# NONLINEAR WAVE EQUATIONS - 2019/0

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

CHENG Bin (Maths)

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

Independent Learning Hours: 117

Lecture Hours: 33

Semester 1

## Module content

The content of the module will involve 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 learned in the previous module "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
Examination EXAMINATION 80
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 20

NA

## Assessment Strategy

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

• Understanding of fundamental concepts and ability to develop and apply them to a new context.

• Subject knowledge through recall of key definitions, formulae and derivations.

• Analytical ability through the solution of unseen problems in the test and examination.

Thus, the summative assessment for this module consists of:

• One two hour examination at the end of the semester, worth 80% of the overall module mark

• A in-semester test worth 20%

Formative assessment and feedback

Students receive written feedback via the marked in-semester test. The solutions to the in-semester test are also reviewed in the lecture. Two un-assessed courseworks are also given to the students for submission, and complete solutions to these are also provided. In addition, verbal feedback is provided during lectures and office hours.

## Module aims

• The main aim of this lecture course is to introduce the study of nonlinear PDEs.  For definiteness the module is restricted to wave equations; that is, hyperbolic or dispersive PDEs.  The students will have studied linear wave equations in detail in MAT2011 and this module will extend the student's knowledge to nonlinear wave equations.

## Learning outcomes

 Attributes Developed 1 Demonstrate understanding of the definition and classification of hyperbolic and dispersion linear and nonlinear PDEs. K 2 Interpret, apply and extend basic concepts and theorems in differential equations and analysis to the nonlinear regime. KCT 3 Develop theory and analysis for nonlinear PDEs and apply them 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 ideas learned in linear PDEs and ODEs, and other related modules, to the nonlinear context.

• Experience (through demonstration) of the methods used to interpret, understand and solve problems in partial differential equations.

The learning and teaching methods include:

• 3 x 1 hour lectures per week for 11 weeks,

• Supplementary notes for topics of significant difficulty or special interest

• Q+A opportunites for students

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.