Signum-Gordon spectral mass from nonlinear Fourier mode mixing

Abstract

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.

Figures (7)

• Figure 1: Comparative Analysis of the field equation's right-hand side. The function $f(x):=-\partial^2_x\varphi+V'(\varphi)$ is plotted at $t=0$ for the field configuration \ref{['eq:ic_eigen']} ($A_0=1$). Solid lines show the SG expression, $f_1(x)=A_0k_0^2\cos(k_0x)+\mathop{\mathrm{sgn}}\nolimits(\cos(k_0x))$, and dashed lines show the free wave case, $f_2(x)=A_0k_0^2\cos(k_0x)$. (a) $A_0k_0^2=10^{-1}$ (Low-Amplitude/Wavenumber). (b) $A_0k_0^2=10$ (High-Amplitude/Wavenumber). The enhanced alignment between the two curves for greater values of $A_0k_0^2$ confirms the convergence to the massless limit (free propagation with $\omega_0\simeq\pm k_0$). Note that the linear behavior, $f_2(x)$, already dominates when $A_0k_0^2=10$.
• Figure 2: Field and Fourier amplitudes for two regimes. Final field profiles $\phi(x, t=30)$ and Fourier mode amplitudes $A(k)$ resulting from the numerical solution of the SG equation, initialized by Equation \ref{['eq:ic_eigen']}. (a) and (b) Field value and spectrum, respectively, for the free-wave (massless) regime ($A_0k_0^2=10^4$). (c) and (d) Corresponding results for the ultra-massive regime ($A_0k_0^2=1$). The transition from single-mode dominance at high $A_0k_0^2$ to multi-mode excitation at low $A_0k_0^2$ is clearly demonstrated.
• Figure 3: Field and Fourier amplitudes at $t=30$ for SG evolution. Field configuration (a, c) and Fourier mode spectrum (b, d) at the final time $t=30$. The simulation was initialized using \ref{['eq:ic_eigen']}. Top row (a, b): Results for $A_0k_0^2=50$ (approaching massless limit). Bottom row (c, d): Results for $A_0k_0^2=25$ (transitional regime). The Fourier spectrum in (d) demonstrates nonlinear mode mixing; however, the initial wavenumber $k=10\pi$ persists as the overwhelmingly dominant component.
• Figure 4: Comparison of the asymptotic function $\lambda(\nu)$ (dashed line) with the scaled explicit coefficients $\frac{2}{N}\ln|\lambda^{(N)}_n|$ (points) for initial wave amplitudes (a) $A_0=1$ and (b) $A_0=2$. This demonstrates the rapid convergence of the NKG potential coefficients to their large-$N$ limit.
• Figure 5: The potential \ref{['eq:NKGpotential']} is plotted with an initial amplitude $A_0=1$ for truncation parameters ${N=1}$, ${N=5}$, and ${N=41}$. The effect of including higher-order coupling constants (increasing $N$) on the potential shape is clearly visible.