# Advanced Mathematical Methods 1: Integrable Systems and Solitons

*20 credits / 30 lectures*

The discovery of the Inverse Scattering Transform and classes of completely integrable nonlinear evolution equations solvable using this technique, is regarded as one of the most important developments in mathematical physics of the twentieth century. Due to a hidden symmetry, the nonlinear integrable system is amenable to comprehensive analysis that would be typical only of linear equations. Integrable equations are mathematically exceptional; however they arise in a broad variety of physical contexts. A special role in the integrable dynamics is played by solitons, localised waves that travel maintaining their shape and speed. Mathematically, solitons dominate asymptotic behavior of generic initial conditions. Physically, they represent stable self-regulating pulses of energy that are observed in nonlinear dispersive systems. Examples include surface waves in shallow water and internal waves in the sea; optical pulses in fibres used in telecommunication lines and spatial solitons utilised in optical instrumentation; ion-acoustic solitons in plasma; magnetic fluxons in superconducting Josephson junctions; kinks in charge-density wave conductors, and so on.

- Solitons without Inverse Scattering: the method of Hirota
- Integration of the KdV using the Inverse Scattering Method
- The direct and inverse scattering problem for the stationary Schroedinger equation on the line
- Reflectionless potentials and exact N-soliton solutions
- Conservation laws of the KdV
- KdV as a Hamiltonian system. Complete Integrability of the KdV
- Inverse scattering for the Dirac equation: the nonlinear Schroedinger and sine-Gordon equations
- The IST-based perturbation theory
- The Darboux and Baecklund transformations
- The Riemann-Hilbert problem, 2D Toda lattices and chiral fields

Prerequisites:

The undergraduate Applied Mathematics modules- Differential Equations (2OD),
- Boundary-Value Problems (2BP),
- Nonlinear Dynamics (2ND),
- Complex Variables (3CV), and,
- Mathematical Physics (3MP)