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In the defocusing case, the ''Miura transformation'' <math> v = \partial_x u + u^2 </math> transforms a solution of defocussing [[mKdV]] to a solution of [[KdV]] <center><math>\partial_t v + \partial_x^3 v = 6 v \partial_x v</math>.</center> Thus one expects the LWP and GWP theory for mKdV to be one derivative higher than that for KdV. In the focusing case, the Miura transform is now <math>v = \partial_x u + i u^2</math>. This transforms <span class="SpellE">focussing</span> mKdV to ''complex-valued'' KdV, which is a slightly less tractable equation. (However, the transformed solution v is still real in the highest order term, so in principle the real-valued theory carries over to this case. The Miura transformation can be generalized. If v and w solve the system <center><math>\partial_t v + \partial_x^3 v = 6(v^2 + w) \partial_x v</math><br /><math>\partial_t w + \partial_x^3 w = 6(v^2 + w) \partial_x w</math></center> Then <math>u = v^2 + \partial_x v + w </math> is a solution of <span class="SpellE">KdV</span>. In particular, if a and b are constants and v solves <center><math>\partial_t v + \partial_x^3 v = 6(a^2 v^2 + bv) \partial_x v</math></center> <span class="GramE">then</span> <math>u = a^2 v^2 + a \partial_x v + bv</math> solves <span class="SpellE">KdV</span> (this is the ''Gardener transform''). [[Category:Integrability]] [[Category:Airy]] [[Category:Transforms]]

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