5M: Solitons MATHS5053

  • Academic Session: 2018-19
  • School: School of Mathematics and Statistics
  • Credits: 20
  • Level: Level 5 (SCQF level 11)
  • Typically Offered: Either Semester 1 or Semester 2
  • Available to Visiting Students: No
  • Available to Erasmus Students: No

Short Description

The aim of introduce the main ideas, both physical and mathematical, underlying the theory of solitons - solutions to non-linear partial differential equations.

Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

Requirements of Entry

Mandatory Entry Requirements

 

Recommended Entry Requirements

Assessment

Assessment

 

100% Final Exam

 

 

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

Course Aims

The aim of the course is to introduce the main ideas, both physical and mathematical, underlying the theory of solitons - exact solutions to non-linear partial differential equations.

 

The topics cover will in include a review of linear and nonlinear wave theory including scattering properties, transform methods for linear PDEs, properties of 1- and 2-soliton solutions of the Korteweg-de Vries (KdV) equation, solving initial value problems for the KdV equation using the inverse scattering transform and a general approach using Lax pairs, finding solutions by Darboux and Bcklund transformations and Hirota's method, Painleve equations and discrete integrable systems. Hamiltonian properties of the KdV equation, together with the construction of infinite families of conservation laws and commuting flows, will also be covered.

Intended Learning Outcomes of Course

By the end of the course students will be able to:

 

i) highlight the major differences between linear and nonlinear waves and the special features of solitons,

 

ii) calculate the phase shift as arising from soliton interaction for KdV-like equations,

 

iii) use the inverse scattering transform as applied to the KdV and related equations and to construct solutions in simple situations,

 

iv) compute the compatibility condition of a Lax pair and construct a Lax pair for a particular equation,

 

v) use Darboux and Bcklund transformations and Hirota's method to construct exact solutions,

 

vi) state the Painleve property and apply the Painleve test to particular equations,

 

vii) find solutions of discrete integrable system using Darboux and Backlund transformations and Hirota's method

 

viii) take continuum limits from discrete integrable systems to continuous ones,

 

ix) construct hierarchies of commuting flows from Lax equations,

 

x) compute the Hamiltonian structure of the KdV hierarchy.

Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.