MAGIC021: Nonlinear Waves

Course details

A core MAGIC course


Spring 2021
Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th


Live lecture hours
Recorded lecture hours
Total advised study hours


12:05 - 12:55
13:05 - 13:55

Course forum

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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 in simple settings, 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)
    • Fermi-Pasta-Ulam (FPU) problem, Zabusky-Kruskal model and Boussinesq equation, derivation of the Korteweg - de Vries (KdV) equation, travelling waves, phase-plane analysis, solitons and cnoidal waves. 
    •  Frenkel-Kontorova model, sine-Gordon equation, travelling waves, phase-plane analysis, Bäcklund transformations, kinks and breathers. 
    •  Nonlinear Schrödinger (NLS) equation, derivation, focusing and defocusing, criterion of modulational instability, bright and dark solitons, breathers. 
    •  Resonant wave interactions (three-wave and four-wave interactions, second harmonic generation, long-short wave resonance). Phase-plane analysis for travelling waves (three-wave interactions). 
  3. Inverse scattering transform (IST) and applications (4 hours)
    • KdV equation: 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 (scheme). Reflectionless potentials and N-soliton solutions. Example: delta-function initial condition. Infinity of conservation laws. Hamiltonian structures. KdV hierarchy. 
    •  AKNS scheme, linear problem, inverse scattering transform (scheme) for the focusing NLS equation, N-soliton solutions. 
    •  Near-integrable equations: perturbed and higher-order KdV equations (waves in variable environment), asymptotic integrability, Gardner equation. 
  4. Nonlinear hyperbolic waves and classical shocks (5 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. Dispersive hydrodynamics and modulation theory (5 hours)
    •  Dispersive hydrodynamics: an overview. 
    •  Whitham's method of slow modulations (linear modulated waves, 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. Resolution of an initial discontinuity for the KdV equation. Gurevich-Pitaevskii problem. 
    •  Integrable turbulence and soliton gas. 

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,

 [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] Braun, O.M., Kivshar, Y.S. 2004 
The Frenkel-Kontorova model. Concepts, methods, and applications. Springer, Berlin. 

 [10] Grimshaw, R. (ed.). 2005 
Nonlinear Waves in Fluids: Recent Advances and Modern Applications. CISM Courses and Lectures, No. 483, Springer, Wien, New York. 

 [11] Grimshaw, R. (ed.) 2007 
Solitary Waves in Fluids. Advances in Fluid Mechanics, Vol 47, WIT Press, UK. 

 [12] Ablowitz, M.J. 2011 
Nonlinear Dispersive Waves. Cambridge University Press, UK. 


  • Professor Gennady El

    Professor Gennady El

    Northumbria University
    Main contact
  • Dr Karima Khusnutdinova

    Dr Karima Khusnutdinova

    Loughborough University


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Assessment for this course will be via a single take-home 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, each worth 10 marks, and to obtain at least 50% (i.e. at least 15 marks). If all 4 questions are answered, the 3 best marks will be included. 

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