Perturbative Approach to Nonlinear Capacitance Matrix Formulations

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 $δ$ 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 $\sqrtδ$. 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 $\sqrtδ$), 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.