Some classical integrable systems with topological solitons

Abstract

This thesis is concerned with some low dimensional non-linear systems of partial differential equations and their solutions. The systems are all in the classical domain and aside from a version of one model in Appendix D, are continuous. To begin with we examine the field equations of motion derived from Hamiltonian and Lagrangian densities, respectively defining the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models, where the metric on the target manifold is indefinite. The models are integrable in the sense that a suitable Lax pair exists, and admit solitonic solutions classifiable by an integer winding number. Such solutions are explicitly derived in both the static and time dependent cases where physical space X is the circle S(^1). The existence of travelling wave solutions of topological type is discussed for each model with X = S(^1) and X = R; explicit solutions are derived for the X = S(^1) case and it is shown for both the Heisenberg and sigma models, that no such travelling wave solutions exist if X is the real line. Nevertheless, time dependent solutions (not of travelling wave type) are possible in each case for X = R, some examples of which are derived explicitly. A further integrable system; the Hyperbolic 'Pivotal' model is proposed as a special case of a more general model on Hermitian symmetric spaces. Of particular interest is the fact that the Pivotal model interpolates between the previous two models. To begin with the integrability of the model is established via a Lax representation. Solutions analogous to some of those of the previous models are then derived and the interpolative limits examined with respect to the Heisenberg and sigma models. Conserved currents for the model are also briefly discussed. Finally, some conclusions and further possibilities are noted including a brief examination of a discrete version of the sigma model where the target manifold is positive definite. A Bogomol'nyi bound is shown to exist for the systems energy in terms of a well defined winding number.