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.