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

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The <span class="SpellE">focussing</span> <span class="SpellE">mKdV</span>
The <span class="SpellE">focussing</span> <span class="SpellE">mKdV</span>
<center><span class="SpellE">u_t</span> + <span class="SpellE">u_xxx</span> = - 6 u^2 <span class="SpellE">u_x</span></center>
<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.
<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'' v = <span class="SpellE">u_x</span> + u^2 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><span class="SpellE">v_t</span> + <span class="SpellE">v_xxx</span> = 6 v <span class="SpellE">v_x</span>.</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 v = <span class="SpellE">u_x</span> + <span class="SpellE">i</span> u^2. 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><span class="SpellE">v_t</span> + <span class="SpellE">v_xxx</span> = 6(v^2 + w) <span class="SpellE">v_x</span><br /><span class="SpellE">w_t</span> + <span class="SpellE">w_xxx</span> = 6(v^2 + w) <span class="SpellE">w_x</span></center>
Then u = v^2 + <span class="SpellE">v_x</span> + w is a solution of <span class="SpellE">KdV</span>. In particular, if a and b are constants and v solves
<center><span class="SpellE">v_t</span> + <span class="SpellE">v_xxx</span> = 6(a^2 v^2 + <span class="SpellE"><span class="GramE">bv</span></span>) <span class="SpellE">v_x</span></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'').

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.