Difference between revisions of "Modified Korteweg-de Vries equation"

From DispersiveWiki
(Miura transform)
(One intermediate revision by the same user not shown)
Line 11: Line 11:
<span class="GramE">is</span> very similar, but admits soliton solutions.
<span class="GramE">is</span> very similar, but admits soliton solutions.
== Miura transform ==
The modified KdV equation is related to the [[KdV]] equation via the [[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><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'').

Latest revision as of 07:41, 31 July 2006

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

It is completely integrable, and has infinitely many conserved quantities. Indeed, for each non-negative integer k, there is a conserved quantity which is roughly equivalent to the H^k norm of u. This equation has been studied on the line, on the circle, and on the half-line.

The focussing mKdV

is very similar, but admits soliton solutions.

The modified KdV equation is related to the KdV equation via the Miura transform.