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MAGIC 021: Nonlinear Waves (20 hours)
Lecturers: R.H.J. Grimshaw, G.A. El, K.R. Khusnutdinova
- 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.
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.).
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).
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,
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).
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:
- 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.
 Whitham, G.B. 1974 Linear and Nonlinear Waves
, Wiley, New
 Ablowitz, M.J. & Segur, H. 1981 Solitons and the Inverse Scattering Transform
 Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. & Morris, H.C. 1982 Solitons and Nonlinear Waves Equations
, Academic Press, Inc.
 Novikov, S.P., Manakov, S.V., Pitaevskii, L.P. & Zakharov, V.E.
1984 The Theory of Solitons: The Inverse Scattering Method
Consultants, New York.
 Newell, A.C. 1985 Solitons in Mathematics and Physics
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Cambridge University Press,
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, Oxford University Press Inc., New York.
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Their Modulations-An Introductory Course
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, Elsevier Science, USA.
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 Grimshaw, R. (ed.). 2005 Nonlinear Waves in Fluids: Recent Advances and Modern Applications
. CISM Courses and Lectures, No. 483, Springer, Wien, New York.
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. Advances in Fluid Mechanics,
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