Electrical ILMs
1) Solitons
2) ILMs in electrical transmission lines
3) Controlling the motion of these ILMs
1) Solitons in an Electrical Transmission Line
A simpe discrete transmission line consist of a line inductors with capacitors to ground at each junction or node. This circuit acts as a low-pass filter. One can create a band-pass filter by adding another set of inductors to ground at each node of the lattice. This lifts the dispersion curve up and creates a gap in the frequency spectrum.
To add nonlinearity to our electrical lattice, we replace the capacitors with diodes which have effective capacitence when reverse-biased. However, this capacitence is voltage dependent.
The lattice now exhibits both nonlinearity and dispersion and can therefore sustain solitons. The following picture depicts a single soliton being launched from one end of the lattice/tranmission line and propagating without dispersing, collected by my student Ryan Stearrett.
Collision of solitons can also be observed. In that case two solitons are launched from opposite ends of the line.
2) ILMs in this electrical lattice
When the two ends of this lattice are connected together, we create a ring with no boundaries. By driving this ring either at all nodes simultaneously, we can generate intrinsic localized modes. So even though the driving is spatially homogenous, the response is very localized. These localized modes are created via an instability of the uniform mode (modulational instability), but once created these localized modes are stable.
The figure above shows experimental data. In the beginning, the uniform mode can be seen to break up into fingers. Eventually, one traveling ILM is formed. A detailed profile of such an ILM can be seen in the next figure:
3) Controlling the motion of these ILMs
By introducing a temporary impurity into the transmission-line lattice shown above (via inductive coupling to an external inductor), we can force the ILM to hop to a neighboring lattice site. In this way, we can use an impurity to shepherd the localized energy through the lattice. ILMs can also be seeded in this fashion by driving below resonance and momentarily introducing an impurity (of the correct sign).
In the figure below, we demonstrate that the ILM can be made to jump to the neighboring site. The left-most profile (centered at site 16) turns into the dotted (red) trace when an external inductor is placed in the direct vicinity of the L2 inductor at site 17. Upon removal of this external inductor, the ILM persists at site 17 (solid black trace). This process can be repeated to move the ILM further to the right, as shown. All data correspond to experiments.
