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A shift map with a discontinuous entropy function
Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states
1. | Department of Mathematics, National University of Singapore, Singapore 119076, Singapore |
2. | Department of Mathematics, Rutgers University, Piscataway, NJ, USA 08854, USA |
In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from instability of a bound state under a nonlinear Fermi's golden rule. In [
References:
[1] |
D. Bambusi and S. Cuccagna,
On dispersion of small energy solutions to the nonlinear Klein-Gordon equation with a potential, Amer. J. Math., 133 (2011), 1421-1468.
doi: 10.1353/ajm.2011.0034. |
[2] |
V.-S. Buslaev and G.-S. Perelman,
On the stability of solitary waves for nonlinear Schrödinger equations, Amer. Math. Soc. Transl. Ser., 164 (1995), 75-98.
doi: 10.1090/trans2/164/04. |
[3] |
V.-S. Buslaev and C. Sulem,
On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 419-475.
doi: 10.1016/S0294-1449(02)00018-5. |
[4] |
S. Cuccagna and T. Mizumachi,
On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys., 284 (2008), 51-77.
doi: 10.1007/s00220-008-0605-3. |
[5] |
J. Fröhlich and Z. Gang,
Emission of Cherenkov radiation as a mechanism for hamiltonian friction, Adv. Math., 264 (2014), 183-235.
doi: 10.1016/j.aim.2014.07.013. |
[6] |
J. Fröhlich, Z. Gang and A. Soffer, Some hamiltonian models of friction, Jour. Math. Phys., 52 (2011), 083508, 13pp.
doi: 10.1063/1.3619799. |
[7] |
Z. Gang, A resonance problem in relaxation of ground states of nonlinear Schrödinger equaions, arXiv: 1505.01107. |
[8] |
Z. Gang and I.-M. Sigal,
Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. Math., 216 (2007), 443-490.
doi: 10.1016/j.aim.2007.04.018. |
[9] |
Z. Gang and M.-I. Weinstein,
Dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations; mass transfer in systems with solitons and degenerate neutral modes, Analysis and PDE, 1 (2008), 267-322.
doi: 10.2140/apde.2008.1.267. |
[10] |
Z. Gang and M.-I. Weinstein,
Equipartition of energy of nonlinear Schrödinger equations, Applied Mathematics Research Express, 2011 (2011), 123-181.
doi: 10.1093/amrx/abr001. |
[11] |
A. Komech, Attractors of nonlinear Hamiltonian PDEs, arXiv: 1409.2009. |
[12] |
M. Kowalczyk, Y. Martel and C. Muñoz,
Kink dynamics in the $\phi^4$ model: Asymptotic stability for odd perturbations in the energy space, J. Amer. Math. Soc., 30 (2017), 769-798.
doi: 10.1090/jams/870. |
[13] |
A. Lawrie, S.-J. Oh and S. Shahshahani,
Gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space, J. Funct. Anal., 271 (2016), 3111-3161.
doi: 10.1016/j.jfa.2016.08.019. |
[14] |
T. Mizumachi,
Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., 48 (2008), 471-497.
doi: 10.1215/kjm/1250271380. |
[15] |
I.-M. Sigal,
Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions, Comm. Math. Phys., 153 (1993), 297-320.
|
[16] |
A. Soffer and M.-I. Weinstein,
Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. math., 136 (1999), 9-74.
doi: 10.1007/s002220050303. |
[17] |
A. Soffer and M.-I. Weinstein,
Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146.
doi: 10.1007/BF02096557. |
[18] |
A. Soffer and M.-I. Weinstein,
Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys., 16 (2004), 977-1071.
doi: 10.1142/S0129055X04002175. |
[19] |
A. Soffer and M.-I. Weinstein, Theory of nonlinear dispersive waves and selection of the ground state, Phys. Rev. Lett., 95 (2005), 213905.
doi: 10.1103/PhysRevLett.95.213905. |
[20] |
T.-P. Tsai,
Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, Jour. Math. Phys., 192 (2003), 225-282.
doi: 10.1016/S0022-0396(03)00041-X. |
[21] |
T.-P. Tsai and H.-T. Yau,
Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not., 2002 (2002), 1629-1673.
doi: 10.1155/S1073792802201063. |
show all references
References:
[1] |
D. Bambusi and S. Cuccagna,
On dispersion of small energy solutions to the nonlinear Klein-Gordon equation with a potential, Amer. J. Math., 133 (2011), 1421-1468.
doi: 10.1353/ajm.2011.0034. |
[2] |
V.-S. Buslaev and G.-S. Perelman,
On the stability of solitary waves for nonlinear Schrödinger equations, Amer. Math. Soc. Transl. Ser., 164 (1995), 75-98.
doi: 10.1090/trans2/164/04. |
[3] |
V.-S. Buslaev and C. Sulem,
On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 419-475.
doi: 10.1016/S0294-1449(02)00018-5. |
[4] |
S. Cuccagna and T. Mizumachi,
On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys., 284 (2008), 51-77.
doi: 10.1007/s00220-008-0605-3. |
[5] |
J. Fröhlich and Z. Gang,
Emission of Cherenkov radiation as a mechanism for hamiltonian friction, Adv. Math., 264 (2014), 183-235.
doi: 10.1016/j.aim.2014.07.013. |
[6] |
J. Fröhlich, Z. Gang and A. Soffer, Some hamiltonian models of friction, Jour. Math. Phys., 52 (2011), 083508, 13pp.
doi: 10.1063/1.3619799. |
[7] |
Z. Gang, A resonance problem in relaxation of ground states of nonlinear Schrödinger equaions, arXiv: 1505.01107. |
[8] |
Z. Gang and I.-M. Sigal,
Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. Math., 216 (2007), 443-490.
doi: 10.1016/j.aim.2007.04.018. |
[9] |
Z. Gang and M.-I. Weinstein,
Dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations; mass transfer in systems with solitons and degenerate neutral modes, Analysis and PDE, 1 (2008), 267-322.
doi: 10.2140/apde.2008.1.267. |
[10] |
Z. Gang and M.-I. Weinstein,
Equipartition of energy of nonlinear Schrödinger equations, Applied Mathematics Research Express, 2011 (2011), 123-181.
doi: 10.1093/amrx/abr001. |
[11] |
A. Komech, Attractors of nonlinear Hamiltonian PDEs, arXiv: 1409.2009. |
[12] |
M. Kowalczyk, Y. Martel and C. Muñoz,
Kink dynamics in the $\phi^4$ model: Asymptotic stability for odd perturbations in the energy space, J. Amer. Math. Soc., 30 (2017), 769-798.
doi: 10.1090/jams/870. |
[13] |
A. Lawrie, S.-J. Oh and S. Shahshahani,
Gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space, J. Funct. Anal., 271 (2016), 3111-3161.
doi: 10.1016/j.jfa.2016.08.019. |
[14] |
T. Mizumachi,
Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., 48 (2008), 471-497.
doi: 10.1215/kjm/1250271380. |
[15] |
I.-M. Sigal,
Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions, Comm. Math. Phys., 153 (1993), 297-320.
|
[16] |
A. Soffer and M.-I. Weinstein,
Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. math., 136 (1999), 9-74.
doi: 10.1007/s002220050303. |
[17] |
A. Soffer and M.-I. Weinstein,
Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146.
doi: 10.1007/BF02096557. |
[18] |
A. Soffer and M.-I. Weinstein,
Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys., 16 (2004), 977-1071.
doi: 10.1142/S0129055X04002175. |
[19] |
A. Soffer and M.-I. Weinstein, Theory of nonlinear dispersive waves and selection of the ground state, Phys. Rev. Lett., 95 (2005), 213905.
doi: 10.1103/PhysRevLett.95.213905. |
[20] |
T.-P. Tsai,
Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, Jour. Math. Phys., 192 (2003), 225-282.
doi: 10.1016/S0022-0396(03)00041-X. |
[21] |
T.-P. Tsai and H.-T. Yau,
Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not., 2002 (2002), 1629-1673.
doi: 10.1155/S1073792802201063. |
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