MATH41420: Solitons

• Prior knowledge of calculus and linear algebra at undergraduate level.

• 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.
• Students will have an ability to read independently to acquire knowledge and understanding in related areas.

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 complex and specialised problems.

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

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