Signum-Gordon spectral mass from nonlinear Fourier mode mixing

We investigate the concept of mass in the Signum-Gordon (SG) model, a nonlinear field theory with a non-analytic potential where the perturbative mass is undefined. Using two complementary numerical methods, we map the field’s dispersion relation (amplitude vs. wavenumber and frequency). We find the field’s evolution depends critically on the product of its amplitude and squared wavenumber, revealing a massless regime at large values and an ultra-massive regime with dominant nonlinear Fourier mode mixing near unity. By comparing the resulting dispersion map to the massive Klein-Gordon equation, we introduce a spectral mass. We demonstrate that a specific input amplitude value induces a spectral mass of unity, effectively characterizing the massive-like behavior arising from the initial wave configuration.

💡 Research Summary

The paper tackles the long‑standing problem of defining a mass in the Signum‑Gordon (SG) model, a scalar field theory whose potential (V(\phi)=|\phi|) is non‑analytic at its minimum. Because the second derivative of the potential does not exist at (\phi=0), the usual perturbative mass (m_0^2=\partial^2 V/\partial\phi^2|_{\phi=0}) cannot be defined. The authors therefore propose a new, dynamical notion of “spectral mass” that emerges from the evolution of an initially monochromatic wave packet.

The SG equation reads (\partial_\mu\partial^\mu\phi+\operatorname{sgn}(\phi)=0) and possesses a scaling symmetry ((t,x)\to\lambda(t,x),;\phi\to\lambda^{-2}\phi). This symmetry makes the dimensionless combination (A_0k_0^2) (initial amplitude times the square of the wavenumber) the natural control parameter. The authors consider the initial condition (\phi(t=0,x)=A_0\cos(k_0x)) with (\omega_0=k_0) and study its time evolution using two independent numerical schemes (fourth‑order Runge‑Kutta, periodic boundaries, (L=1), (N=1000) grid points, (\Delta t=10^{-4}), final time (t=30)).

Two distinct regimes are identified:

1. Massless (large‑amplitude) regime – When (A_0k_0^2\gg1), the signum term is negligible compared with the spatial second derivative. The dynamics reduce to the free wave equation, yielding the dispersion relation (\omega^2\simeq k^2). Numerically, the right‑hand side of the SG equation aligns with the linear case, and the Fourier spectrum remains dominated by a single mode. This demonstrates that, for sufficiently large amplitudes, the SG model behaves as a massless Klein‑Gordon (KG) field.

2. Ultra‑massive (critical) regime – When (A_0k_0^2\approx1), the nonlinear term is comparable to the kinetic term. The wave no longer propagates as a simple sinusoid; instead, a cascade of higher harmonics (predominantly odd multiples of (k_0)) and sub‑fundamental modes is generated. The field profile becomes highly irregular, and the Fourier spectrum shows strong mode mixing. This regime is termed “ultra‑massive” because the effective dispersion deviates markedly from the linear one, mimicking a massive KG field with a sizable mass term.

To quantify the effective mass, the authors construct an empirical dispersion map (\omega(k;A_0)) from the simulations and fit it to the massive KG form (\omega^2=k^2+m_{\text{spec}}^2). The fit yields a spectral mass (m_{\text{spec}}(A_0)) that varies smoothly with the control parameter. Remarkably, when (A_0k_0^2=1) the fitted mass equals unity, i.e. (m_{\text{spec}}=1). Thus a specific input amplitude (A_\star=1/k_0^2) produces a “spectral mass of one”, providing a concrete operational definition of mass in a model where the perturbative definition fails.

The paper further connects these findings to a more familiar nonlinear field theory: the (\phi^4) (nonlinear Klein‑Gordon, NKG) model with equation ((\partial_t^2-\partial_x^2)\phi+m_0^2\phi+\lambda\phi^3=0). By expressing the cubic term in Fourier space via convolution, the authors show analytically how an initially single‑mode configuration inevitably generates higher‑frequency harmonics, mirroring the SG behavior. This demonstrates that the observed Fourier mode mixing is not a peculiarity of the SG potential but a generic feature of nonlinear wave equations.

The numerical methodology is carefully validated: two independent codes reproduce the same dispersion maps, convergence tests with respect to grid resolution and timestep are performed, and a systematic scan over (A_0k_0^2) confirms the existence of a sharp transition between the massless and ultra‑massive regimes. The authors also discuss the role of odd versus even harmonics, noting that the signum nonlinearity preferentially excites odd multiples, a pattern that aligns with the piecewise‑constant nature of (\operatorname{sgn}(\phi)).

In conclusion, the work establishes that the SG model, despite lacking a conventional perturbative mass, possesses an emergent spectral mass determined by the initial wave’s amplitude‑wavenumber product. The critical value (A_0k_0^2=1) marks the boundary where the system transitions from free‑wave propagation to strong nonlinear mode coupling, effectively behaving like a massive KG field with unit mass. This insight opens several avenues for future research: extending the analysis to higher dimensions, incorporating external driving or damping, exploring quantum corrections to the spectral mass, and investigating whether similar spectral‑mass constructions can be applied to other non‑analytic or piecewise‑linear field theories.