Durham University
Programme and Module Handbook

Undergraduate Programme and Module Handbook 2005-2006 (archived)

Module MATH3231: SOLITONS III

Department: MATHEMATICAL SCIENCES

MATH3231: SOLITONS III

Type Open Level 3 Credits 20 Availability Available in 2006/07 and alternate years thereafter Module Cap None. Location Durham

Prerequisites

  • Complex Analysis II (MATH2011), Linear Algebra II (MATH2021) and Analysis in Many Variables II (MATH2031); alternatively, Contours and Symmetries II (MATH2111) or Contours and Hyperbolic Geometry II (MATH2**1), Linear Algebra (MATH2021) and Analysis in Many Variables II (MATH2031).

Corequisites

  • None.

Excluded Combination of Modules

  • Solitons IV (MATH4121).

Aims

  • To provide an introduction to solvable problems in nonlinear partial differential equations which have a physical application.
  • This is an area of comparatively recent development which still possesses potential for growth.

Content

  • Nonlinear wave equations.
  • Progressive wave solutions.
  • Baklund transformations for Sine Gordon equation.
  • Baklund transformations for KdV equation.
  • Conservation laws in integrable systems.
  • Hirota's method.
  • The Nonlinear Schrodinger equation.
  • The inverse scattering method.
  • The inverse scattering method: two component equations.
  • Toda equations.
  • Integrability.

Learning Outcomes

Subject-specific Knowledge:
  • By the end of the module students will: be able to solve novel and/or complex problems in Solitons.
  • have a systematic and coherent understanding of theoretical mathematics in the field of Solitons.
  • have acquired coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Nonlinear wave equations.
  • Progressive wave solutions.
  • Backlund transformations for the sine-Gordon equation and the KdV equation.
  • Conservation laws in integrable systems.
  • Hirota's method.
  • The nonlinear Schrodinger equation.
Subject-specific Skills:
  • In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling, spatial awareness.
Key Skills:

    Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

    • Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
    • Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
    • Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
    • The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

    Teaching Methods and Learning Hours

    Activity Number Frequency Duration Total/Hours
    Lectures 40 2 per week for 19 weeks and 2 in term 3 1 Hour 40
    Preparation and Reading 160
    Total 200

    Summative Assessment

    Component: Examination Component Weighting: 100%
    Element Length / duration Element Weighting Resit Opportunity
    three hour written examination 100%

    Formative Assessment:

    Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

    ■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University