Continuum (PDE) extension of the spatiotemporal zeta formulation

Show that the spatiotemporal zeta-function formulation 1/ζ[β,z] applies to continuous spacetime systems by defining spacetime periodicities over R^d, replacing the generating variable z with a Laplace parameter s in Z[β,s], and evaluating stability exponents as functional integrals over the Brillouin zone; in particular, establish applicability to the Kuramoto–Sivashinsky and Navier–Stokes partial differential equations.

Background

The paper develops the theory on discrete lattices for pedagogical and technical convenience, with stability exponents computed via Bloch theory on the reciprocal lattice. The authors outline how an analogous construction might work in the continuum by treating periodicities as continuous vectors and using a Laplace transform parameter.

Demonstrating the continuum extension would bridge the formalism to widely studied PDEs, such as Kuramoto–Sivashinsky and Navier–Stokes, showing that the zeta-function approach can handle genuinely continuous spatiotemporal dynamics.