A study of finite gap solutions to the nonlinear Schrödinger equation

The vector nonlinear Schrödinger equation is an envelope equation which models the propagation of ultra-short light pulses and continuous-wave beams along optical fibres. Previous work has focused almost entirely on soliton solutions to the equation using a Lax representation originally developed by Manakov. We prove recursion formulae for the family of higher-order nonlinear Schrödinger equations, along with its associated Lax hierarchy, before investigating finite gap solutions using an algebrogeometric approach which introduces Baker-Akhiezer functions defined upon the Riemann surface of the relevant spectral curve. We extend this approach to account for solutions of arbitrary genus and compare it with an alternative method describing solutions of genus two. The scalar nonlinear Schrödinger and Heisenberg ferromagnet equations were shown to be equivalent following work by Lakshmanan; we generalise this idea by introducing the Heisenberg ferromagnet hierarchy and show it is entirely gauge equivalent to the scalar nonlinear Schrödinger hierarchy in the attractive case. We also investigate the polarisation state evolution of general solutions to the vector nonlinear Schrödinger equation and study possible degenerations to the Heisenberg ferromagnet equation