|
|
| (9 intermediate revisions by 2 users 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><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'').
| |
|
| |
|
| | [[Category:Integrability]] |
| | [[Category:Airy]] |
| [[Category:Equations]] | | [[Category:Equations]] |
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.
