# [SOLVED]KdV modified and complex modified from AKNS

#### dwsmith

##### Well-known member
Let $$L_R = \begin{pmatrix} \partial - 2q\partial^{-1}r & 2q\partial^{-1}q\\ -2r\partial^{-1}r & -\partial + 2r\partial^{-1}q \end{pmatrix}$$.
We know that
$\begin{pmatrix} q\\ -r \end{pmatrix}_t = -iL_R^n \begin{pmatrix} q\\ r \end{pmatrix}$
When $$n = 2$$, the RHS is
$- \begin{pmatrix} q_{xx} & -2q^2r\\ r_{xx} & -2r^2q \end{pmatrix}$
For $$n = 3$$, the RHS is
$i \begin{pmatrix} q_{xxx} - 2q\partial^{-1}rq_{xx} + 2q\partial^{-1}qr_{xx} & -2(q^2)_xr - 2q^2r_x\\ -r_{xxx} - 2r\partial^{-1}rq_{xx} + 2r\partial^{-1}qr_{xx} & 2(r^2)_xq + 2r^2q_x \end{pmatrix}$
Thus,
$q_t = i[q_{xxx} - 6qq_xr].$
I am supposed to be able to obtain the modified and complex modified KdV equations by the substitutions $$r = -q$$ and $$r = -q^*$$, respectively. However, I get
\begin{align}
q_t - iq_{xxx} - 6iq^2q_x &= 0\\
q_t - iq_{xxx} - 6i\lvert q\rvert^2q_x &= 0
\end{align}
but the equations should be
\begin{align}
q_t + q_{xxx} + 6q^2q_x &= 0\\
q_t + q_{xxx} + 6\lvert q\rvert^2q_x &= 0
\end{align}
Where should I have picked up another multiple of $$i$$?

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