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Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit
1. | Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, 44322 Nantes Cedex 03, France, France |
2. | INRIA & ENS Cachan Bretagne, Avenue Robert Schuman, 35170 Bruz, France |
References:
[1] |
D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), 253-285.
doi: 10.1007/s00220-002-0774-4. |
[2] |
D. Bambusi, J.-M. Delort, B. Grébert and J. Szeftel, Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math., 60 (2007), 1665-1690.
doi: 10.1002/cpa.20181. |
[3] |
D. Bambusi and B. Grébert, Birkhoff normal form for PDEs with tame modulus, Duke Math. J., 135 (2006), 415-615.
doi: 10.1215/S0012-7094-06-13534-2. |
[4] |
M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Annales scientifiques de l'ENS, fascicule 2, 46 (2013), 299-373. |
[5] |
M. Berti and P. Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbbT^d$ and a multiplicative potential, J. European Math. Society, 15 (2013), 229-286.
doi: 10.4171/JEMS/361. |
[6] |
M. Berti and C. Carminati, Chaotic dynamics for perturbations of infinite dimensional Hamiltonian systems, Nonlinear Analysis, 48 (2002), 481-504.
doi: 10.1016/S0362-546X(00)00200-5. |
[7] |
J. Bourgain, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., 229 (2005), 62-94.
doi: 10.1016/j.jfa.2004.10.019. |
[8] |
J.-M. Delort, Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle, Trans. Amer. Math. Soc., 361 (2009), 4299-4365.
doi: 10.1090/S0002-9947-09-04747-3. |
[9] |
L. H. Eliasson et S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[10] |
B. Grébert, T. Jézéquel and L. Thomann, Stability of large periodic solutions of Klein-Gordon near a homoclinic orbit, arXiv:1312.1851. |
[11] |
B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427.
doi: 10.1007/s00220-011-1327-5. |
[12] |
M. Groves and G. Schneider, Modulating pulse solutions for a class of nonlinear wave equations, Comm. Math. Phys., 219 (2001), 489-522.
doi: 10.1007/s002200100423. |
[13] |
M. Groves and G. Schneider, Modulating pulse solutions for quasilinear wave equations, J. Differential Equations, 219 (2005), 221-258.
doi: 10.1016/j.jde.2005.01.014. |
[14] |
M. Groves and G. Schneider, Modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales, Comm. Math. Phys., 278 (2008), 567-625.
doi: 10.1007/s00220-007-0400-6. |
[15] |
B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, 139, Cambridge University Press, 2013. |
[16] |
P.-F. Hsieh and Y. Sibuya, Basic Theory of Ordinary Differential Equations, Universitext. Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1506-6. |
[17] |
G. Iooss and M.-C. Pérouème, Perturbed homoclinic solutions in 1:1 resonance vector fields, J. Differential Equations, 102 (1993), 62-88.
doi: 10.1006/jdeq.1993.1022. |
[18] |
T. Jézéquel, P. Bernard and E. Lombardi, Homoclinic orbits with many loops near a $0^2i\omega$ resonant fixed point of Hamiltonian systems, To appear. |
[19] |
R. Joly and G. Raugel, A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations, Confluentes Mathematici, 3 (2011), 471-493.
doi: 10.1142/S1793744211000369. |
[20] |
E. Lombardi, Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rationnal Mech. Anal., 137 (1997), 227-304.
doi: 10.1007/s002050050029. |
[21] |
E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics , 1741, 2000. Springer.
doi: 10.1007/BFb0104102. |
[22] |
A. Mielke, Hamiltonian and Lagrangian Flows on Centre Manifolds, Lecture Notes in Mathematics. 1489, Springer, 1991. |
[23] |
A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Differential Equations, 65 (1986), 68-88.
doi: 10.1016/0022-0396(86)90042-2. |
[24] |
K. Nakanishi and W. Schlag, Invariant manifolds and dispersive hamiltonian evolution equations, Zürich Lectures in Advanced Mathematics, (2010) EMS.
doi: 10.4171/095. |
[25] |
O. Perron, Über ein vermeintliches Stabilitätskriterium, Gött. Nachr., (1930), 128-129. |
[26] |
J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[27] |
J. Shatah and C. Zeng, Orbits homoclinic to centre manifolds of conservative PDEs, Nonlinearity, 16 (2003), 591-614.
doi: 10.1088/0951-7715/16/2/314. |
[28] |
C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
show all references
References:
[1] |
D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys., 234 (2003), 253-285.
doi: 10.1007/s00220-002-0774-4. |
[2] |
D. Bambusi, J.-M. Delort, B. Grébert and J. Szeftel, Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math., 60 (2007), 1665-1690.
doi: 10.1002/cpa.20181. |
[3] |
D. Bambusi and B. Grébert, Birkhoff normal form for PDEs with tame modulus, Duke Math. J., 135 (2006), 415-615.
doi: 10.1215/S0012-7094-06-13534-2. |
[4] |
M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Annales scientifiques de l'ENS, fascicule 2, 46 (2013), 299-373. |
[5] |
M. Berti and P. Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbbT^d$ and a multiplicative potential, J. European Math. Society, 15 (2013), 229-286.
doi: 10.4171/JEMS/361. |
[6] |
M. Berti and C. Carminati, Chaotic dynamics for perturbations of infinite dimensional Hamiltonian systems, Nonlinear Analysis, 48 (2002), 481-504.
doi: 10.1016/S0362-546X(00)00200-5. |
[7] |
J. Bourgain, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., 229 (2005), 62-94.
doi: 10.1016/j.jfa.2004.10.019. |
[8] |
J.-M. Delort, Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle, Trans. Amer. Math. Soc., 361 (2009), 4299-4365.
doi: 10.1090/S0002-9947-09-04747-3. |
[9] |
L. H. Eliasson et S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.
doi: 10.4007/annals.2010.172.371. |
[10] |
B. Grébert, T. Jézéquel and L. Thomann, Stability of large periodic solutions of Klein-Gordon near a homoclinic orbit, arXiv:1312.1851. |
[11] |
B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys., 307 (2011), 383-427.
doi: 10.1007/s00220-011-1327-5. |
[12] |
M. Groves and G. Schneider, Modulating pulse solutions for a class of nonlinear wave equations, Comm. Math. Phys., 219 (2001), 489-522.
doi: 10.1007/s002200100423. |
[13] |
M. Groves and G. Schneider, Modulating pulse solutions for quasilinear wave equations, J. Differential Equations, 219 (2005), 221-258.
doi: 10.1016/j.jde.2005.01.014. |
[14] |
M. Groves and G. Schneider, Modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales, Comm. Math. Phys., 278 (2008), 567-625.
doi: 10.1007/s00220-007-0400-6. |
[15] |
B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, 139, Cambridge University Press, 2013. |
[16] |
P.-F. Hsieh and Y. Sibuya, Basic Theory of Ordinary Differential Equations, Universitext. Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1506-6. |
[17] |
G. Iooss and M.-C. Pérouème, Perturbed homoclinic solutions in 1:1 resonance vector fields, J. Differential Equations, 102 (1993), 62-88.
doi: 10.1006/jdeq.1993.1022. |
[18] |
T. Jézéquel, P. Bernard and E. Lombardi, Homoclinic orbits with many loops near a $0^2i\omega$ resonant fixed point of Hamiltonian systems, To appear. |
[19] |
R. Joly and G. Raugel, A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations, Confluentes Mathematici, 3 (2011), 471-493.
doi: 10.1142/S1793744211000369. |
[20] |
E. Lombardi, Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rationnal Mech. Anal., 137 (1997), 227-304.
doi: 10.1007/s002050050029. |
[21] |
E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics , 1741, 2000. Springer.
doi: 10.1007/BFb0104102. |
[22] |
A. Mielke, Hamiltonian and Lagrangian Flows on Centre Manifolds, Lecture Notes in Mathematics. 1489, Springer, 1991. |
[23] |
A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Differential Equations, 65 (1986), 68-88.
doi: 10.1016/0022-0396(86)90042-2. |
[24] |
K. Nakanishi and W. Schlag, Invariant manifolds and dispersive hamiltonian evolution equations, Zürich Lectures in Advanced Mathematics, (2010) EMS.
doi: 10.4171/095. |
[25] |
O. Perron, Über ein vermeintliches Stabilitätskriterium, Gött. Nachr., (1930), 128-129. |
[26] |
J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296.
doi: 10.1007/BF02566420. |
[27] |
J. Shatah and C. Zeng, Orbits homoclinic to centre manifolds of conservative PDEs, Nonlinearity, 16 (2003), 591-614.
doi: 10.1088/0951-7715/16/2/314. |
[28] |
C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.
doi: 10.1007/BF02104499. |
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