MATH5175 is a special topic in the Applied Mathematics course for honours and postgraduate coursework students.  The Topic title is Calculus of Variations

Units of credit: 6

Cycle of offering: Topics rotate; Term 2, 2024

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

More information:  The Course outline will be made available closer to the start of term - please visit this website: https://www.unsw.edu.au/course-outlines

Important additional information as of 2023

UNSW Plagiarism Policy

The University requires all students to be aware of its policy on plagiarism.

For courses convened by the School of Mathematics and Statistics no assistance using generative AI software is allowed unless specifically referred to in the individual assessment tasks.

If its use is detected in the no assistance case, it will be regarded as serious academic misconduct and subject to the standard penalties, which may include 00FL, suspension and exclusion.

The Online Handbook entry contains information about the course. 

If you are currently enrolled in MATH5175, you can log into UNSW Moodle for this course.

Course overview

The discovery of the Inverse Scattering Transform (IST) method and its application to the celebrated Korteweg-de Vries (KdV) equation in 1967 by Gardner, Green, Kruskal and Miura is regarded as one of the most important developments in mathematical physics in the past 50 years. It followed the numerical discovery in 1965 by Kruskal and Zabusky of a groundbreaking nonlinear phenomenon, namely the soliton interaction. The IST method is a nonlinear analogue of the Fourier transform method for privileged systems of nonlinear partial differential equations which are known as soliton equations or integrable systems. The latter term refers to the fact that these systems have properties which generalise those of classical integrable systems in the sense of classical Hamiltonian mechanics.

Remarkably, in the past 50 years, it has been demonstrated that the applicability of integrable systems is not confined to the original physical setting of the important Fermi-Pasta-Ulam problem. Indeed, the ubiquitous nature of integrable systems is reflected in their (apparent and disguised) presence in a wide range of both mathematical fields (e.g. partial differential and difference equations, differential and algebraic geometry, Galois theory, representation theory, special functions, quantum and cluster algebras, number theory, Nevanlinna theory in complex analysis) and physical fields (e.g. general relativity, twistor theory, field theory, (continuum) mechanics, nonlinear optics, Josephson junctions, Bose-Einstein condensates, biophysics, surface and water waves, plasma physics).

By means of specific examples, this course serves as a gentle introduction to the world of integrable systems. Topics may include:

  • Classical Liouville-Arnold integrability of ODEs based on Hamiltonian/Lagrangian mechanics, symmetries, conservation laws (Noether’s theorem).
  • Integrability of infinite-dimensional systems: PDEs, soliton theory, Lax pairs, Bäcklund  transformations, (differential) geometric aspects.
  • Basic theory of (linear) difference equations.
  • Discrete (nonlinear) integrable systems: (algebraic) geometry and features of integrable maps and partial difference equations, discrete Painlevé equations, integrability detectors.
  • Applications in mathematics and physics.