Nonlinear wave dynamics in discrete space-time systems

Moorcroft, Thomas Taylor (2026) Nonlinear wave dynamics in discrete space-time systems. Doctoral thesis, University of East Anglia.

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Abstract

Nonlinear wave dynamics are ubiquitous across many fields of mathematics and physics, and are typically studied in the context of continuous space-time. In this thesis a different setting is considered, in which wave evolution is governed by discrete space-time.

In particular, an experimentally realisable discrete space-time optical lattice model, here referred to as the photonic mesh lattice, is studied. The lattice is comprised of two distinct dispersion bands, each equipped with a unique set of normal modes. The thermalisation processes among these bands are investigated in-depth, to understand the role of discrete-time on modal interactions and thermal equilibrium.

Through numerical simulations, it is shown that the system evolves toward a universal thermal equilibrium characterised by equal modal occupation among both bands, independent of lattice parameters and initial conditions. This behaviour arises from additional resonant wave interactions that exist through duplicate dispersion bands permitted by discrete-time, violating the conservation of linear energy. Consequently, the thermal equilibrium state is no longer constrained by linear energy conservation, leading to a deviation from the Rayleigh–Jeans distribution typically predicted by weak wave turbulence theory.

Furthermore, these additional discrete-time resonances are also shown to accelerate thermodynamic processes via a four-wave resonant interaction. A systematic exploration of lattice parameters reveals a clear relationship between the size of the resonant manifold and thermalisation timescales, with larger manifolds corresponding to substantially faster modal equilibration.

Lastly, the thermal equilibrium state is explored for a driven/damped lattice model, by driving one band and dissipating the other. In doing so, the effects of discrete time on modal interaction and thermal equilibrium are diminished, returning the system to the typical Rayleigh–Jeans distribution, reminiscent of continuous-time multi-band wave systems.

To the best of the author’s knowledge, this thesis demonstrates the first observations of the effect of discrete-time on thermalisation processes and weak wave turbulence, showing that discrete-time is a novel setting for the study of thermalisation and weak wave turbulence theory.

Item Type:	Thesis (Doctoral)
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
Depositing User:	Chris White
Date Deposited:	20 Oct 2026 10:17
Last Modified:	20 Oct 2026 10:17
URI:	https://ueaeprints.uea.ac.uk/id/eprint/104824
DOI:

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