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.

See also a movie of this collision:    SolitonCollision  (thanks, Ryan!)

 

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:

 

Multiple locked ILMs can also be generated by judiciously selecting the driver's frequency.  Click on the link to see a movie clip of experimental data showing a couple  stationary ILM locked to the driving signal. The frequency of oscillation here is at around 300 MHz.

Hopping ILMs

To find out more, see the paper:   L.Q. English, R. Basu Thakur, R. Stearrett, "Patterns of Traveling Intrinsic Localized Modes in a driven electrical lattice", Phys. Rev. E 77, 066601 (2008).