Announcements


These lectures will begin on Wednesday February 1, and then continue as scheduled.
Due to late start, extra lectures have been timetabled for Fridays, MAGIC weeks 6 - 9, as follows:
24 Feb (13.00 - 14.00)
2 March (13.00 - 14.00)
9 March (13.00 - 15.00) - 2 lectures
16 March (13.00 - 15.00) - 2 lectures.
There will be no lectures in MAGIC week 10.

Forum

General


This course is part of the MAGIC core.

Description

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

Spring 2012 (Monday, January 16 to Friday, March 23)

Timetable
  • Wed 12:05 - 12:55
  • Thu 13:05 - 13:55

Lecturers


Roger Grimshaw (main contact)
Email R.H.J.Grimshaw@lboro.ac.uk
Phone (01509) 223480
Interests Fluid dynamics and nonlinear waves
vcard
Photo of Roger Grimshaw
Gennady El
Email G.El@lboro.ac.uk
Phone (01509) 222869
Interests Nonlinear waves, solitons, fluid dynamics
vcard
Photo of Gennady El
Karima Khusnutdinova
Email K.Khusnutdinova@lboro.ac.uk
Phone (01509) 228202
Interests Nonlinear waves, continuum mechanics
vcard
Photo of Karima Khusnutdinova


Students


Photo of Azwani  Alias
Azwani Alias
(Loughborough)
Photo of Gokcen Cekic
Gokcen Cekic
(Birmingham)
Photo of Neslihan Delice
Neslihan Delice
(Leeds)
Photo of Aiman Elragig
Aiman Elragig
(Exeter)
Photo of Mark Esson
Mark Esson
(Birmingham)
Photo of Maxwell Fennelly
Maxwell Fennelly
(Southampton)
Photo of William   Haese-Hill
William Haese-Hill
(Loughborough)
Photo of Esen hanac
Esen hanac
(Birmingham)
Photo of Julian Mak
Julian Mak
(Leeds)
Photo of Thorpe Maria
Thorpe Maria
(Manchester)
Photo of Charlotte Page
Charlotte Page
(East Anglia)
Photo of Pearce Philip
Pearce Philip
(Manchester)
Photo of Harvind Rai
Harvind Rai
(Birmingham)
Photo of Greg Roddick
Greg Roddick
(Loughborough)
Photo of Ilia Roustemoglou
Ilia Roustemoglou
(Loughborough)
Photo of Jonathan SIMMONS
Jonathan SIMMONS
(Birmingham)
Photo of Jonathan Stone
Jonathan Stone
(Southampton)
Photo of Wei King  Tiong
Wei King Tiong
(Loughborough)
Photo of Daniel Wacks
Daniel Wacks
(Newcastle)
Photo of Derek Watson
Derek Watson
(Southampton)
Photo of Francis Watson
Francis Watson
(Manchester)
Photo of Xizheng Zhang
Xizheng Zhang
(Loughborough)


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
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, description in elliptic functions).
    • 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, 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.
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.

Bibliography


Linear and Nonlinear WavesWhitham
Solitons and the Inverse Scattering TransformAblowitz and Segur
Solitons and Nonlinear Wave EquationsDodd, Eilbeck, Gibbon and Morris
Theory of Solitons: The Inverse Scattering MethodNovikov
Solitons in mathematics and physicsNewell
Solitons: an introductionDrazin and Johnson
Nonlinear science: emergence and dynamics of coherent structuresScott
Nonlinear Periodic Waves and Their Modulations: An Introductory CourseKamchatnov
Optical Solitons from Fibers to Photonic Crystals: From Fibers to Photonic CrystalsKivshar and Agrawal
The Frenkel-Kontorova model: concepts, methods, and applicationsBraun and Kivshar
Nonlinear waves in fluids: recent advances and modern applicationsGrimshaw
Solitary waves in fluidsGrimshaw
Waves in FluidsJames Lighthill


Note:

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Assessment


Assessment for this course will be through a single take-home exam. The exam paper will be put online towards the end of the course (23 March). The deadline for completion will be 22 April. To pass the exam one is required to complete at least 4 out of 6 questions and to obtain at least 50%.

Assignments


MAGIC021 EXAM 2012

Deadline: Monday 23 April 2012 (914.0 days ago)
Instructions:This is the exam. You are required to do at least 4 of the 6 problems. Each problem will be marked out of 10, and the maximum mark that can be obtained is 40. A pass requires at least a mark of 20, that is 50 per cent. The problems need to submitted online on or before April 23 2012.