I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation:
∂tU−∂xV+[U,V]=0
where
U=U(x,t,λ) and
V=V(x,t,λ) are matrix-valued functions and
λ is a parameter. The motivation behind this question is that the Lax pairs for the KdV equation:
ut+6uux−uxxx=0
is given by:
U=(01λ+u0) and V=(ux4λ−2u4λ2+2λu+uxx−2u2−ux)
Now, it is not too difficult to verify that this indeed satisfies the zero-curvature representation, but I'm trying to figure out why we cannot use the Lax pairs:
U=(00λ+u0) and V=(00λ+3u2−uxx0)
These matrices clearly satisfy the zero-curvature representation, but for some reason none of the notes I've been reading use them. What is the reason that they are not a valid Lax pair for the KdV equation?
I've also asked this question here (I hope that's ok):
http://mathoverflow.net/questions/194986/integrability-conditions-of-lax-pairs