Module MATH3231: Solitons III

Department: Mathematical Sciences

MATH3231: Solitons III

Prerequisites

• Calculus I (MATH1061) OR Calculus I (Maths Hons) (MATH1081) OR Single Mathematics B (MATH1571).

Corequisites

• None.

Excluded Combination of Modules

• None.

Aims

• To provide an introduction to solvable problems in nonlinear partial differential equations which have a physical application.
• To expose students to an area of comparatively recent development which is still the subject of active research.

Content

• Dispersion and dissipation in linear wave equations.
• Nonlinear wave equations.
• Travelling wave solutions.
• Topological lumps and the Bogomolnyi bound.
• Conservation laws in integrable systems.
• Backlund transformations for the sine-Gordon
• equation.
• Hirota's method.
• The Lax formalism.
• The inverse scattering method.
• The KdV hierarchy and conservation laws.
• Finite-dimensional integrable systems and Toda equations.

Learning Outcomes

• 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.
• travelling wave solutions.
• Backlund transformations for the sine-Gordon equation.
• conservation laws in integrable systems.
• Hirota's method.
• the Lax formalism.
• the inverse scattering method.
• the KdV hierarchy and conservation laws.
• finite-dimensional integrable systems.

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling, spatial awareness.

• Students will have enhanced problem solving and abstract reasoning 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

Summative Assessment

Formative Assessment:

Eight assignments to be submitted.

■ 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