Deriving the Continuous $β$-FPUT Model for Decayless Solar Coronal Oscillations Directly from Resistive MHD

Abstract

We present an analytical framework deriving the fluid-continuum $β$-Fermi-Pasta-Ulam-Tsingou ($β$-FPUT) system directly from a third-order resistive magnetohydrodynamics (MHD) expansion to explain persistent, decayless transverse oscillations observed in solar coronal loops. For a continuous wave field propagating along a longitudinal magnetic field with cross-field density variations, a multi-scale expansion demonstrates that quadratic $\mathcal{O}(2)$ advective and Maxwell tension terms vanish identically due to out-of-plane polarization symmetry. Instead, the quadratic longitudinal ponderomotive force drives field-aligned density modulations that back-react on the primary transverse wave, yielding a cubic $\mathcal{O}(3)$ restoring force. In the long-wavelength limit, the governing equations reduce to a fluid-continuum Modified Korteweg-de Vries (mKdV) system where cross-field profile variation provides the non-linear dispersive mechanism. We prove that this internal fluid mechanism establishes stable, amplitude-dependent frequency self-focusing, which phase-locks adjacent elements and prevents phase-mixing wave attenuation without requiring external steady driving parameterizations. This framework enables macro-scale coronal seismology, providing closed-form expressions to invert observed decayless wave-packet durations to compute the unresolved cross-field density gradient scale length ($L_x$) and the internal volumetric filling factor ($f_V$) without relying on traditional emission measure or spectroscopic line-ratio diagnostics. Much like famous Enrico Fermi's historic use of macro-scale displacements to estimate complex atomic explosion yields, this framework establishes a paradigm where monitoring macroscopic oscillation durations empowers observers to extract fine-scale, otherwise unresolvable waveguide properties.