:10.1090/jams/870
Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Am. Math. Soc. 19(4), 815–920 (2006)
Kruskal, M.D., Segur, Harvey: Nonexistence of small-amplitude breather solutions in \(\phi ^4\) theory. Phys. Rev. Lett. 58(8), 747–750 (1987)
Lamb, G.L.: Elements of Soliton Theory, Pure and Applied Mathematics. Wiley, New York (1980)
Lindblad, H., Soffer, A.: A Remark on long range scattering for the nonlinear Klein–Gordon equation. J. Hyperb. Differ. Equ. 2(1), 77–89 (2005)
Lindblad, H., Soffer, A.: A remark on asymptotic completeness for the critical nonlinear Klein–Gordon equation. Lett. Math. Phys. 73(3), 249–258 (2005)
Lindblad, H., Soffer, A.: Scattering for the Klein–Gordon equation with quadratic and variable coefficient cubic nonlinearities. Trans. Amer. Math. Soc. 367(12), 8861–8909 (2015). arXiv:1307.5882
Lindbland, H., Tao, T.: Asymptotic decay for a one-dimensional nonlinear wave equation. Anal. PDE 5(2), 411–422 (2012). doi:10.2140/apde.2012.5.411
Lohe, M.A.: Soliton structures in \(P(\phi )_2\). Phys. Rev. D. 20(12), 3120–3130 (1979)
Martel, Y., Merle, F.: A Liouville theorem for the critical generalized Korteweg–de Vries equation. J. Math. Pures Appl. (9) 79(4), 339–425 (2000)
Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157(3), 219–254 (2001)
Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical gKdV equations revisited. Nonlinearity 18(1), 55–80 (2005)
Merle, F., Raphael, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. (2) 161(1), 157–222 (2005)
Peskin, M.E., Schroeder, D.V.: An introduction to quantum field theory. In: Advanced Book Program. Addison-Wesley Publishing Company, Reading (1995) (Edited and with a foreword by David Pines)
Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS, preprint. arXiv:math/0309114
Segur, H.: Wobbling kinks in \(\varphi ^{4}\) and sine-Gordon theory. J. Math. Phys. 24(6), 1439–1443 (1983)
Soffer, A., Weinstein, M.I.: Time dependent resonance theory. Geom. Funct. Anal. 8(6), 1086–1128 (1998)
Soffer, A., Weinstein, M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136(1), 9–74 (1999)
Sterbenz, J.: Dispersive decay for the 1D Klein–Gordon equation with variable coefficient nonlinearities. Trans. Amer. Math. Soc. 368(3), 2081–2113 (2016). arXiv:1307.4808
Titchmarsh, E.C.: Eigenfunction Expansions Associated With Second-order Differential Equations. Oxford University Press, Oxford (1946)
Vachaspati, T.: Kinks and Domain Walls: An Introduction to Classical and Quantum Solitons. Cambridge University Press, New York (2006)
Tsai, T.-P., Yau, H.T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Commun. Pure Appl. Math. 55(2), 153–216 (2002)
Vuillermot, P.-A.: Nonexistence of spatially localized free vibrations for a class of nonlinear wave equations. Comment. Math. Helv. 64, 573–586 (1987)
Witten, E.: From superconductors and four-manifolds to weak interactions. Bull. Am. Math. Soc. (N.S.) 44(3), 361–391 (2007) (electronic)
Acknowledgements
M. Kowalczyk was partially supported by Chilean research Grants FONDECYT 1130126, Fondo Basal CMM-Chile. C. Muñoz was partly funded by Chilean research Grants FONDECYT 1150202, Fondo Basal CMM-Chile, and Millennium Nucleus Center for Analysis of PDE NC130017 Part of this work was done, while the authors were hosted at the IHES during the program Nonlinear Waves 2016. Their stay was partly funded by the Grant ERC 291214 BLOWDISOL.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kowalczyk, M., Martel, Y. & Muñoz, C. Nonexistence of small, odd breathers for a class of nonlinear wave equations. Lett Math Phys 107, 921–931 (2017). https://doi.org/10.1007/s11005-016-0930-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-016-0930-y