Multi-travelling waves for the nonlinear Klein-Gordon equation

Côte, Raphaël; Martel, Yvan

doi:10.1090/tran/7303

Multi-travelling waves for the nonlinear Klein-Gordon equation
HTML articles powered by AMS MathViewer

by Raphaël Côte and Yvan Martel PDF

Trans. Amer. Math. Soc. 370 (2018), 7461-7487 Request permission

Corrigendum: Trans. Amer. Math. Soc. 375 (2022), 3755-3757.

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.

References

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 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 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 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, DOI 10.1090/coll/046
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, DOI 10.1090/cln/010
Vianney Combet, Multi-soliton solutions for the supercritical gKdV equations, Comm. Partial Differential Equations 36 (2011), no. 3, 380–419. MR 2763331, DOI 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 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 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 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 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 10.1002/cpa.21452
Wei Yue Ding, On a conformally invariant elliptic equation on $\textbf {R}^n$, Comm. Math. Phys. 107 (1986), no. 2, 331–335. MR 863646, DOI 10.1007/BF01209398
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 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 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 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 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 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, DOI 10.1007/BF01221125
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
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 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 10.1016/0022-1236(87)90044-9
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 10.1016/0022-1236(90)90016-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 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 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 10.1007/s00209-011-0934-3
Man Kam Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. MR 969899, DOI 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 10.1016/S0021-7824(02)01274-6
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, DOI 10.1353/ajm.2005.0033
Kevin McLeod, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\textbf {R}^n$. II, Trans. Amer. Math. Soc. 339 (1993), no. 2, 495–505. MR 1201323, DOI 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, DOI 10.1007/BF02096981
Yvan Martel and Frank Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 23 (2006), no. 6, 849–864 (English, with English and French summaries). MR 2271697, DOI 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 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 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 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 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 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 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, DOI 10.2977/prims/1145477225
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, DOI 10.4171/095
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 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 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, DOI 10.1007/BFb0073054
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 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, DOI 10.1007/BF01212446
Terence Tao, Low regularity semi-linear wave equations, Comm. Partial Differential Equations 24 (1999), no. 3-4, 599–629. MR 1683051, DOI 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 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 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

AMS Website Down

Transactions of the American Mathematical Society

Multi-travelling waves for the nonlinear Klein-Gordon equation
HTML articles powered by AMS MathViewer

Abstract: