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.