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<p>In this paper, we consider the Klein-Gordon equations with cubic nonlinearity in three spatial dimensions, which are Hamiltonian perturbations of the linear one with potential. It is assumed that the corresponding linear Schrodinger operator admits an arbitrary number of possibly degenerate eigenvalues. By analyzing the resonance mechanisms between multiple discrete and continuous spectral modes, we determine the precise rate of energy transfer and radiation damping. Compared to [<CitationRef CitationID="CR5">5</CitationRef>], our results provide the first quantitative answer to the general multiple bound state problem proposed by Soffer and Weinstein in their seminal work [<CitationRef CitationID="CR52">52</CitationRef>]. Our proof leverages a <i>pseudo-one-dimensional cancellation structure</i> within each eigenspace, a <i>renormalized damping mechanism</i>, and an <i>enhanced damping effect</i> arising from interactions among discrete modes. Additionally, the analysis incorporates a refined Birkhoff normal form transformation and an extended version of Fermi’s Golden Rule, building on the foundational work of Bambusi and Cuccagna [<CitationRef CitationID="CR5">5</CitationRef>].</p>

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Resonances, Energy Transfer and Radiation in Hamiltonian Nonlinear Wave Equations with Multiple Internal Modes

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Zhen Lei,
  • Jie Liu,
  • Zhaojie Yang

摘要

In this paper, we consider the Klein-Gordon equations with cubic nonlinearity in three spatial dimensions, which are Hamiltonian perturbations of the linear one with potential. It is assumed that the corresponding linear Schrodinger operator admits an arbitrary number of possibly degenerate eigenvalues. By analyzing the resonance mechanisms between multiple discrete and continuous spectral modes, we determine the precise rate of energy transfer and radiation damping. Compared to [5], our results provide the first quantitative answer to the general multiple bound state problem proposed by Soffer and Weinstein in their seminal work [52]. Our proof leverages a pseudo-one-dimensional cancellation structure within each eigenspace, a renormalized damping mechanism, and an enhanced damping effect arising from interactions among discrete modes. Additionally, the analysis incorporates a refined Birkhoff normal form transformation and an extended version of Fermi’s Golden Rule, building on the foundational work of Bambusi and Cuccagna [5].