Laurie, Jason Paul (2010) Six-wave systems in one-dimensional wave turbulence. PhD thesis, University of Warwick.

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Abstract

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

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Nonlinear waves, Hamiltonian systems, Solitons
Official Date:	September 2010
Dates:	Date Event September 2010 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Nazarenko, Sergey
Extent:	xvii, 169 leaves : charts
Language:	eng

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