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Miura transformation

  1. Aug 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Show how the Miura transformation u = -(v2 + vx) connects the KdV ut + 6uux + uxxx = 0 to the modified KdV vt - 6v2vx + vxxx = 0
    Which of the following is true, and why?
    a) u solves the KdV if v solves the MKdV
    b) v solves the MKdV if u solves the KdV
    c) u solves the KdV if and only if v solves the MKdV.

    2. Relevant equations

    3. The attempt at a solution
    Let u = -(v2 + vx) therefore ut + 6uux + uxxx = vt + 6(-(v2 + vx))(2v+d/dx) + vxxx = 0, I'm lost after this bit, any ideas?
    I know a) is true because of application to the Miura transformation.
  2. jcsd
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