Modified Korteweg-de Vries equation

From DispersiveWiki
Revision as of 19:31, 28 July 2006 by Wangk (talk | contribs) (Miura transform)

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.

Miura transform

In the defocusing case, the Miura transformation 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

Then is a solution of KdV. In particular, if a and b are constants and v solves

\partial_t v + \partial_x^3 v = 6(a^2 v^2 + bv) \partial_x v</math>

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