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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

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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Grébert, T. Jézéquel and L. Thomann, Stability of large periodic solutions of Klein-Gordon near a homoclinic orbit,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, 139, Cambridge University Press, 2013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics , 1741, 2000. Springer. doi: 10.1007/BFb0104102.  Google Scholar

[22]

A. Mielke, Hamiltonian and Lagrangian Flows on Centre Manifolds, Lecture Notes in Mathematics. 1489, Springer, 1991.  Google Scholar

[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.  Google Scholar

[24]

K. Nakanishi and W. Schlag, Invariant manifolds and dispersive hamiltonian evolution equations, Zürich Lectures in Advanced Mathematics, (2010) EMS. doi: 10.4171/095.  Google Scholar

[25]

O. Perron, Über ein vermeintliches Stabilitätskriterium, Gött. Nachr., (1930), 128-129. Google Scholar

[26]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Grébert, T. Jézéquel and L. Thomann, Stability of large periodic solutions of Klein-Gordon near a homoclinic orbit,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

B. Helffer, Spectral Theory and Its Applications, Cambridge Studies in Advanced Mathematics, 139, Cambridge University Press, 2013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics , 1741, 2000. Springer. doi: 10.1007/BFb0104102.  Google Scholar

[22]

A. Mielke, Hamiltonian and Lagrangian Flows on Centre Manifolds, Lecture Notes in Mathematics. 1489, Springer, 1991.  Google Scholar

[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.  Google Scholar

[24]

K. Nakanishi and W. Schlag, Invariant manifolds and dispersive hamiltonian evolution equations, Zürich Lectures in Advanced Mathematics, (2010) EMS. doi: 10.4171/095.  Google Scholar

[25]

O. Perron, Über ein vermeintliches Stabilitätskriterium, Gött. Nachr., (1930), 128-129. Google Scholar

[26]

J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996), 269-296. doi: 10.1007/BF02566420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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