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).


Higher Order Dissipative Partial Differential Equations And
Positivity Preservation of Their Solutions

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.