MAGIC021: Nonlinear Waves

Course details

Semester

Spring 2008
Monday, January 21st to Friday, March 14th; Monday, April 28th to Friday, May 16th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
0

Timetable

Wednesdays
10:05 - 10:55
Thursdays
10:05 - 10:55

Description

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

Prerequisites

No prerequisites information is available yet.

Syllabus



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 (integrable cases, solitary waves, modulational instability).
    • Perturbed sine-Gordon equation (soliton and multisoliton perturbatioin theory, effects of disorder in crystals, kink-impurity interaction, nonlinear impurity modes, resonant interactions with impurities) [2 hours].
  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,
London.
[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.

Lecturers

  • RG

    Professor Roger Grimshaw

    University
    Loughborough University
    Role
    Main contact
  • Professor Gennady El

    Professor Gennady El

    University
    Northumbria University
  • Dr Karima Khusnutdinova

    Dr Karima Khusnutdinova

    University
    Loughborough University

Bibliography

No bibliography has been specified for this course.

Assessment

Attention needed

Assessment information will be available nearer the time.

Files

Only consortium members have access to these files.

Please log in to view course materials.

Lectures

Please log in to view lecture recordings.