# The non-commutativ KdV equations

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The non-commutativ KdV equations are

$u_t +u_{xxx}-3(u u_x+ u_x u)=0$

They can be written by a pair of Lax :

$L(u)=-\partial^2 +u$

$A(u)=4 \partial^3-6u \partial -3 u_x$

$\frac {\partial L}{\partial t}=[L,A]$

Is it an integrable system? For example, $u=a +bi +cj +dk$ has values in the Hamilton numbers $R [i,j,k]$; it is then equivalent to a set of four coupled real differential equations.

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