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# Department of Mathematical Sciences

# MATH3231/4121 Solitons III/IV

Solitons occur as solutions of certain special nonlinear partial differential equations. In general, nonlinear equations cannot be handled analytically. However, soliton equations (which arise naturally in mathematical physics) have beautiful properties; in particular, they admit solutions that one can write down explicitly in terms of simple functions. Such solutions include solitons: localized lumps which can move around and interact with other lumps without changing their shape. The course will describe a number of these soliton equations, explore their solutions, and study solution-generating techniques such as Bäcklund transformations and inverse scattering theory.

## Outline of Course

Aim: To provide an introduction to solvable problems in nonlinear partial differential equations which have a physical application. This is an area of comparatively recent development which still possesses potential for growth.

### Term 1

• Nonlinear Wave Equations: Historical introduction; John Scott Russell and the first experimental observation of a solitary wave (1834); Korteweg and de Vries' wave equation (1905). Soliton scattering. Properties of nonlinear wave equations; dispersion, dissipation: dispersion law for linear equations; group velocity and phase velocity. Examples.
• Travelling Wave Solutions: Derivation of the sine-Gordon equation as the limit of the motion under gravity of a set of rigid pendulums suspended from a torsion wire.
• Travelling wave solutions: D'Alembert's solution recalled, and travelling wave solutions of KdV and sine-Gordon equations.
• Topological lumps and the Bogomolnyi argument: sine-Gordon and ϕ⁴ examples.
• Conservation Laws in Integrable Systems: Conservation laws for the wave equation, sine-Gordon and KdV equations. Relation to conservation of soliton number and momentum in scattering.
• Bäcklund Transformations for Sine-Gordon Equation: The Liouville equation and its solution by Bäcklund transformations of solutions of the free field equation. Generation of multisoliton solutions of sine-Gordon by Bäcklund transformations. Theorem of permutability. Two-soliton and breather solutions of sine-Gordon. Lorentz-invariant properties. Asymptotic limits of two-soliton solutions. Discussion of scattering.
• Bäcklund Transformations for KdV Equation: The same analysis repeated for the KdV equation.
• Hirota's Method: Hirota's method for multisoliton solutions of the wave and KdV equations.

### Term 2

• The Inverse Scattering Method: Discussion of the initial value problem; motivation for the inverse scattering method. The inverse scattering method for the KdV equation. Lax pairs. Discussion of spectrum of Schrödinger operator for potential; Bargmann potentials. Scattering theory; asymptotic states, reflection and transmission coefficients, bound states. Marchenko equations. Multisoliton solutions of KdV.
• Integrability: Hamiltonian structures for the KdV equation. Integrability. Hierarchies of equations determined by conservation laws.
• Toda equations: The Toda molecule and the Toda chain. Conservation laws for the Toda molecule.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook: MATH3231, MATH4121.