Perturbative Approach to Nonlinear Capacitance Matrix Formulations

Technical Summary: Perturbative Approach to Nonlinear Capacitance Matrix Formulations

Problem Setting
This paper investigates a nonlinear Helmholtz system in three dimensions featuring high-contrast inclusions embedded in a homogeneous medium. The system is governed by a cubic nonlinearity (Kerr-type effect) within the inclusions, while the exterior remains linear. The primary focus is the "high-contrast limit," where the contrast parameter $\delta$ (representing the ratio of material parameters between the inclusion and the background) tends to zero. In this regime, the system supports subwavelength resonances—localized modes where the wavelength significantly exceeds the inclusion size.

The central mathematical challenge addressed is the behavior of these resonances when nonlinearity is introduced. While the linear high-contrast limit is well-understood via the capacitance matrix formalism (where resonant frequencies are determined by the eigenvalues of a finite-dimensional matrix), the nonlinear case presents significant difficulties. Previous work established the existence of solutions only at the first order in $\delta$. This paper seeks to rigorously determine whether the discrete nonlinear capacitance model faithfully captures the full continuous nonlinear soliton solutions to all orders and whether every continuous solution with natural subwavelength scaling reduces to a discrete one.

Methodology
The authors employ a perturbative approach based on the Dirichlet-to-Neumann (DtN) operator. The transmission problem is recast as a weak equation on the interior domain, utilizing the exterior DtN operator to handle the unbounded exterior.

1. Fractional Power Series Expansion: The frequency $\omega$ and the field $u$ are expanded as fractional power series in $\sqrt{\delta}$:
   $$u = \sum_{n=0}^{\infty} \delta^{n/2} u_n, \quad \omega = \sum_{n=1}^{\infty} \delta^{n/2} \omega_n.$$
2. Cascade of Equations: Substituting these ansatzes into the weak formulation generates a recursive cascade of equations. The authors decompose the solution space into a subspace of resonator-wise constant functions ($V_d$) and mean-zero fluctuations ($V_0$).
3. Recursive Solution:
   • The leading order ($n=0, 1$) yields constant fields on each inclusion.
   • At order $n=2$, the projection onto $V_d$ recovers the discrete nonlinear capacitance system: $(C - \omega_1^2 M)a - \sigma \omega_1^2 N(a) = 0$, where $C$ is the capacitance matrix, $M$ is the mass matrix, and $N(a)$ represents cubic interactions.
   • Higher-order terms are solved iteratively. The authors utilize a combinatorial concentration estimate to control the convolution sums arising from the coupling of frequency and field, ensuring the convergence of the series.
4. Extension to Non-Subwavelength Regime: The methodology is extended to regimes where $\omega = O(1)$. In this case, the expansion is in integer powers of $\delta$, and the relevant object is a frequency-dependent capacitance matrix. The nonlinearity is treated as a perturbation only for small-amplitude solitons (where the leading order field is zero).

Key Contributions and Results

• Two-Way Correspondence (Theorems 3.1 & 3.2):

  • Lifting (Discrete to Continuous): Under a non-degeneracy assumption (Assumption 1), every solution of the discrete nonlinear capacitance system lifts to a continuous soliton. This continuous solution is given by a convergent expansion in $\sqrt{\delta}$ and is analytic in $\sqrt{\delta}$.
  • Reduction (Continuous to Discrete): Conversely, every continuous family of solutions satisfying natural subwavelength scaling bounds reduces, after passing to a subsequence, to a solution of the discrete nonlinear capacitance system.
  • Significance: This establishes a rigorous, two-way correspondence, proving that the discrete model captures the nonlinear resonances faithfully without adding or missing solution branches.

• Algorithmic Construction: The construction is algorithmic, providing explicit recursive coefficients for higher-order corrections in both subwavelength and non-subwavelength regimes. This allows for the computation of asymptotic expansions to arbitrary order.

• Non-Subwavelength Regime: The paper extends the perturbative method to the non-subwavelength regime (where frequency is $O(1)$). For isolated eigenvalues of the frequency-dependent capacitance matrix, the authors obtain asymptotic expansions and prove the existence of small-amplitude solitons.

• Symmetry Breaking: The reduction to a finite discrete system allows the application of techniques from discrete self-trapping literature. The authors illustrate this by characterizing a symmetry-breaking bifurcation in a symmetric dimer. They derive a criterion for this bifurcation expressed through the eigenvalues of the generalized capacitance matrix, identifying symmetric, antisymmetric, and asymmetric branches.

Significance and Claims
The paper claims to provide the first rigorous existence results for both subwavelength and non-subwavelength solitons in the high-contrast regime with cubic nonlinearity. Unlike previous approaches that established existence only at first order or relied on Lyapunov-Schmidt reduction without explicit higher-order algorithms, this work offers:

1. Analyticity: Proof that the solutions are analytic in $\sqrt{\delta}$ (subwavelength) or $\delta$ (non-subwavelength).
2. Faithfulness: A proof that the discrete capacitance model is not merely an approximation but a faithful representation of the continuous system's solution structure in the high-contrast limit.
3. Constructive Nature: The derivation of an iterative algorithm for computing corrections to arbitrary order, bridging the gap between continuous scattering problems and discrete nonlinear Schrödinger-type models.

The authors emphasize that their results are constructive and rigorous, relying on combinatorial estimates to control the nonlinear terms, and they validate their theoretical findings through numerical illustrations, including convergence rates and bifurcation diagrams.