Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit

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September 2014, 34(9): 3485-3510. doi: 10.3934/dcds.2014.34.3485

Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit

Benoît Grébert ^1, , Tiphaine Jézéquel ^2, and Laurent Thomann ^1,

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

Received March 2013 Revised December 2013 Published March 2014

Abstract
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We consider the Klein-Gordon equation (KG) on a Riemannian surface $M$ $$ \partial^{2}_t u-\Delta u-m^{2}u+u^{2p+1} =0,\quad p\in \mathbb{N}^{*},\quad (t,x)\in \mathbb{R}\times M,$$ which is globally well-posed in the energy space. This equation has a homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we get the existence of a large family of heteroclinic connections to the center manifold that are close to the homoclinic orbit during all times. We point out that the solutions we construct are not small.

Keywords: wave equation, Klein-Gordon equation, center manifold., homoclinic orbit.

Mathematics Subject Classification: 37K45, 35Q55, 35Bx.

Citation: Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485

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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