Fermi's Golden Rule and H¹ Scattering for Nonlinear Klein-Gordon Equations with Metastable States

2015/10/22 by An, Xinliang, Soffer, Avy
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.1510.06485

Abstract

In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from the 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)^(1)/(4). Here we develop a new method to study the L²_x norm of solutions to Klein-Gordon equations. With this method, we prove the first H¹ scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, we also give another more robust and more intuitive approach to derive the sharp decay rate 1/(1+t)^(1)/(4).

Related

• Energy transfer, weak resonance, and Fermi's golden rule in Hamiltonian nonlinear Klein-Gordon equations
• Energy transfer and radiation in Hamiltonian nonlinear Klein-Gordon equations: general case
• Scattering of rough solutions of the nonlinear Klein-Gordon equations in 3D
• A survey on asymptotic stability of ground states of nonlinear Schrodinger equations II
• Scattering threshold for the focusing nonlinear Klein-Gordon equation