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Nonlinear Modal Reduction for Subwavelength Dielectric Scattering

by H. Ammari and I. Gorea and B. Li

(Report number 2026-31)

Abstract
We study three-dimensional wave scattering by high-index dielectric resonators with Kerr-type nonlinearity under plane-wave incidence, together with the associated nonlinear dielectric scattering resonances, through a nonlinear Lippmann--Schwinger equation. Using a Lyapunov--Schmidt reduction near a simple eigenmode of the Newtonian potential, we obtain a local decomposition of the scattered wave into a resonant contribution and a controlled remainder and derive an explicit nonlinear equation for the resonant modal coefficient for sufficiently small contrast parameter and locally small resonant amplitude and incident field. A second reduction covers the regime of incident fields of order one and weak nonlinearity. Our results extend for the first time the linear modal decomposition to nonlinear wave-scattering problems. We complement them by developing a numerical framework based on Nystr\"om discretization, real Newton iteration, and pseudo-arclength continuation. For single resonators of several geometries, our computations confirm the predicted high-contrast resonance scaling and the nonlinear modal approximation, and exhibit the multivalued incident-wave response. For a mirror-symmetric resonator dimer, we track symmetric, antisymmetric, and symmetry-broken resonance branches over a broad range of separations. Our computations show that the symmetry-breaking threshold increases as the gap between the resonators decreases, in agreement with the leading-order bifurcation theory.

Keywords: nonlinear scattering, modal reduction, subwavelength physics, bifurcation theory, high contrast index material.

@Techreport{AGL26_1186,
  author = {H. Ammari and I. Gorea and B. Li},
  title = {Nonlinear Modal Reduction for Subwavelength Dielectric Scattering},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-31},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-31.pdf },
  year = {2026}
}

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