Multi-travelling waves for the nonlinear Klein-Gordon equation
Authors:
Raphaël Côte and Yvan Martel
Journal:
Trans. Amer. Math. Soc. 370 (2018), 7461-7487
MSC (2010):
Primary 35Q51; Secondary 35L71, 35Q40
DOI:
https://doi.org/10.1090/tran/7303
Published electronically:
June 20, 2018
MathSciNet review:
3841855
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For the nonlinear Klein-Gordon equation in $\mathbb {R}^{1+d}$, we prove the existence of multi-solitary waves made of any number $N$ of decoupled bound states. This extends the work of Côte and Muñoz (Forum Math. Sigma 2 (2014)) which was restricted to ground states, as were most previous similar results for other nonlinear dispersive and wave models.
- Weiwei Ao, Monica Musso, Frank Pacard, and Juncheng Wei, Solutions without any symmetry for semilinear elliptic problems, J. Funct. Anal. 270 (2016), no. 3, 884–956. MR 3438325, DOI https://doi.org/10.1016/j.jfa.2015.10.015
- H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI https://doi.org/10.1007/BF00250555
- H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), no. 4, 347–375. MR 695536, DOI https://doi.org/10.1007/BF00250556
- J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, Providence, RI, 1999. MR 1691575
- Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047
- Vianney Combet, Multi-soliton solutions for the supercritical gKdV equations, Comm. Partial Differential Equations 36 (2011), no. 3, 380–419. MR 2763331, DOI https://doi.org/10.1080/03605302.2010.503770
- Vianney Combet, Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension, Discrete Contin. Dyn. Syst. 34 (2014), no. 5, 1961–1993. MR 3124722, DOI https://doi.org/10.3934/dcds.2014.34.1961
- Raphaël Côte and Stefan Le Coz, High-speed excited multi-solitons in nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 96 (2011), no. 2, 135–166 (English, with English and French summaries). MR 2818710, DOI https://doi.org/10.1016/j.matpur.2011.03.004
- Raphaël Côte, Yvan Martel, and Frank Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam. 27 (2011), no. 1, 273–302. MR 2815738, DOI https://doi.org/10.4171/RMI/636
- Raphaël Côte and Claudio Muñoz, Multi-solitons for nonlinear Klein-Gordon equations, Forum Math. Sigma 2 (2014), Paper No. e15, 38. MR 3264254, DOI https://doi.org/10.1017/fms.2014.13
- Raphaël Côte and Hatem Zaag, Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math. 66 (2013), no. 10, 1541–1581. MR 3084698, DOI https://doi.org/10.1002/cpa.21452
- Wei Yue Ding, On a conformally invariant elliptic equation on ${\bf R}^n$, Comm. Math. Phys. 107 (1986), no. 2, 331–335. MR 863646
- Manuel del Pino, Monica Musso, Frank Pacard, and Angela Pistoia, Large energy entire solutions for the Yamabe equation, J. Differential Equations 251 (2011), no. 9, 2568–2597. MR 2825341, DOI https://doi.org/10.1016/j.jde.2011.03.008
- Manuel del Pino, Monica Musso, Frank Pacard, and Angela Pistoia, Torus action on $S^n$ and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 1, 209–237. MR 3088442
- G. H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Mathematical Phys. 5 (1964), 1252–1254. MR 174304, DOI https://doi.org/10.1063/1.1704233
- T. Duyckaerts, H. Jia, C. E. Kenig, and F. Merle, Soliton resolution along a sequence of times for the focusing energy critical wave equation, preprint arXiv:1601.01871.
- Thomas Duyckaerts, Carlos Kenig, and Frank Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math. 1 (2013), no. 1, 75–144. MR 3272053, DOI https://doi.org/10.4310/CJM.2013.v1.n1.a3
- Thomas Duyckaerts and Frank Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 (2009), no. 6, 1787–1840. MR 2491692, DOI https://doi.org/10.1007/s00039-009-0707-x
- Thomas Duyckaerts and Frank Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP , posted on (2008), Art ID rpn002, 67. MR 2470571, DOI https://doi.org/10.1093/imrp/rpn002
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR 0473443
- J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z. 189 (1985), no. 4, 487–505. MR 786279, DOI https://doi.org/10.1007/BF01168155
- Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. MR 901236, DOI https://doi.org/10.1016/0022-1236%2887%2990044-9
- Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), no. 2, 308–348. MR 1081647, DOI https://doi.org/10.1016/0022-1236%2890%2990016-E
- Emmanuel Hebey and Michel Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth, J. Funct. Anal. 119 (1994), no. 2, 298–318. MR 1261094, DOI https://doi.org/10.1006/jfan.1994.1012
- Joachim Krieger, Yvan Martel, and Pierre Raphaël, Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math. 62 (2009), no. 11, 1501–1550. MR 2560043, DOI https://doi.org/10.1002/cpa.20292
- J. Krieger, K. Nakanishi, and W. Schlag, Global dynamics above the ground state energy for the one-dimensional NLKG equation, Math. Z. 272 (2012), no. 1-2, 297–316. MR 2968226, DOI https://doi.org/10.1007/s00209-011-0934-3
- Man Kam Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in ${\bf R}^n$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. MR 969899, DOI https://doi.org/10.1007/BF00251502
- Mihai Mariş, Existence of nonstationary bubbles in higher dimensions, J. Math. Pures Appl. (9) 81 (2002), no. 12, 1207–1239 (English, with English and French summaries). MR 1952162, DOI https://doi.org/10.1016/S0021-7824%2802%2901274-6
- Yvan Martel, Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math. 127 (2005), no. 5, 1103–1140. MR 2170139
- Kevin McLeod, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in ${\bf R}^n$. II, Trans. Amer. Math. Soc. 339 (1993), no. 2, 495–505. MR 1201323, DOI https://doi.org/10.1090/S0002-9947-1993-1201323-X
- Frank Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys. 129 (1990), no. 2, 223–240. MR 1048692
- Yvan Martel and Frank Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 6, 849–864 (English, with English and French summaries). MR 2271697, DOI https://doi.org/10.1016/j.anihpc.2006.01.001
- Yvan Martel and Frank Merle, Stability of two soliton collision for nonintegrable gKdV equations, Comm. Math. Phys. 286 (2009), no. 1, 39–79. MR 2470923, DOI https://doi.org/10.1007/s00220-008-0685-0
- Yvan Martel and Frank Merle, Construction of multi-solitons for the energy-critical wave equation in dimension 5, Arch. Ration. Mech. Anal. 222 (2016), no. 3, 1113–1160. MR 3544324, DOI https://doi.org/10.1007/s00205-016-1018-7
- Yvan Martel, Frank Merle, and Tai-Peng Tsai, Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations, Duke Math. J. 133 (2006), no. 3, 405–466. MR 2228459, DOI https://doi.org/10.1215/S0012-7094-06-13331-8
- Mei Ming, Frederic Rousset, and Nikolay Tzvetkov, Multi-solitons and related solutions for the water-waves system, SIAM J. Math. Anal. 47 (2015), no. 1, 897–954. MR 3315224, DOI https://doi.org/10.1137/140960220
- Frank Merle and Hatem Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math. 134 (2012), no. 3, 581–648. MR 2931219, DOI https://doi.org/10.1353/ajm.2012.0021
- Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 404890, DOI https://doi.org/10.1137/1018076
- Makoto Nakamura and Tohru Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces, Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, 255–293. MR 1855424
- Kenji Nakanishi and Wilhelm Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2011. MR 2847755
- K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differential Equations 250 (2011), no. 5, 2299–2333. MR 2756065, DOI https://doi.org/10.1016/j.jde.2010.10.027
- Robert L. Pego and Michael I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A 340 (1992), no. 1656, 47–94. MR 1177566, DOI https://doi.org/10.1098/rsta.1992.0055
- Peter Cornelis Schuur, Asymptotic analysis of soliton problems, Lecture Notes in Mathematics, vol. 1232, Springer-Verlag, Berlin, 1986. An inverse scattering approach. MR 874343
- James Serrin and Moxun Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 (2000), no. 3, 897–923. MR 1803216, DOI https://doi.org/10.1512/iumj.2000.49.1893
- Jalal Shatah and Walter Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), no. 2, 173–190. MR 804458
- Terence Tao, Low regularity semi-linear wave equations, Comm. Partial Differential Equations 24 (1999), no. 3-4, 599–629. MR 1683051, DOI https://doi.org/10.1080/03605309908821435
- Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. MR 783974, DOI https://doi.org/10.1137/0516034
- Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 51–67. MR 820338, DOI https://doi.org/10.1002/cpa.3160390103
- N.J. Zabusky and M.D. Kruskal, Interaction of “solitons” in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240–243.
- V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118–134 (Russian, with English summary); English transl., Soviet Physics JETP 34 (1972), no. 1, 62–69. MR 0406174
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35Q51, 35L71, 35Q40
Retrieve articles in all journals with MSC (2010): 35Q51, 35L71, 35Q40
Additional Information
Raphaël Côte
Affiliation:
Université de Strasbourg, CNRS, IRMA UMR 7501, F-67000 Strasbourg, France
Email:
cote@math.unistra.fr
Yvan Martel
Affiliation:
CMLS, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
MR Author ID:
367956
Email:
yvan.martel@polytechnique.edu
Keywords:
Klein-Gordon equation,
multi-soliton,
ground states,
excited states,
instability
Received by editor(s):
December 6, 2016
Received by editor(s) in revised form:
May 11, 2017
Published electronically:
June 20, 2018
Additional Notes:
The authors were supported in part by the ERC advanced grant 291214 BLOWDISOL. The first author was also supported in part by the ANR contract MAToS ANR-14-CE25-0009-01.
Article copyright:
© Copyright 2018
American Mathematical Society