Gross-Pitaevskii Dynamics as a Higgs-Gap-Controlled Slow Manifold
Year: 2026
Type: preprint
Abstract: A framework paper on the matching-fixed slow-manifold sector of a broken-symmetry nonlinear second-order field equation. The Gross-Pitaevskii (GP) equation has organized the description of coherent nonlinear bosonic matter for more than six decades, across condensates, superfluids, nonlinear optical systems, and related scalar-field models. Its nonlinear coefficient is typically fixed by system-specific microscopic input, while the dynamical scale controlling departures from the GP slow manifold is usually not isolated as a single gap-controlled variable. Here we construct a GP-matched slow-manifold sector of a broken-symmetry nonlinear second-order field equation (1/c²) ∂²Ψ/∂t² = ∇²Ψ + λΨ − α|Ψ|²Ψ, with the coupling and effective mass fixed — not fitted — by background stationarity and long-wavelength sound-speed matching: g_eff A_0² = μ_H and m_eff = μ_H / c², where A_0 = √(λ/α) is the broken-symmetry vacuum. The boundary of the slow-manifold sector is set by the amplitude (Higgs) gap of the parent field, μ_H = c√(2λ), and is controlled by a single dimensionless ratio ε = k²/(4λ). Three independent operational tests: Sector Identity tested Range Result Radial / Higgs gap μ_H = c√(2λ) 21 runs, λ ∈ {0.5–4}, δ ∈ {0.005–0.02} max |R_i−1| = 3.22 × 10⁻⁴ Phase / Goldstone–Bogoliubov (ω_GP² − ω_parent²)/μ_H² = ε² 22 well-resolved points, ε ∈ [0.05, 5.5] log-log slope 1.99915, R² = 0.99999766 Radial / inertial residual Ω_KG/Ω_GP = 2/(1+√(1+2ε)) 2.7 decades of ε sub-10⁻³ pointwise residuals What is and is not claimed. The paper does not replace microscopic derivations of GP in specific materials, nor does it claim that every system described by GP must realise a sharp amplitude resonance. It identifies μ_H as the dimensionless scale that controls the slow-manifold boundary whenever the parent field's broken-symmetry structure is present. Numerical methodology. The parent field is integrated with a kick–drift symplectic scheme on a 256×256 periodic lattice with an exact phase-bias correction ω = (2/dt) sin(ω̃ dt/2). The matched-GP envelope, where needed for cross-comparison, is integrated by a Strang split-step Fourier scheme using the finite-difference-consistent kinetic eigenvalue. All simulations use c = α = 1. Files included: main.pdf, main.tex — manuscript fig2_*.png, fig2b_*.png, fig3_*.png, fig4_*.png — figures fig*_*.py — self-contained simulation scripts (NumPy with optional CuPy GPU fallback) Citation:J.-A. Shin, "Gross–Pitaevskii Dynamics as a Higgs-Gap-Controlled Slow Manifold" (preprint, 2026).
Author: Shin, Jae-Ahn
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161592476
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