View source for Modified Korteweg-de Vries equation

The (defocusing) '''modified Korteweg-de Vries (mKdV) equation''' is

<center><math>\partial_t u + \partial_x^3 u = 6 u^2 \partial_x u</math></center>

It is completely <span class="SpellE">integrable</span>, and has infinitely many conserved quantities. Indeed, for each non-negative integer k, there is a conserved quantity which is roughly equivalent to the <span class="SpellE">H^k</span> norm of u. This equation has been studied [[modified Korteweg-de Vries on R|on the line]], [[modified Korteweg-de Vries on T|on the circle]], and [[modified Korteweg-de Vries on R|on the half-line]].

The <span class="SpellE">focussing</span> <span class="SpellE">mKdV</span>

<center><math>\partial_t u + \partial_x^3 u = - 6 u^2 \partial_x u</math></center>

<span class="GramE">is</span> very similar, but admits soliton solutions.

== Miura transform ==

In the defocusing case, the ''Miura transformation'' <math> v = \partial_x u + u^2 </math> transforms a solution of <span class="SpellE">defocussing</span> <span class="SpellE">mKdV</span> to a solution of [#kdv <span class="SpellE">KdV</span>]

<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 <span class="SpellE">mKdV</span> to be one derivative higher than that for <span class="SpellE">KdV</span>.

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> <span class="SpellE">mKdV</span> to ''complex-valued'' <span class="SpellE">KdV</span>, 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>\partial_t v + \partial_x^3 v = 6(a^2 v^2 + bv) \partial_x v</math></center>

<span class="GramE">then</span> u = a^2 v^2 + <span class="SpellE">av_x</span> + <span class="SpellE">bv</span> solves <span class="SpellE">KdV</span> (this is the ''Gardener transform'').

[[Category:Equations]]