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[SOLVED] KdV modified and complex modified from AKNS

dwsmith

Well-known member
Feb 1, 2012
1,673
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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