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.
A repeating lattice is used, so a blip reappears on the left when it crosses the right boundary. There is a broad low "col" wherever a tall, faster blip overtakes a small, slower one; see the four-blip thumbnail. Blips reappear with the same shape and direction, but their tracks are displaced laterally, as if an attractive force exists between them. Each blip maintains direction through many interactions. The multiple interactions at the top of the thumbnail disprove the suggestion that interacting solitons do not cross but "exchange identities".
Use the controls to select different plot types (which involve progressively more computation) and starting conditions. The first case is a single blip that retains its shape; it is followed by two sets of points that split into two and four blips (with some residual noise). Other cases are solitonic solutions to various equations. They split into multiple blips.
The final (squarewave) case shows a series of blips developing from the flat section. These attain steady heights and directions when they separate. Just as in Fourier and wavelet analysis, "blip analysis" appears to characterize different shapes. Is there an analogous "blip synthesis"?
Blips are unusual, being very restricted in width. They decrease sharply to zero, unlike compactons (even powers of sinusoidal pulses such as the cos squared case) that decay smoothly to zero at and have Gaussian pulses as the high power limit.
Partial differentiations with respect to the time variable and the space variables , , ... can be expressed using subscripts as and .
Solitons are localized disturbances (solitary waves) that survive interactions: when two solitons meet they appear to pass through each other and recover their shapes and directions, though their tracks are displaced sideways. Conservative solitons are solutions to "conservative nonlinear PDEs" that have several conserved properties. One definition of solitons requires that solutions to their equations can be found by "inverse scattering". Does this apply to these equations?
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.