We investigate one-dimensional (1D) wave turbulence (WT) systems that are\ud characterised by six-wave interactions. We begin by presenting a brief introduction\ud to WT theory - the study of the non-equilibrium statistical mechanics of nonlinear\ud random waves, by giving a short historical review followed by a discussion on the physical\ud applications.\ud We implement the WT description to a general six-wave Hamiltonian system that\ud contains two invariants, namely, energy and wave action. This enables the subsequent\ud derivations for the evolutions equations of the one-mode amplitude probability density\ud function (PDF) and kinetic equation (KE). Analysis of the stationary solutions of these\ud equations are made with additional checks on their underlying assumptions for validity.\ud Moreover, we derive a differential approximation model (DAM) to the KE for super-local\ud wave interactions and investigate the possible occurrence of a \ud fluctuation relation. We\ud then consider these results in the context of two physical systems - Kelvin waves in\ud quantum turbulence (QT) and optical wave turbulence (OWT).\ud We discuss the role of Kelvin waves in decaying QT, and show that they can be\ud described by six-wave interactions. We explicitly compute the interaction coefficients for\ud the Biot-Savart equation (BSE) Hamiltonian and represent the Kelvin wave dynamics in\ud the form of a KE. The resulting non-equilibrium Kolmogorov-Zakharov (KZ) solutions\ud to the KE are shown to be non-local, thus a new non-local theory for Kelvin wave\ud interactions is discussed. A local equation for the dynamics of Kelvin waves, the local\ud nonlinear equation (LNE), is derived from the BSE in the asymptotic limit of one long\ud Kelvin wave. Numerical computation of the LNE leads to an agreement with the nonlocal\ud Kelvin wave theory.\ud Finally, we consider 1D OWT. We present the first experimental implementation\ud of OWT and provide a comparable decaying numerical simulation for verification. We\ud show that 1D OWT is described by a six-wave process and that the inverse cascade\ud state leads to the development of coherent solitons at large scales. Further investigation\ud is conducted into the behaviour of solitons and their impact to the WT description.\ud Analysis of the \ud fluxes and intensity PDFs lead to the development of a wave turbulence\ud life cycle (WTLC), explaining the coexistence between coherent solitons and incoherent\ud waves. Additional numerical simulations are performed in non-equilibrium stationary\ud regimes to determine if a pure KZ state can be realised
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