Linear and Nonlinear Waves - MA691

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Module delivery information

Location Term Level1 Credits (ECTS)2 Current Convenor3 2020 to 2021
Autumn 6 15 (7.5) PROF P Clarkson checkmark-circle


Linear PDEs. Dispersion relations. Review of d'Alembert’s solutions of the wave equation. Review of Fourier transforms for solving linear diffusion equations.

Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.

Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.

General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.

Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).


This module appears in the following module collections.

Contact hours

38 hours

Method of assessment

80% examination and 20% coursework.

Indicative reading

M.J. Ablowitz, Nonlinear Dispersive Waves, Cambridge (2011)
J. Bellingham and A.C. King, Wave Motion, Cambridge (2000)
P.G. Drazin and R.S. Johnson, Solitons: an Introduction, Cambridge (1989)
R. Knobel, An Introduction to the Mathematical Theory of Waves, A.M.S. (2000)
J.D Logan, An Introduction to Partial Differential Equations, Wiley (1994)
I.N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill (1957)

See the library reading list for this module (Canterbury)

Learning outcomes

The intended subject specific learning outcomes. On successfully completing the level 6 module students will be able to:
1 demonstrate knowledge and critical understanding of the well-established principles within linear and nonlinear partial differential equations (PDEs);
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of the material in the following areas: Fourier transforms for linear differential equations, shock waves, exact solutions of nonlinear PDEs;
3 apply the concepts and principles in PDEs in well-defined contexts beyond those in which they were first studied, showing the ability to evaluate critically the appropriateness of different tools and techniques;
4 make appropriate use of MAPLE.

The intended generic learning outcomes. On successfully completing the level 6 module students will be able to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make competent use of information technology skills such as online resources (Moodle);
7 demonstrate an increased level of skill in numeracy and computation.


  1. Credit level 6. Higher level module usually taken in Stage 3 of an undergraduate degree.
  2. ECTS credits are recognised throughout the EU and allow you to transfer credit easily from one university to another.
  3. The named convenor is the convenor for the current academic session.
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