Energy growth for a nonlinear oscillator coupled to a monochromatic wave

A system consisting of a chaotic (billiard-like) oscillator coupled to a linear wave equation in the threedimensional space is considered. It is shown that the chaotic behaviour of the oscillator can cause the transfer of energy from a monochromatic wave to the oscillator, whose energy can grow without bound. 1 Setting the problem and result The system we consider (a linear wave coupled to an oscillator) is formally defined by the Hamiltonian H = 1 2 ( py + p 2 z ) + V (y, z) + ǫk(y, z) ∫ ‖x‖≤1 u(x, t) dx+ 1 2 ∫ ( ut + (∇xu) ) dx, (1) where u(x, t), the massless Klein-Gordon field, is a scalar function on R ×R1, and (y, z) ∈ R are coordinates in the configuration space of the oscillator ((py = ẏ, pz = ż) ∈ R are the corresponding momenta). The smooth potential V (y, z) is bounded from below and tends to infinity as ‖y, z‖ → ∞. To avoid technicalities, we assume that V equals to infinity outside a bounded domain in the (y, z) plane. The interaction coefficient k(y, z) is smooth and bounded along with the first derivatives, and ǫ is small. The corresponding equations of motion are utt −∆u = −ǫk(y(t), z(t))ξ(x), (2) where ξ(x) = { 1 ‖x‖ ≤ 1 0 ‖x‖ > 1 (the characteristic function of the unit ball in the x-space), and ÿ + ∂ ∂y (V (y, z)) = −ǫk y ∫