v , 
\label{nls-Hop}
\\
\D & =\s\inum( D_x^2 +vD\inv_x\Re\bar v D_x +\inum D_xv D\inv_x\Im\bar v)
\label{nls-Dop}\end{aligned}$$ are mutually compatible Hamiltonian operators, with $\Re$ and $\Im$ viewed as algebraic operators, and where $$\begin{aligned}
D & =\int_{-\infty}^{\infty} |v|^2\,dx , 
\\
H & =\int_{-\infty}^{\infty} \Im(\bar v v_x)\,dx , 
\\
I & =\int_{-\infty}^{\infty} \tfrac{1}{4} |v|^4 - |v_x|^2 \,dx\end{aligned}$$ are the corresponding Hamiltonians.

The two lowest order Hamiltonian operators – yield the hereditary recursion operator $$\label{nls-Rop}
\R=\H\I\inv=\s\inum(D_x+v D\inv_x \Re\bar v) .$$ This operator and all three of the Hamiltonian operators are invariant under $x$-translations applied to $v$, as well as phase rotations applied to $v$, corresponding to the invariance of the NLS equation under infinitesimal $x$-translations and infinitesimal phase rotations, which are represented by the symmetry operators $\X=-v_x\partial_v$ and $\X=\s\inum v\partial_v$. These operators yield the respective flows $$\label{nls-root-flows}
P^{(1)} = -v_x,
\quad
P^{(0)} = \s\inum v$$ which are related by the NLS recursion operator : $$\label{nls-root-flow-rel}
P^{(1)} = \R P^{(0)} .$$ The NLS equation corresponds to a higher flow $$v_t=-\R({-v_x})=-\R^2(\s\inum v)= \s\inum(\tfrac{1}{2}|v|^2 v +v_{xx})$$ where the square of the recursion operator is given by $$\label{nls-Rsqop}
\R^2=-\D\I\inv=-(D_x^2+D_xv D\inv_x \Re\bar v +\inum v D\inv_x \Im\bar v D_x) .$$ Note that the reduction of this operator under the 