Nonlinear physics

Theoretical nonlinear physics II: Spatially extended systems and pattern formation

Summer semester 2019

Lecture with tutorials

Course number in the course overview: 116291 (lecture), 116292 (tutorials)

  • Lecturers, organization, dates & exams

    Lecturer

    Dates & Locations

    Lecture Tue., 14:00–16:00 KP/TP 304
    Start: Tue., 02.04.2019
    Tutorials Wed., 08:00–10:00 (every second week) KP/TP 303
    Beginn: by arrangement
  • Contents & literature

    Contents of the lecture

    1. Introduction
    2. Repetition of dynamical systems [attractors and their stability, local bifurcations and their normal forms]
    3. Systems and models – from transport equations to order parameter equations [concept of von balance and transport equations, selected physical applications, basic model equations of structure formation]
    4. Stationary states in spatially extended systems and their linear stability [Turing instability, Hopf instability, codimension 2 instabilities, classification of instabilities of one-dimensional patterns]
    5. Bifurcations and weakly nonlinear theory
    6. Secondary instabilities, defects and chaos [space–time chaos in structure formation]

    In the course of the lectures, the topics

    • Reduction methods for order parameter equations
    • Multi-scale analysis
    • Galerkin approximation
    • Role of symmetries
    • Selection mechanisms for patterns
    • Models such as Swift-Hohenberg/Ginzburg-Landau/Kuramoto-Sivashinsky/reaction-diffusion equations

    will also be mentioned.

    Literature

    Chapter 2:

    • Argyris, J.; Faust, G.; Haase, M. & Friedrich, R., An Exploration of Dynamical Systems and Chaos: Completely Revised and Enlarged Second Edition, Springer Berlin Heidelberg, 2015
    • Strogatz, S. H., Nonlinear Dynamics and Chaos Addison-Wesley, 1994

    Chapters 3–6:

    • G. Nicolis, Introduction to nonlinear science, Cambridge University Press, 1999
    • M. Bestehorn, Hydrodynamik und Strukturbildung: Mit einer kurzen Einfuhrung in die Kontinuumsmechanik (Springer-Lehrbuch) (German Edition) Springer, 2006
    • R. Hoyle: Pattern formation, Cambridge Univ. Press 2007
    • P. Colinet, J. C. Legros & M. G. Velarde, Surface-Tension-Driven Instabilities, Wiley-VCH, 2001
    • M.C. Cross, H. Greenside: Pattern formation and dyanmics in nonequilibrium systems, Cambridge Univ. Press 2009
    • M.C. Cross , P.C. Hohenberg, Pattern formation outside equilibrium, Rev. Mod. Phys. 65, 854 (1993)
    • A.C. Newell, T. Passot, J. Lega, Order parameter equations for pattern, Annu. Rev. Fluid Mech. 25, 399 (1993)
    • D. Walgraef: Spatio-temporal pattern formation, Springer 1997

    more literature in the lecture

  • Lecture material

    The script and other materials are provided in Learnweb.

  • Tutorials

    Coordination: Steffen Richters-Finger

  • Problem sheets

    The exercise sheets are provided in Learnweb.