Integrable dynamical systems associated with Kac-Moody algebras

PhD thesis, February 1988

## Abstract

In Chapter 2 it is shown how to construct infinitely many conserved quantitites for the classical non-linear Schrodinger equation associated with an arbitrary Hermitian symmetric space G/K. These quantities are non-local in general, but include a series of local quantities as a special case. Their Poisson bracket algebra is studied, and is found to be a realization of the "half" Kac-Moody algebrakx C[l] consisting of polynomials in positive powers of a complex parameterlwhich have coefficients ink(the Lie algebra of K).

In Chapter 3 the construction is extended to provide a realization of the Kac-Moody algebrakx C[l,l^{-1}]. One can then define auxiliary quantities to obtain the full algebragx C[l,l^{-1}]. This leads to the formal linearization of the system.

In Chapter 4 the procedure is generalized so as to enable one to construct realizations of centre-free Kac-Moody algebras as hierarchies of 1+1 dimensional classical dynamical systems. The equations of motion (which are, in general, non-local) have Hamiltonians which form realizations of the same algebra. The Cartan subalgebra provides infinitely many conserved quantities in involution, while a sub-class of the step operators (which may be interpreted as generators of translations in "internal dimensions") enable the systems to be linearized. The system can be regarded as having a "gauge symmetry" which includes momentum.

## Preface

The work in this thesis was carried out in the Theoretical Physics group at the Department of Physics, Imperial College, London, between October 1983 and October 1987 under the supervision of Professor D.I. Olive. Financial support was provided by the SERC from October 1983 until September 1986. Unless otherwise stated, the work is original, and it has not been submitted before for a degree of this or any other university.

The thesis consists of three papers. The first two appeared in Communications in Mathematical Physics108, 631-646 (1987);111, 167-179 (1987), and the third has been submitted for publication (Imperial Preprint TP/87-88/2).

I should like to express my gratitude to Professor Olive for introducing me to so fruitful an area of research, and to thank him for his help.

"of the said to made it has a good road"## Contents

Chapter 1: Introductory remarks

Chapter 2: Local and non-local conserved quantities for generalized non-linear Schrodinger equations

Chapter 3: Kac-Moody symmetry of generalized non-linear Schrodinger equations

Chapter 4: Integrable dynamical systems and Kac-Moody algebras

Chapter 5: Concluding remarks