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

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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 />< | + | <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

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