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Evolution of the EPDiff Equation

Patterns and Simulations

By Martin Staley and Darryl Holm
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In the natural world, spectacular examples of singular wavefronts are seen in the form of the transbasin oceanic internal waves, whose characteristic feature is that their strongly nonlinear wavefronts reconnect when two collide transversely. The EPDiff equation has been proposed as a model for this phenomenon allowing for elastic collisions between singular wavefronts (solitons) whose momentum is distributed along curves moving in the plane. A series of 2d and 3d numerical simulations demonstrate the rich dynamics of solutions of the EPDiff equation in various scenarios.
Author Institutional AffiliationLANL (Staley) and Imperial College (Holm)
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KeywordsPatterns, wavefronts, epdiff

Documents to download

  • Evolution of parallel initial velocity profiles(.jpg, 70.57 KB) - 3888 download(s) Evolution of two parallel initial peakon distributions in the 2d EPDiff equation.
  • Evolution of star configuration(.jpg, 98.72 KB) - 3885 download(s) These frames show the evolution of an initial star-shaped velocity profile under the 2d EPDiff equation.
  • Evolution of 2d In-Out Configuration(.jpg, 90.45 KB) - 3898 download(s) Evolution of an initial Gaussian ring of width alpha in speed, with a distribution of initial velocities having motion inwards along the positive diagonal and outward along the negative diagonal. Outward motion breaks into a sequence of curved peakon segments, while the inward motion produces peakon segments which undergo head-on collisions and therefore annihiliate, re-create, and re-emerge moving along the positive diagonal.

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