Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

A geometric approach to stationary defect solutions in one space dimension

Analysis and Stochastic

Localised excitations in long Josephson junctions with phase-shifts with time-varying drive

In this project, we consider a variety of ac-driven, inhomogeneous sine-Gordon equations describing an infinitely long Josephson junctions with phase shifts, driven by a microwave field. First, the case of a small driving amplitude and a driving frequency close to the natural (defect) frequency is considered. We construct a perturbative expansion for the breathing mode to obtain equations for the slow time evolution of the oscillation amplitude. We show that, in the absence of an ac-drive, a breathing mode oscillation decays with a rate of at least \mathcal{O}(t^{-1/4}) and \mathcal{O}(t^{-1/2}) for 0-\pi-0 and 0-\kappa junctions, respectively. Multiple scale expansions are used to determine whether, e.g., an external drive can excite the defect mode of a junction (a breathing mode), to switch the junction into a resistive state. Next, we extend the study to the case of large oscillation amplitude with a high frequency drive. Considering the external driving force to be rapidly oscillating, we apply an asymptotic procedure to derive an averaged nonlinear equation, which describes the slowly varying dynamics of the sine-Gordon field. We discuss the threshold distance of 0-\pi-0 junctions and the critical bias current in $0-\kappa$ junctions in the presence of ac drives. Then, we consider a spatially inhomogeneous sine-Gordon equation with two regions in which there is a \pi-phase shift, and a time periodic drive, modelling 0-\pi-0-\pi-0 long Josephson junctions. We discuss the interactions of symmetric and antisymmetric defect modes in long Josephson junctions. We show that the amplitude of the modes decay in time. In particular, exciting the two modes at the same time will increase the decay rate. The decay is due to the energy transfer from the discrete to the continuous spectrum. For a small drive amplitude, there is an energy balance between the energy input given by the external drive and the energy output due to radiative damping experience by the coupled mode. Finally, we consider spatially inhomogeneous coupled sine-Gordon equations with a time periodic drive, modelling stacked long Josephson junctions with a phase shift. We derive coupled amplitude equations considering weak coupling and strong coupling in the absence of ac-drive. Next, by considering the strong coupling with time periodic drive, we expect that the amplitude of oscillation tends to constant for long times

Localised excitations in long Josephson junctions with phase-shifts with time-varying drive

In this project, we consider a variety of ac-driven, inhomogeneous sine-Gordon equations describing an infinitely long Josephson junctions with phase shifts, driven by a microwave field. First, the case of a small driving amplitude and a driving frequency close to the natural (defect) frequency is considered. We construct a perturbative expansion for the breathing mode to obtain equations for the slow time evolution of the oscillation amplitude. We show that, in the absence of an ac-drive, a breathing mode oscillation decays with a rate of at least \mathcal{O}(t^{-1/4}) and \mathcal{O}(t^{-1/2}) for 0-\pi-0 and 0-\kappa junctions, respectively. Multiple scale expansions are used to determine whether, e.g., an external drive can excite the defect mode of a junction (a breathing mode), to switch the junction into a resistive state. Next, we extend the study to the case of large oscillation amplitude with a high frequency drive. Considering the external driving force to be rapidly oscillating, we apply an asymptotic procedure to derive an averaged nonlinear equation, which describes the slowly varying dynamics of the sine-Gordon field. We discuss the threshold distance of 0-\pi-0 junctions and the critical bias current in $0-\kappa$ junctions in the presence of ac drives. Then, we consider a spatially inhomogeneous sine-Gordon equation with two regions in which there is a \pi-phase shift, and a time periodic drive, modelling 0-\pi-0-\pi-0 long Josephson junctions. We discuss the interactions of symmetric and antisymmetric defect modes in long Josephson junctions. We show that the amplitude of the modes decay in time. In particular, exciting the two modes at the same time will increase the decay rate. The decay is due to the energy transfer from the discrete to the continuous spectrum. For a small drive amplitude, there is an energy balance between the energy input given by the external drive and the energy output due to radiative damping experience by the coupled mode. Finally, we consider spatially inhomogeneous coupled sine-Gordon equations with a time periodic drive, modelling stacked long Josephson junctions with a phase shift. We derive coupled amplitude equations considering weak coupling and strong coupling in the absence of ac-drive. Next, by considering the strong coupling with time periodic drive, we expect that the amplitude of oscillation tends to constant for long times