Signum-Gordon spectral mass from nonlinear Fourier mode mixing

TL;DR

This paper explores how the signum-Gordon nonlinear field model exhibits an effective spectral mass through nonlinear Fourier mode mixing, despite its non-analytic potential, by analyzing wave evolution and dispersion relations.

Contribution

It demonstrates that the signum-Gordon model can mimic a massive theory, establishing a method to quantify spectral mass in non-analytic scalar field models.

Findings

The model exhibits a spectral mass of unity for specific initial amplitudes.

Nonlinear Fourier mode mixing populates higher harmonics due to potential's non-analyticity.

Dispersion maps show effective mass behavior similar to Klein-Gordon theory.

Abstract

We investigate the emergence of a spectral mass in the signum-Gordon model, a nonlinear field theory characterized by a non-analytic, V-shaped potential where standard perturbative mass definitions are inapplicable. By analyzing the evolution of monochromatic wave trains, we identify two distinct dynamical regimes governed by the relationship between the wave's amplitude and its wavenumber. In the nonlinear regime, the model exhibits nonlinear Fourier mode mixing, where the potential's lack of analyticity acts as a source that populates higher-order harmonics. Using two complementary numerical methods -- tracking frequency distributions from initial wavenumbers and measuring spatial responses to boundary signals -- we construct comprehensive dispersion maps in energy-momentum space. Our results demonstrate that the signum-Gordon field effectively mimics a massive theory. Specifically, we show that a particular initial wave amplitude induces a spectral mass of unity, perfectly matching the behavior of the massive Klein-Gordon equation and providing a robust framework for quantifying mass in non-analytic scalar models.