# Blip Solitons

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Integration of two simultaneous partial differential equations (PDEs) converts initial disturbances (the points on the blue line) into stable solitonic "blips" (the red points) with speeds related to heights. One, two, or more distinct blips are generated from different initial conditions. The filled red line attempts quadratic interpolation of the red points, but sometimes disappears or becomes negative because the points are on a spiky function. Both equations give similar results. They are related to a third "modulated Gaussian" equation that is shown but that requires further research.

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Contributed by: Roger Beresford (August 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Solitonic behavior often arises when a conservative functional relationship such as is solved. The two-blip equations have , , where is an arbitrary multiplier; this simplifies after differentiation to . As both equations conserve the same properties (differing only by the scaling factor ) they can be adjusted to give similar results at high values of . They were discovered as a stable variation on a third equation that was expected to have solitonic "modulated Gaussian" pulses. (This equation shows the development of modulations in the last snapshot).

The integration uses a very simple "forward Euler" procedure. Equispaced values of the initial function are labeled , , , , and . The value of after one time step is stored in , using and so on for the derivatives; , , , and are then cycled left and a new value (allowing for wraparound) found for . At the end of a cycle is replaced by ; a copy is saved at intervals for later output. The integration step length is , and must be kept small to keep the integration stable. (It has been optimized to avoid instability when demonstrating the longest integrations.)

This leads to the expressions that are built into the function Euler5. It should be noted that the two related equations are the result of changing the parametric multipliers within the functionals and , and not in their derivatives. Changing a multiplier for either or would completely alter the results.

For more information, see the related web pages on the soliton, Euler model, compacton, and wavelet.

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