**The**

^{Nonlinear Partial Differential}

** Equations Group**

The Department of Mathematics, University of Surrey

MEMBERS

**Michele Bartuccelli
Tom Bridges
Stephen Gourley
**

Ian Roulstone Bjorn Sandstede Anne Skeldon Claudia Wulff

**Research Topics of Interest Include**

**
**

Qualitative Analysis of PDEs.

** In particular the
study 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.**

Y. Kyrychko, M. Bartuccelli: Length scales for the Navier-Stokes equations on a rotating sphere. Physics Letters A, Volume 324, 179-184 (2004).

M. Bartuccelli, K.B. Blyuss, Y. Kyrychko: Length Scales and Positivity of Solutions of a Class of Reaction-Diffusion Equations. Communications on Pure and Applied Analysis, 3, No. 1, 25 - 40 (2004).

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

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).**

**Coherent Structures and Defects
**

**Coherent structures are interfaces
between stable, spatially periodic
structures with possibly different spatial wave numbers. These
interfaces can also be thought of as defects at which the underlying
perfectly periodic structure is broken. Surprisingly, the defects
observed in experiments and numerical simulations appear to be
time-periodic when viewed in an appropriate reference frame. Our goal
is to investigate the existence and stability properties of such
defects: What happens to a defect if we change the wave numbers of the
asymptotic periodic pattern or external system parameters? How does it
behave if we subject it to small perturbations? How do defects arise
in the first place from stationary patterns? Our attempts to answer
these questions employ dynamical-systems techniques and concepts such
as group velocities. More information can be found on my
personal
homepage.
**

**B Sandstede and A Scheel.
Absolute instabilities of standing pulses. Nonlinearity (accepted).
**

**B Sandstede and A Scheel.
Defects in oscillatory media: toward a classification.
SIAM Journal of Applied Dynamical Systems 3 (2004) 1-68.
**

**B Sandstede and A Scheel.
Evans function and blow-up methods in critical eigenvalue problems.
Discrete and Continuous Dynamical Systems 10 (2004) 941-964.
**

**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''. Physica D, 178 , 83--102 (2003).**

**Symmetry Reduction for Nonlinear Waves**

** Symmetry plays an important role for pattern formation.
On extended domains the symmetry group is often non-compact (due to the presence of translational
symmetries) and non-abelian (an example is the Euclidean symmetry of the plane). To analyze the
dynamics near nonlinear waves and their bifurcations it is useful to split the dynamics into drift
dynamics and symmetry-reduced dynamics, also called shape dynamics. Often the system in
question only has approximate symmetries so that forced symmetry-breaking
(due to inhomogeneities or boundaries of the domain) has to be studied.**

**J.Lamb, C.Wulff. Pinning and locking of discrete waves.
Physics Letter A 267, 167-173, 2000.**

**
V. LeBlanc, C. Wulff. Translational symmetry breaking for
spiral waves. J. Nonlinear Sci. 10, 569 - 601, 2000.**

** C. Wulff. Spiral Waves and Euclidean
Symmetries. Zeitschrift fur physikalische Chemie, 216, 535-550, 2002.**