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Lecture Titles

LMS Invited Lecturer

Professor Knobloch will deliver ten 45 minute lecures on spatially localised structures. The content of these will be as follows:


Lecture 1: Localised patterns in Nature

Introduces observations of: Ferrofluid peaks, spot formation in chemical systems, vortices, oscillons, etc and discusses defects, holes and fronts as localized structures (LS).

Lecture 2 & 3: Simple physical models of LS in one spatial dimension

The Swift-Hohenberg equation (SH23): numerics of the snaking region, origin of pinning re- gion/analogy with phase transitions, geometrical explanation of snaking and its relation to pinning, other inhabitants of the pinning region (multipulse states etc), wavenumber selection, depinning and front motion. Swift-Hohenberg equation (SH35): brief discus- sion, including collapsed snaking.

Lecture 4: Origin of snaking

Bifurcations in spatially reversible systems, 1:1 re- versible Hopf bifurcation, exponential asymptotics, termination of the snaking branches and finite domain effects.

Lecture 5: Swift-Hohenberg equation in 2D

Stripes, targets vs spots, localized hexagons and lozenge-like structures (worms).

Lecture 6: Broken symmetry

Broken translation invariance: finite domain effects; Broken reversibility: drifting pulses; Broken variational structure.

Lecture 7: Forced complex Ginzburg-Landau equation

Oscillons, holes and the forced complex Ginzburg-Landau equation for the 1:1 and 2:1 temporal resonance, higher order resonances, defect-mediated snaking.

Lecture 8: Localised structures in Fluid Mechanics

Doubly diffusive sys- tems, Binary fluid convection, Marangoni convection, Plane Couette flow, Generalized Korteweg-de Vries equation.

Lecture 9: Localised structures in Chemistry

Reaction-diffusion systems, Grey-Scott model, spots and spot replication in 1D and 2D, snaking of stationary and traveling pulses.

Lecture 10: Other applications and unsolved problems

Solitons in nonlinear optics, melting of soliton lattices, writing and erasing solitons, and buckling in structural mechanics. Novel applications: liquid films, vortices, defects etc.