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January  2020, 40(1): 331-373. doi: 10.3934/dcds.2020013

Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states

Xinliang An 1,, and  Avy Soffer 2,

1. 

Department of Mathematics, National University of Singapore, Singapore 119076, Singapore
2. 

Department of Mathematics, Rutgers University, Piscataway, NJ, USA 08854, USA

* Corresponding author: Xinliang An

Received  January 2019 Revised  June 2019 Published  October 2019

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 [16], Soffer and Weinstein studied the instability mechanism and obtained an anomalously slow-decaying rate $ 1/(1+t)^{ \frac14} $. Here we develop a new method to study the evolution of $ L^2_x $ norm of solutions to Klein-Gordon equations. With this method, we prove a $ H^1 $ scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, with a dynamical system approach we also find a more robust and more intuitive way to derive the sharp decay rate $ 1/(1+t)^{ \frac14} $.

Keywords: Nonlinear Klein-Gordon equation, Fermi's golden rule, metastable states, $ H^1 $ scattering.
Mathematics Subject Classification: 35Q40, 35B34, 35B40, 35L70.
Citation: Xinliang An, Avy Soffer. Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Gang, A resonance problem in relaxation of ground states of nonlinear Schrödinger equaions, arXiv: 1505.01107. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Komech, Attractors of nonlinear Hamiltonian PDEs, arXiv: 1409.2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

I.-M. Sigal, Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions, Comm. Math. Phys., 153 (1993), 297-320.   Google Scholar

[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.  Google Scholar

[17]

A. Soffer and M.-I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146.  doi: 10.1007/BF02096557.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Gang, A resonance problem in relaxation of ground states of nonlinear Schrödinger equaions, arXiv: 1505.01107. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Komech, Attractors of nonlinear Hamiltonian PDEs, arXiv: 1409.2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

I.-M. Sigal, Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions, Comm. Math. Phys., 153 (1993), 297-320.   Google Scholar

[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.  Google Scholar

[17]

A. Soffer and M.-I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146.  doi: 10.1007/BF02096557.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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