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Perturbative Approach to Nonlinear Capacitance Matrix Formulations

by H. Ammari and C. Thalhammer

(Report number 2026-23)

Abstract
We study a nonlinear Helmholtz system with cubic nonlinearity on high-contrast inclusions in three dimensions, and the solitons that emerge as the contrast \(\delta\) tends to zero. Using the Dirichlet-to-Neumann operator and a capacitance formalism, we develop a perturbative cascade that expands the resonant frequency and field in powers of \(\sqrt{\delta}\). Our main result is a rigorous two-way correspondence with a finite discrete nonlinear capacitance system: every discrete solution lifts to a continuous soliton (a convergent expansion, analytic in \(\sqrt{\delta}\)), and every continuous family with the natural subwavelength scaling reduces to a discrete one. The construction is algorithmic, giving higher-order corrections in both the subwavelength and non-subwavelength regimes, the latter via a frequency-dependent capacitance matrix. We illustrate the theory numerically and characterise a symmetry-breaking bifurcation in a symmetric dimer.

Keywords: Nonlinear subwavelength resonance, nonlinear discrete approximation, Helmholtz equation, perturbation theory

@Techreport{AT26_1178,
  author = {H. Ammari and C. Thalhammer},
  title = {Perturbative Approach to Nonlinear Capacitance Matrix Formulations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-23.pdf },
  year = {2026}
}

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