MATH3231: Solitons III

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

• None.

• None.

• 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.

• 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.

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.
• 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.

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:

• Students will have enhanced problem solving and abstract reasoning skills.

• 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.

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