Nonlinear Waves (MAGIC021)

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

No specific requirements. Standard undergraduate courses in analysis, mathematical methods and partial differential equations are desirable.

The aim of this module is to introduce major ideas and techniques of modern nonlinear wave theory, with an emphasis on asymptotic methods for nonlinear dispersive PDEs and applied aspects of integrability and inverse scattering transform.

1. Introduction and general overview (2 hours)
   • Wave motion, linear and nonlinear dispersive waves, non-dispersive waves, shocks, canonical linear and nonlinear wave equations, integrability and inverse scattering transform (IST), asymptotic and perturbation methods.
2. Dispersive wave models: derivation techniques and basic properties (4 hours)
   • Korteweg-de Vries (KdV) and related equations.
   • Nonlinear Schrödinger (NLS) equation, and generalizations with applications to modulational instability of periodic wavetrains.
   • Resonant interactions of waves (general three-wave and four-wave interactions, second harmonic generation, long-short wave resonance; phase-plane analysis).
   • Second order models: Boussinesq and sine-Gordon equations (Fermi-Pasta-Ulam problem, Zabusky-Kruskal model, solitons; Frenkel-Kontorova model, Bäcklund transformations, kinks and breathers).
3. Inverse scattering thansform and solitons (6 hours)
   • KdV equation (conservation laws, Miura transformation, Lax pair, discrete and continuous spectrum of the time-independent Schrödinger operator, direct and inverse scattering problems, initial-value problem by the inverse scattering transform. Reflectionless potentials and N-soliton solutions. Hamiltonian structures).
   • NLS equation (symmetries, focusing and defocusing, bright and dark solitons, breathers, AKNS scheme, linear problem, scheme of inverse scattering transform for the focusing NLS equation, N-soliton solutions).
   • Perturbed and higher-order KdV equations (effects of inhomogeneity, asymptotic integrability, Gardner equation).
4. Nonlinear hyperbolic waves and classical shocks (3 hours)
   • Kinematic waves, solution via characteristics, hodograph transformation, Riemann invariants, gradient catastrophe.
   • Hyperbolic conservation laws, weak solutions and shock waves. Rankine-Hugoniot conditions. Lax entropy condition.
   • Structure of the viscous shock wave, Burgers equation, Cole-Hopf transformation, Taylor's shock profile, N-wave.
5. Whitham modulation theory and dispersive shock waves (5 hours)
   • Whitham's method of slow modulations (nonlinear WKB, averaging of conservation laws, Lagrangian formalism).
   • Generalised hodograph transform and integrability of the Whitham equations. Connection with the inverse scattering transform.
   • Formation of a dispersive shock wave. Decay of an initial discontinuity for the KdV equation. Gurevich-Pitaevskii problem.

[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.

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