The Structure of Multipulse Solitons in the Generalised Nonlinear Schrödinger Equation

Abstract

In this thesis, we perform a detailed study of the generalised nonlinear Schrödinger equation (GNLSE) with quartic dispersion and show that it supports infinitely many multipulse solitons for a wide parameter range of the dispersion terms. These solitons exist through the balance between the quartic and quadratic dispersions with the Kerr nonlinearity, and they come in infinite families with different signatures. We consider a stationary wave solution ansatz, where the optical pulse does not undergo a change in shape while propagating. This allows us to transform the quartic GNLSE into a fourth-order nonlinear ordinary differential equation (ODE), which we use to find bi-asymptotic trajectories, known as homoclinic solutions, corresponding to solitons. We take advantage of the mathematical properties of reversibility and existence of a Hamiltonian of the ODE to show that there exist infinite families of symmetric and non-symmetric homoclinic solutions to the origin. Furthermore, connections between equilibrium solutions and periodic solutions (EtoP connections) are organizing centres for the existence of homoclinic solutions of the ODE. Therefore, by finding connections between the origin and different periodic solutions, we are able to find families of homoclinic solutions. Specifically, we make use of continuation algorithms for two-point boundary value problems to compute a representative number of such homoclinic solutions. Due to the vital role that periodic solutions of the ODE play in the organization of homoclinic solutions, we also investigate the periodic solution structure of the ODE. We show that each new homoclinic solution generates infinitely many periodic solutions, which are organised as surfaces in phase space, as parameterised by the Hamiltonian energy. The geometry of different surfaces changes as a single parameter is varied via degenerate bifurcations of periodic solutions. Since there are infinitely many periodic solutions in the ODE, we also find connections between different periodic solutions, which are referred to as PtoP connections. These PtoP connections together with EtoP connections organise an incredible zoo of homoclinic solutions to the origin, which corresponds to solitons of the GNLSE. We also briefly investigate the stability of these solitons by integrating a perturbation of them as solutions of the GNLSE. This suggests that some of these solitons may be observable experimentally in photonic crystal wave-guides over several dispersion lengths.