American Institute of Mathematical Sciences

Advanced
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 - A, 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. 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. 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. 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, 46 (2013), 299. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00220-007-0400-6. Google Scholar

[15]

B. Helffer, Spectral Theory and Its Applications,, Cambridge Studies in Advanced Mathematics, (2013). Google Scholar

[16]

P.-F. Hsieh and Y. Sibuya, Basic Theory of Ordinary Differential Equations,, Universitext. Springer-Verlag, (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. 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. 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. doi: 10.1007/s002050050029. Google Scholar

[21]

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

[22]

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

[23]

A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces,, J. Differential Equations, 65 (1986), 68. 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). doi: 10.4171/095. Google Scholar

[25]

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

[26]

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

[27]

J. Shatah and C. Zeng, Orbits homoclinic to centre manifolds of conservative PDEs,, Nonlinearity, 16 (2003), 591. 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. 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. 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. 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. 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, 46 (2013), 299. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00220-007-0400-6. Google Scholar

[15]

B. Helffer, Spectral Theory and Its Applications,, Cambridge Studies in Advanced Mathematics, (2013). Google Scholar

[16]

P.-F. Hsieh and Y. Sibuya, Basic Theory of Ordinary Differential Equations,, Universitext. Springer-Verlag, (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. 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. 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. doi: 10.1007/s002050050029. Google Scholar

[21]

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

[22]

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

[23]

A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces,, J. Differential Equations, 65 (1986), 68. 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). doi: 10.4171/095. Google Scholar

[25]

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

[26]

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

[27]

J. Shatah and C. Zeng, Orbits homoclinic to centre manifolds of conservative PDEs,, Nonlinearity, 16 (2003), 591. 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. doi: 10.1007/BF02104499. Google Scholar

[1]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[2]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679
[3]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076
[4]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359
[5]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233
[6]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[7]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
[8]

Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085
[9]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251
[10]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215
[11]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235
[12]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071
[13]

Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315
[14]

Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389
[15]

Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889
[16]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
[17]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573
[18]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925
[19]

Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711
[20]

Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences