Course details
A core MAGIC course
Semester
 Spring 2019
 Monday, January 21st to Friday, March 29th
Hours
 Live lecture hours
 20
 Recorded lecture hours
 0
 Total advised study hours
 80
Timetable
 Wednesdays
 12:05  12:55
 Thursdays
 13:05  13:55
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 specific requirements. Standard undergraduate courses in analysis, mathematical methods and partial differential equations are desirable.
Syllabus
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, nondispersive waves, shocks, canonical linear and nonlinear wave equations, integrability and inverse scattering transform (IST), asymptotic and perturbation methods.

Dispersive wave models: derivation techniques and basic properties (4 hours)
 Kortewegde Vries (KdV) and related equations.
 Nonlinear Schrödinger (NLS) equation, and generalizations with applications to modulational instability of periodic wavetrains.
 Resonant interactions of waves (general threewave and fourwave interactions, second harmonic generation, longshort wave resonance; phaseplane analysis).
 Second order models: Boussinesq and sineGordon equations (FermiPastaUlam problem, ZabuskyKruskal model, solitons; FrenkelKontorova model, Bäcklund transformations, kinks and breathers).

Inverse scattering thansform and solitons (6 hours)
 KdV equation (conservation laws, Miura transformation, Lax pair, discrete and continuous spectrum of the timeindependent Schrödinger operator, direct and inverse scattering problems, initialvalue problem by the inverse scattering transform. Reflectionless potentials and Nsoliton 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, Nsoliton solutions).
 Perturbed and higherorder KdV equations (effects of inhomogeneity, asymptotic integrability, Gardner equation).

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. RankineHugoniot conditions. Lax entropy condition.
 Structure of the viscous shock wave, Burgers equation, ColeHopf transformation, Taylor's shock profile, Nwave.

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. GurevichPitaevskii problem.
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 ModulationsAn 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 FrenkelKontorova 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
Prof Roger Grimshaw
 University
 Loughborough University
 Role
 Main contact

Professor Gennady El
 University
 Northumbria University

Dr Karima Khusnutdinova
 University
 Loughborough University
Bibliography
Follow the link for a book to take you to the relevant Google Book Search page
You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library'  this sometimes works well, but not always  you will need to enter your location, but it will be saved after you do that for the first time.
 Solitons and the Inverse Scattering Transform (Ablowitz and Segur, book)
 Solitons and Nonlinear Wave Equations (Dodd, Eilbeck, Gibbon and Morris, book)
 Theory of Solitons: The Inverse Scattering Method (Novikov, book)
 Solitons in mathematics and physics (Newell, book)
 Solitons: an introduction (Drazin and Johnson, book)
 Nonlinear science: emergence and dynamics of coherent structures (Scott, book)
 Nonlinear Periodic Waves and Their Modulations: An Introductory Course (Kamchatnov, book)
 Optical Solitons from Fibers to Photonic Crystals: From Fibers to Photonic Crystals (Kivshar and Agrawal, book)
 The FrenkelKontorova model: concepts, methods, and applications (Braun and Kivshar, book)
 Nonlinear waves in fluids: recent advances and modern applications (Grimshaw, book)
 Solitary waves in fluids (Grimshaw, book)
 Waves in Fluids (James Lighthill, book)
 Linear and Nonlinear Waves (Whitham, book)
Assessment
Description
Assessment for this course will be via a single takehome paper with 2 weeks to complete and submit online. To pass the exam one is required to complete at least 3 out of 4 questions and to obtain at least 50
Assessment not available
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Lectures
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