Announcements


The lectures are on Wednesdays 12 and Thursdays at 1. For the firstvtwo weeks, the lectures will be given from the node at Exeter

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 2017 (Monday, January 23 to Friday, March 31)

Timetable

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

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
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.

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 Meshari Alesemi
Meshari Alesemi
(Loughborough)
Photo of Noura Alharthi
Noura Alharthi
(Loughborough)
Photo of Timothy Burchell
Timothy Burchell
(Surrey)
Photo of Matthew Charles
Matthew Charles
(East Anglia)
Photo of Martina Cracco
Martina Cracco
(Cardiff)
Photo of Duc Lam Duong
Duc Lam Duong
(Sussex)
Photo of Rudy Kusdiantara
Rudy Kusdiantara
(*Ext_Assessed)
Photo of Amal Mohammed
Amal Mohammed
(*Ext_Assessed)
Photo of Marianthi Moschou
Marianthi Moschou
(Manchester)
Photo of Junaid Mustafa
Junaid Mustafa
(*Ext_Assessed)
Photo of Chinedu Nwaigwe
Chinedu Nwaigwe
(*External)
Photo of Joseph Oloo
Joseph Oloo
(Keele)
Photo of Anthony Williams
Anthony Williams
(Manchester)
Photo of Guangxiu Zhao
Guangxiu Zhao
(Loughborough)


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 via a single take-home paper in April 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

MAGIC021 Exam 2017

Files:Exam paper
Released: Monday 24 April 2017 (5.0 days ago)
Deadline: Sunday 7 May 2017 (9.0 days to go)


Recorded Lectures


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