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Analysis of nonlinear resonances in resonator crystals: Tight-binding approximation and existence of subwavelength soliton-like solutions

by H. Ammari and J. Qiu

(Report number 2025-25)

Abstract
This work provides a mathematical framework for elucidating physical mechanisms for confining waves at subwavelength scales in periodic systems of nonlinear resonators. A discrete approximation in terms of the linear capacitance operator is provided to characterize the nonlinear subwavelength resonances. Moreover, the existence of subwavelength soliton-like localized waves in periodic systems of nonlinear resonators is proven. As a by-product, a tight-binding approximation of the capacitance operator is shown to be valid for crystals of subwavelength resonators. Both full- and half-space crystals are considered. The framework developed in this work opens the door to the study of topological properties of periodic lattices of subwavelength nonlinear resonators, such as the emergence of nonlinearity-induced topological edge states, and to elucidate the interplay between nonlinearity and disorder.

Keywords: Nonlinear subwavelength resonance, nonlinear periodic structure, nonlinear discrete approximation, Kerr nonlinearity, Helmholtz equation, soliton-like solution.

@Techreport{AQ25_1146,
  author = {H. Ammari and J. Qiu},
  title = {Analysis of nonlinear resonances in resonator crystals: Tight-binding approximation and existence of subwavelength soliton-like solutions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-25},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-25.pdf },
  year = {2025}
}

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