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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 ''complexvalued'' <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 realvalued 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'').
 
   
 +  [[Category:Integrability]] 
 +  [[Category:Airy]] 
 [[Category:Equations]]   [[Category:Equations]] 
Latest revision as of 07:41, 31 July 2006
The (defocusing) modified Kortewegde Vries (mKdV) equation is
$\partial _{t}u+\partial _{x}^{3}u=6u^{2}\partial _{x}u$
It is completely integrable, and has infinitely many conserved quantities. Indeed, for each nonnegative 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 halfline.
The focussing mKdV
$\partial _{t}u+\partial _{x}^{3}u=6u^{2}\partial _{x}u$
is very similar, but admits soliton solutions.
The modified KdV equation is related to the KdV equation via the Miura transform.