MAGIC021: Nonlinear Waves

Course details

A specialist MAGIC course


Spring 2009
Monday, January 19th to Friday, March 27th; Monday, April 27th to Monday, April 27th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55
10:05 - 10:55


The aim of this module is to introduce students to the major ideas and techniques in the nonlinear wave theory (see the Syllabus).


No prerequisites information is available yet.


MAGIC 021: Nonlinear Waves (20 hours)
Lecturers: R.H.J. Grimshaw, G.A. El, K.R. Khusnutdinova
  1. Introduction and general overview (2 hours):
    Wave motion, linear and nonlinear dispersive waves, canonical nonlinear wave equations, integrability and inverse scattering transform (IST), asymptotic and perturbation methods, solitary waves as homoclinic orbits.
  2. Derivation and basic properties of some important nonlinear wave models (4 hours):
    • Korteweg-de Vries (KdV) and related equations (surface water waves, internal waves, etc.).
    • Nonlinear Schrodinger (NLS) equation, and generalizations with applications to modulational instability of periodic wavetrains (optics, water waves, etc.).
    • Resonant interactions of waves (general three-wave and four-wave interactions, second harmonic generation in optics, long-short wave resonance, etc.).
    • Second order models: Boussinesq and sine-Gordon equations and generalizations (Fermi-Pasta-Ulam problem, long longitudinal waves in an elastic rod, Frenkel-Kontorova model, etc.).
  3. Properties of integrable models (4 hours):
    • KdV equation (conservation laws, inverse scattering transform (IST), solitons, Hamiltonian structure) [2 hours].
    • NLS equation (focusing and defocusing, bright and dark solitons, breathers, IST).
    • Sine-Gordon equation (Bäcklund transformations, kinks and breathers).
  4. Extension to non-integrable nonlinear wave equations (5 hours):
    • Perturbed KdV equation (effects of variable environment and damping).
    • Higher-order KdV equations (integrability issues, Gardner equation, solitary waves).
    • Coupled NLS systems (modulational instability, solitary waves, integrable cases).
    • Perturbed sine-Gordon equation (soliton and multisoliton perturbation theory, kink-impurity interaction, nonlinear impurity modes).
    • Boussinesq equation with piecewise-constant coefficients (scattering of long longitudinal waves in elastic waveguides).
  5. Whitham theory and dispersive shock waves (5 hours):
    • Whitham's method of slow modulations (nonlnear WKB, averaging of conservation laws, Lagrangian formalism) [2 hours].
    • Decay of an initial discontinuity for the KdV equation: Gurevich-Pitaevskii problem.
    • Generalised hodograph transform and integrability of the Whitham equations.
    • Applications of the Whitham theory: undular bores, dispersive shock waves in plasma, nonlinear optics and Bose-Einstein condensates.
Main references:
[1] Whitham, G.B. 1974 Linear and Nonlinear Waves, Wiley, New York.
[2] Ablowitz, M.J. & Segur, H. 1981 Solitons and the Inverse Scattering Transform, SIAM.
[3] Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. & Morris, H.C. 1982 Solitons and Nonlinear Waves Equations, Academic Press, Inc.
[4] Novikov, S.P., Manakov, S.V., Pitaevskii, L.P. & Zakharov, V.E. 1984 The Theory of Solitons: The Inverse Scattering Method, Consultants, New York.
[5] Newell, A.C. 1985 Solitons in Mathematics and Physics, SIAM.
[6] Drazin, P.G. & Johnson R.S. 1989 Solitons: an Introduction, Cambridge University Press,
[7] Scott, A. 1999 Nonlinear Science: Emergence and Dynamics of Coherent Structures, Oxford University Press Inc., New York.
[8] Kamchatnov, A.M. 2000 Nonlinear Periodic Waves and Their Modulations-An Introductory Course, World Scientific, Singapore.
[9] Kivshar, Y.S., Agrawal, G. 2003 Optical Solitons: From Fibers to Photonic Crystals, Elsevier Science, USA.
[10] Braun, O.M., Kivshar, Y.S. 2004 The Frenkel-Kontorova model. Concepts, methods, and applications. Springer, Berlin.
[11] Grimshaw, R. (ed.). 2005 Nonlinear Waves in Fluids: Recent Advances and Modern Applications. CISM Courses and Lectures, No. 483, Springer, Wien, New York.
[12] Grimshaw, R. (ed.) 2007 Solitary Waves in Fluids. Advances in Fluid Mechanics, Vol 47, WIT Press, UK.


  • RG

    Professor Roger Grimshaw

    Loughborough University
    Main contact
  • Professor Gennady El

    Professor Gennady El

    Northumbria University
  • Dr Karima Khusnutdinova

    Dr Karima Khusnutdinova

    Loughborough University


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