Practical relevance of temporal fractional derivatives (Caputo) in wave-like lattice systems

Determine whether incorporating temporal fractional derivatives, specifically the Caputo derivative, provides practically relevant modeling advantages for wave-like systems modeled by nonlinear lattices, including Fermi–Pasta–Ulam–Tsingou (FPUT)–type chains, beyond purely mathematical interest. Ascertain contexts and observables where temporal fractional modeling yields predictive value compared to models employing only spatial fractional derivatives or long-range interactions.

Background

The paper discusses fractional media in which integer-order derivatives in governing equations are replaced with fractional ones, highlighting their relevance across fluid and quantum mechanics. It notes connections to fractional discrete variants of the nonlinear Schrödinger and sine-Gordon equations and anticipates analogous developments for FPUT lattices and fractional KdV-type continuum limits.

While spatial fractional derivatives can effectively model long-range interactions in lattices (e.g., magnetic/electrical lattices) and have been shown to influence thresholds for self-trapping and induce bistability, the authors explicitly raise the unresolved issue of whether temporal fractional derivatives, such as the Caputo derivative, will have practical utility in wave-like systems. This question targets the potential empirical and engineering relevance of time-fractional modeling beyond theoretical interest.