9722

Travelling Pulses (Wave Packets)

Various travelling pulses (or wave packets) are displayed; they have similar central shapes but different decay rates. They are given different velocities, so they separate when the time, , is not zero. The Gaussian curve (with velocity zero) is always shown; the amplitudes of the others can be 0, .5, or 1. The rising arm is labelled if the amplitude is 1.
The "rational pulse" has a square-law decay with an effectively infinite range. The "power sinusoid" approaches the Gaussian curve as ; (using 3.56 in place of ) can be seen to provide a close approximation in the central region (though it is a train of pulses separated by "almost zero" regions—try , ). The and pulses are "single soliton" solutions to the MKdV and KdV equations; they are the limiting cases of the and pulse trains (the parameter has been adjusted to give suitable periods). Their "skirts" are wider than the rapidly decaying (short range) Gaussian pulse, but they have a limited range as the decay is exponential.

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DETAILS

These pulses are solutions to nonlinear partial differential equations that will be developed in other Demonstrations. The and pulses are constant amplitude solitonic solutions to the Modified-Korteweg–deVries equation; and are solutions to the original Korteweg–deVries equation. These equations include nonlinearities that cancel the dispersive effect of the change of velocity with frequency (so the pulses do not become lower and wider with time). Different amplitude solitons have different velocities, "passing through" each other with a "change of phase" (not shown in this Demonstration). The Schrödinger equation has Gaussian wave packet solutions that do become lower and wider as time passes—it describes "information about particles". Presumably there is an undiscovered equation that describes sets of stable particles as multi-dimensional wave packets.
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