Numerical solution of the non-linear Schroedinger equation : the half-line problem and dynamical systems and bifurcations of vector fields

2014-02-17T15:08:35Z (GMT) by Maria S. Zakynthinaki
Solutions to the nonlinear Schrodinger equation with potential V(u) = -λulul2 have been theoretically and numerically calculated, revealing the formation of solitons. In this study the finite element method with linear basis functions, distinguished for its simplicity and effective applicability, is considered and a predictor-corrector scheme is applied to simulate the propagation in time. Numerical experiments include the propagation of a single soliton form, a two-soliton collision, as well as the formation of more than one solitons from non-soliton initial data. The important problem of boundary reflections has been successfully overcome by the implementation of absorbing boundaries, a method that in practice achieves a gradual reduction of the wave amplitude at the end of each time step. The second part of this work deals with dynamical systems of the form [see file]. The dynamics of such systems near their equilibrium point depends strongly on the adjustable parameter μ, as it is possible for the system to lose its hyperbolicity and a bifurcation to occur. After reviewing aspects of linearisation, the prospect of change in the equilibrium solutions has been studied, both for flows and maps, in terms of the eigenvalues of the linearised system. In the study of steady-state bifurcation, elements of saddle-node, transcritical, pitchfork, as well as period-doubling bifurcation are considered. Finally, the case when equilibrium solutions persist, known as Hopf bifurcation, has also been included.