Qualitative Analysis of Solutions of Nonlinear Dissipative Partial Differential Equations

Examples of such systems are the ComplexGinzburg-Landau
(CGL),  the Navier-Stokes (NS) equations and equations governing
population dynamics in biology.  Particular attention is given to regularity, global existence and length scales of solutions.  Problems such as stability, instability and turbulence are also addressed.  In recognition of the importance of the work on qualitative functional analysis, a three year grant was awarded by EPSRC in 1994 to support a project on the turbulent behaviour of solutions of the CGL equation.  Higher-order GL equations are of interest.  For example, in recent work an analysis of the instabilities in a quintic order nonvariational Ginzburg-Landau equation has been given.
It was found that the equation permits both sub-and super-critical
zigzag and Eckhaus instabilities, and that the zigzag instability may occur for patterns with wavenumber larger than critical, in contrast
to the usual case.
A new approach for determining the stability of turbulent attractors of a PDE to symmetry-breaking perturbations has been devised based on the computation of dominant Lyapunov exponents associated with particular isotypic components.   The effect of perturbations which break a reflectional symmetry as well as subharmonic perturbations were considered and a spatial period-doubling blowout bifurcation was observed in the CGL equation. This approach shows that while period boundary conditions may be convenient mathematically and numerically, they are not necessarily physically relevant for turbulent solutions.

M.V. Bartuccelli, J.D. Gibbon, M. Oliver, ``Length Scales in Solutions of the
Complex Ginzburg-Landau Equation''. Physica D 89, 267-286  (1996).

M.V. Bartuccelli, S.A. Gourley, C.J. Woolcock, ``Length Scales in Solutions of
a Generalized Diffusion Model''.  Physica Scripta  57, 9-19  (1998).

P.J. Aston, C.R. Laing , ``Symmetry and chaos in the complex Ginzburg-Landau equation. I Reflectional symmetries''. Dyn. Stab. Sys. 14, 233-253  (1999).

P.J. Aston, C.R. Laing, ``Symmetry and chaos in the complex Ginzburg-Landau equation. II Translational symmetries''. Physica D 135, 79-97  (2000).

The group interests centre on dissipative partial differential equations of reaction diffusion type; in particular equations containing a fourth order spatial derivative. Such equations do not, in general, exhibit positivity preservation but may do so under some further restriction on the initial data and parameters; recent results in this area include the study of positivity and convergence of solutions of such equations, by employing generalized energy methods and sharp interpolation inequalities.
 

M.V. Bartuccelli, S.A. Gourley, A.A. Ilyin, ``Positivity and the Attractor Dimension in a Fourth Order Reaction-Diffusion Equation''. Proc. Roy.
Soc. Lond.,  Ser. A 458, 1431 - 1446 ,  (2002).

M.V. Bartuccelli  ``On the asymptotic positivity of solutions for the extended
Fisher-Kolmogorov  equation with nonlinear diffusion''.
Mathematical Methods in the Applied Sciences, 25, 701-708 (2002).

M. Bartuccelli and C.J. Woolcock,  ``On the Positivity of Solutions for a Generalized Diffusion Model''.  Nuovo Cimento, 116 B, 1365 - 1373 (2001).

Regular and Irregular Hexagonal Patterns

The study of hexagonal patterns has traditionally been confined to the case where the hexagons fit almost exactly onto a lattice. In this work, distorted hexagonal patterns in large aspect-ratio systems are investigated. The Cross-Newell partial differential equations for hexagons and triangles are derived for general real gradient systems, and are found to be in flux-divergence form, in contrast to the general nongradient case. Specific examples of complex governing equations that give rise to hexagons and triangles and which have Lyapunov functionals are also considered and explicit forms of the Cross-Newell equations are found in these cases. The Cross-Newell equations permit the recovery of the phase stability boundaries and modes of instability for general distorted hexagons and triangles.

R.B. Hoyle, Phys Rev E 63 pp 2506-2512 (2000).

R.B. Hoyle, Phys Rev E 58 pp. 7315-7318 (1998).

Faraday Waves

When a container of fluid is shaken up and down, waves may appear on the surface of the fluid. The pattern of the waves is dependent on the amplitude and the frequency components of the shaking. Faraday first noted the phenomena in the 1830's, but there has been renewed interest in the problem over the last 20 years firstly because of the occurrence of chaos and secondly because of the huge variety of patterns that are observed. We are interested in understanding some of the principles that govern why some patterns are observed and not others.

M. Silber and A.C. Skeldon, ``Parametrically excited surface waves:
two-frequency forcing, normal form symmetries and pattern selection''.
Phys. Rev. E,  59,  5446--5456,  (1999).

M. Silber, C. Topaz and A.C. Skeldon, ``Two-frequency forced Faraday waves: weakly damped modes and pattern selection''. Physica D, 143, 205--225, (2000).

Influence of Boundaries

Frequently, mathematical progress is made by making an assumption that a physical problem has an infinite domain or the domain is periodic. While these may be reasonable approximations in some situations, it is important to understand the effect that boundaries impose. For example, the Rayleigh-Benard experiment in which a layer of fluid is heated from below is necessarily carried out in, what may be a large, but is nevertheless finite, container. Assuming the container is infinite leads to particular predictions as to when and with what wavelength convection patterns onset. Weakly nonlinear analysis then allows one to study pattern selection. The inclusion of realistic boundary conditions alter the prediction as to when convection onsets and restricts the range of allowable wavelengths of the patterns. We are investigating how realistic boundary conditions affect the pattern selection process.

P.G. Daniels, D. Ho and A.C. Skeldon, ``Solutions for nonlinear convection
in the presence of a lateral boundary''.  Submitted.