| Abstract: |
| Exploring the configuration space of solutions to a complex nonlinear
dynamical system requires the use of continuation and bifurcation techniques.
In particular, continuation methods are numerical algorithmic procedures
for tracing out branches of fixed points/roots to nonlinear (algebraic)
equations as one (or more) of the free parameters of the underlying system
is varied. Among the plethora of continuation methods available, a powerful
continuation technique called the deflated continuation method (DCM) was
recently introduced which itself is capable of finding/constructing undiscovered/disconnected
branches of solutions by eliminating known branches through a penalty technique.
The primary aim of this talk is to apply the DCM to the one- and two-component
Nonlinear Schr\odinger (NLS) models in two spatial dimensions. We will
present novel nonlinear steady states that have not been reported before
in the physics of ultracold atoms as well as discuss bifurcations involving
such states. Finally, we will present recent developments in the one-component
NLS equation in 3D by employing the DCM where the landscape of solutions in
such a higher-dimensional system is far richer. The computation of the associated
spectrum of the solutions (in the realm of linear stability analysis) revealed
challenges in some cases and state-of-the art eigenvalue solvers were employed
and will be discussed if time permits. |
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