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

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} $.

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 - A, 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. KowalczykY. 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. LawrieS.-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. KowalczykY. 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. LawrieS.-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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