# NONLINEAR WAVE EQUATIONS - 2025/6

Module code: MATM027

## Module Overview

The module is an introduction to nonlinear partial differential equations (PDEs) with a focus on hyperbolic and dispersive PDEs. The module takes key classes of equations as the organising centre. Each class of PDEs is considered and the properties, analytical techniques, and analysis of each is taken in turn.

### Module provider

Mathematics & Physics

CHENG Bin (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Semester 2

None.

## Module content

The content of the module will include analysis of some nonlinear wave equations as classical examples of nonlinear PDEs. It starts with linear theory since, upon linearisation, normal mode analysis and Fourier transform can be suitably applied. Techniques in this module also include linear and nonlinear theories covered in a standard module on Ordinary Differential Equations.

Scalar conservation law. Introduction to shock waves. Jump conditions. Regularization with dispersion (KdV) and dissipation (Burgers equation).

Systems of hyperbolic conservation laws. Role of constant solutions, and criticality. Conservation laws. Characteristics and Riemann invariants. Reduction techniques. Example: shallow water equations, and hydraulic jumps.

Korteweg-DeVries equation. Linear analysis of the dispersion relation, phase and group velocity. Travelling wave solutions such as solitons. Conservation laws.

Semi-linear wave equations. Linear wave equations. Periodic and solitary travelling waves of nonlinear problem. Conservation laws and energy inequalities. Weakly nonlinear normal mode.

Nonlinear Schrodinger equation. Dispersion relation, phase and group velocities (linear). Plane wave solutions of nonlinear problem: existence and stability. Bright and dark solitary wave solutions and their stability. Conservation laws. Singularity formation.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 mins) 20
Examination End-of-Semester Examination (2 hrs) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of subject knowledge through recall of key definitions, formulae and derivations using the language of nonlinear PDE theory.

• Understanding of the connection and difference between nonlinear and linear systems encountered in the examples, both theoretically and phenomenologically.

• Ability to apply this knowledge and these critical analysis skills through the solution of unseen problems in the test and examination.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 and 2.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 3.

Formative assessment
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback
Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal feedback given in lectures. Students also receive verbal feedback in office hours.

## Module aims

• The main aim of this module is to introduce the study of nonlinear PDEs. For definiteness the module is restricted to wave equations: that is, hyperbolic or dispersive PDEs. This module will extend the student¿s pre-existing knowledge of linear wave equations to encompass nonlinear wave equations.

## Learning outcomes

 Attributes Developed 001 Students will be able to demonstrate knowledge and understanding of the definition and classification of hyperbolic and dispersion linear and nonlinear PDEs. KC 002 Students will interpret, apply and extend basic concepts and theorems in differential equations and analysis to the nonlinear regime. KC 003 Students will be able to develop theory and analysis for nonlinear PDEs and apply these to a range of examples, using the theory developed in the module. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A detailed introduction to techniques and analysis of nonlinear wave equations, extending the concepts of linear PDEs and ODEs to the nonlinear context.

• Experience (through demonstration) of the methods and techniques used to solve problems in hydrodynamic stability.

The learning and teaching methods include:

• Three one-hour lectures per week for eleven weeks, with supplementary notes for topics of significant difficulty or special interest to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.

• There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

• Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MATM027

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM027 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: Through the module, students cultivate advanced problem-solving skills applicable and valued across diverse industries, such as engineering, finance, and research.

Global and Cultural Capabilities: Student engagement in discussions during lectures naturally cultivates the sharing of the different cultures from which the students originate.

Resourcefulness and Resilience: MATM027 fosters resourcefulness and resilience by immersing students in intricate problem-solving scenarios. Dealing with problems involving the unpredictability of nonlinear systems hones adaptability and perseverance.

Sustainability: The skills gained in addressing systems involving complex dynamics is vital in fields such as climate science and renewable energy, enabling innovative solutions for a more sustainable future. For example, understanding ocean flows aids in predicting environmental changes. Likewise, understanding systems of traffic flow aids in informing sustainable practices.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.