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Question

I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as:

$ X = \begin{pmatrix} \lambda & i u\\ - i u & - \lambda\\ \end{pmatrix}$

I have been told that $T_{22} = - T_{11}$, so let T:

$ T = \begin{pmatrix} a & b\\ c & - a\\ \end{pmatrix}$

Attempted solution.

So the using the compatibility condition $ \frac{\partial X}{\partial t} -\frac{\partial T}{\partial x} + [X,T] =0$ we get the following three equations.

$-a_x+i u (b+c) = 0$

$-2 i a u+2 b \lambda -b_x+i u_t = 0$

$-c_x-i \left(2 a u-2 i c \lambda +u_t\right) = 0$

However when I try to solve this I keep getting contradictions. Am I on the right path and if not can someone please help me understand how to find $T$.

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  • $\begingroup$ Thinking about this would this be better suited for math.stackexchange.com ? $\endgroup$ – User Sep 18 '14 at 18:28
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    $\begingroup$ A Lax pair is a pair of operators $P, L$ (here you would take $L=-D^2+u$) so that the equation becomes $\dot{L}=[P,L]$. What you are asking looks more like the zero curvature formulation. $\endgroup$ – Christian Remling Sep 18 '14 at 20:12
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The point is that you should assume that $a,b,c$ are polynomials in $\lambda$ and then equate to zero the coefficients at various powers of $\lambda$. Similar analysis can, I think, be found in many books, see e.g. Solitons in mathematics and physics by Alan Newell.

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