I'm quite new to Wolfram Mathematica. I need to create an animation of a Burgers Hopf equation's solution which becomes multivalued (somewhat of a breaking wave). Is it possible to do so in Mathematica?

All I seem to obtain is a very sharp series of oscillations which I assume are a numerical correction by the software.

Any help will be appreciated.

  {D[u[x, t], t] + 1/2 D[u[x, t]^2, x] == 0, 
   u[x, 0] == Sech[x]^2, u[-5, t] == u[5, t]}, 
  u[x, t], {x, -5, 5}, {t, 0, 5}]`
  • 1
    $\begingroup$ There are examples of plots, even dynamic with Manipulate, on NDSolve documentation page. Have you seen them? $\endgroup$ – Kuba Aug 6 '18 at 12:41
  • $\begingroup$ i have seen them but nothing seems to create the animation i need, which is a c- shaped wave $\endgroup$ – ddot Aug 6 '18 at 12:54
  • $\begingroup$ I don't think you can get c-shaped wave, nor wiki page shows them. But you create a steep one with your solution. $\endgroup$ – Kuba Aug 6 '18 at 12:58
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    $\begingroup$ What do you mean by c-shaped wave? Can you be more specific? $\endgroup$ – xzczd Aug 6 '18 at 13:04
  • $\begingroup$ i mean an animation of a gaussian like wave in which the upper portion goes faster tham the lower one, so it creates sort of a breaking wave, but kuba confirmed my suspects, it's probably not possible to obtain in Mathematica $\endgroup$ – ddot Aug 6 '18 at 13:12

I don't think NDSolve is the right tool for the task, because it can only solve for single-valued functions. (Well, is there any mathematical software that can deal with multi-valued functions natively? ) Fortunately, your goal is just creating an animation for the multi-valued solution, then we can simply refer to the analytic solution of this problem:

Animate[ParametricPlot[{t f[ξ] + ξ, f[ξ]}, {ξ, -5, 5}, 
  AspectRatio -> 1/GoldenRatio], {t, 0, 5}]

enter image description here

  • $\begingroup$ thank you very much :) $\endgroup$ – ddot Aug 6 '18 at 15:40

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