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

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(Miura transform)
(Miura transform)
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The Miura transformation can be generalized. If v and w solve the system
 
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 /><span class="SpellE">w_t</span> + <span class="SpellE">w_xxx</span> = 6(v^2 + w) <span class="SpellE">w_x</span></center>
+
<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 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
 
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

Revision as of 19:29, 28 July 2006

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t u + \partial_x^3 u = 6 u^2 \partial_x u}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t u + \partial_x^3 u = - 6 u^2 \partial_x u}

is very similar, but admits soliton solutions.

Miura transform

In the defocusing case, the Miura transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v = \partial_x u + u^2 } transforms a solution of defocussing mKdV to a solution of [#kdv KdV]

.

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 . This transforms focussing 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


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t w + \partial_x^3 w = 6(v^2 + w) \partial_x w}

Then u = v^2 + v_x + w is a solution of KdV. In particular, if a and b are constants and v solves

v_t + v_xxx = 6(a^2 v^2 + bv) v_x

then u = a^2 v^2 + av_x + bv solves KdV (this is the Gardener transform).