KL7020 - Nonlinear Waves and Extreme Events

What will I learn on this module?

Wave phenomena appear everywhere in nature, from water waves to magnetic materials, from optics to weather forecasts, hence their description and understanding is of fundamental importance both from the theoretical and the applicative points of view.
You will learn the mathematical theory of nonlinear wave motion. Applications include the understanding of shock waves and coherent structures, such as solitons, of some famous integrable partial differential equations that arise from standard modelling processes and the mechanisms of formation and propagation of anomalous waves such as tsunamis and rogue waves. The module is designed to give you a flavour of modern research in this actively developing area of applied mathematics.

Topics will include theory of linear dispersive waves (wave propagation, elements of Fourier analysis, modulated waves), nonlinear hyperbolic waves and integrable nonlinear wave equations and their applications.

How will I learn on this module?

You will learn through a series of lectures and seminars including problem-solving sessions, classroom discussions and presentations. Workshop will be scheduled at regular intervals to allow exploration of the theoretical background to the techniques covered in the lectures as well as attempt the practical analysis of selected problems. Lectures allow you to witness the development of the relevant theoretical aspects behind nonlinear wave phenomena and understand how to apply the techniques and interpret the results through many examples.

Formative feedback is available in the classes as you get to grips with new techniques and solve problems. In addition, we operate an open door policy where you can meet with your module tutor to seek further advice or help if required. Your ability to use the relevant theory to identify and evaluate solutions to set problems is assessed via a coursework and a closed-book exam at the end of the semester.

General feedback on the exam will be given in a specially-arranged feedback session at the end of semester and individual feedback will be provided. An opportunity to discuss work further will be avilable on an individual basis when work is returned and also through the open door policy.

How will I be supported academically on this module?

Direct contact with the module team during the lectures and seminars will involve participation in both general class discussions as well as one to one discussions during the problem-solving workshops. This gives you a chance to get immediate feedback pertinent to your particular needs in this session. Further feedback and discussion with the module team are also available at any time through our open-door policy. In addition, all teaching materials, selected computer programmes and supplementary material (such as relevant research articles) are available through the e-learning portal.

What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk
(Reading List service online guide for academic staff this containing contact details for the Reading List team – http://library.northumbria.ac.uk/readinglists)

What will I be expected to achieve?

Knowledge & Understanding:
1. Analyse one dimensional hyperbolic nonlinear wave equations in terms of simple waves and discontinuities

2. Solve selected one dimensional nonlinear dispersive wave propagation problems using methods of soliton theory.

3. Apply nonlinear wave theory to a range of environmental and technological problems.

Intellectual / Professional skills & abilities:
3. Develop efficient solutions for advanced problems in nonlinear wave theory.

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
4. Manage your own learning, through knowledge of available reading sources, including advanced texts and research papers.
5. Effectively and concisely communicate complex ideas in the theory of nonlinear waves in written form.

How will I be assessed?

Coursework (30%) – 1,3
(Assignment with set questions and problems - wordcount: max 1000 words + derivations + graphs + plots + tables)

Examination (70%) – 1, 2, 3, 4, 5
(3-hour closed book examination with set questions and problems - wordcount: max 2000 words + derivations + graphs + plots + tables)

FORMATIVE – 1, 2, 3, 4, 5
Formative assessment will be available during problem-solving workshops through normal lecturer-student interactions and discussions around the set questions, allowing students to extend, consolidate and evaluate their knowledge.

Formative feedback will be provided on student work and errors in understanding will be addressed reactively using individual discussion. Solutions to problems will be provided after the students have attempted the questions, allowing students to receive feedback on the correctness of their solutions and to seek help if matters are still not clear.





Module abstract

Wave phenomena appears everywhere in the natural world, from water waves to magnetic materials, from optics to weather forecasts, hence their description and understanding is of fundamental importance both from the theoretical and the applicative point of view. ‘Nonlinear Waves and Extreme Events’ is designed to introduce you to the mathematical theory of nonlinear wave motion. The waves of small amplitude are mathematically described by linear partial differential equations (PDEs) but the propagation of sufficiently strong disturbances is modelled by nonlinear PDEs. Applications of nonlinear wave theory range from traffic flow to extreme events in the ocean such as tsunamis or rogue waves. One of the most interesting types of nonlinear waves are solitons---the ``parcels’’ of matter or energy propagating without distortion over large distances. These and other waves are often described by solutions of certain `integrable’ nonlinear PDEs arising via standard mathematical modelling techniques.

You will learn through a combination of lectures and seminar sessions. Lectures give a formal introduction to theoretical aspects while the seminars offer the opportunity to deepen the knowledge by applying the theory to problems coming from physics, biology, chemistry and engineering. Workshops will be an opportunity to address open research problems; they will often address topics with links beyond the discipline, thus also strengthening your transferable skills and employability.

The module is assessed with a coursework and a formal closed-book examination, which will cover all aspects of the module and will assess your problem solving abilities when applied to new and unseen problems.

Course info

Credits 20

Level of Study Postgraduate

Mode of Study 1 year Full Time

Department Mathematics, Physics and Electrical Engineering

Location Coach Lane Campus, Northumbria University

City Newcastle

Start September 2024

Fee Information

Module Information

All information is accurate at the time of sharing. 

Full time Courses are primarily delivered via on-campus face to face learning but could include elements of online learning. Most courses run as planned and as promoted on our website and via our marketing materials, but if there are any substantial changes (as determined by the Competition and Markets Authority) to a course or there is the potential that course may be withdrawn, we will notify all affected applicants as soon as possible with advice and guidance regarding their options. It is also important to be aware that optional modules listed on course pages may be subject to change depending on uptake numbers each year.  

Contact time is subject to increase or decrease in line with possible restrictions imposed by the government or the University in the interest of maintaining the health and safety and wellbeing of students, staff, and visitors if this is deemed necessary in future.


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