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
- 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.
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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.).
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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.).
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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).
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Extension to non-integrable nonlinear wave equations (5 hours):
- Perturbed KdV equation (effects of variable environment and damping).
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Higher-order KdV equations (integrability issues, Gardner equation,
solitary waves).
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Coupled NLS systems (modulational instability, solitary waves, integrable cases).
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Perturbed sine-Gordon equation (soliton and multisoliton perturbation theory, kink-impurity interaction, nonlinear impurity modes).
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Boussinesq equation with piecewise-constant coefficients (scattering of long longitudinal waves in elastic waveguides).
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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.