Undergraduate Programme and Module Handbook 2010-2011 (archived)

# Module MATH4121: SOLITONS IV

## Department: Mathematical Sciences

### MATH4121: SOLITONS IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2010/11 and alternate years thereafter | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Complex Analysis II (MATH2011) AND Linear Algebra II (MATH2021) AND Analysis in Many Variables II (MATH2031); OR Contours and Symmetries II (MATH2111) AND Linear Algebra (MATH2021) AND Analysis in Many Variables II (MATH2031); OR Contours and Hyperbolic Geometry II (MATH2121) AND Linear Algebra (MATH2021) AND Analysis in Many Variables II (MATH2031); OR Contours and Actuarial Mathematics II (MATH2171) AND Linear Algebra (MATH2021) AND Analysis in Many Variables II (MATH2031); OR Contours and Probability II (MATH2**1) AND Linear Algebra (MATH2021) AND Analysis in Many Variables II (MATH2031). In addition, a minimum of 40 credits of Mathematics modules at Level 3.

#### Corequisites

- None.

#### Excluded Combination of Modules

- Solitons III (MATH3231).

#### 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.
- Backlund transformations for Sine Gordon equation.
- Backlund transformations for KdV equation.
- Conservation laws integrable systems.
- Hirota's method.
- The Nonlinear Schrodinger equation.
- The inverse scattering method.
- The inverse scattering method: two component equations.
- Toda equations.
- Integrability.
- (NB the syllabus is identical to SOLITONS III (A) which is taught in parallel).

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will:
- be able to solve complex, unpredictable and specialised problems in Solitons.
- have an understanding of specialised and complex theoretical mathematics in the field of Solitons.
- have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas:
- Nonlinear wave equations.
- Progressive wave equations.
- 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 highly specialised and advanced mathematical skills in the following areas: 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 complex and specialised problems.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 40 | 2 per week | 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 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