% File: integr2.tex
% Section: INT
% Title: Integrable Hierarchies II
% Last modified: 08.07.2002
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\begin{center}
\large\textbf{INTEGRABLE HIERARCHIES \ II}
\end{center}

\vspace{.0cm}

\begin{center}
\large\textbf{Adler-Kostant-Symes scheme}
\end{center}

\vspace{.1cm}

The Lax representation, which underlies classical integrability, can in many
cases be naturally derived by the use of two other fundamental concepts:
classical $r$-matrix and Kirillov bracket. When taken together, these two
tools form the Adler-Kostant-Symes scheme.

Let $\mathfrak{g}$ be a Lie algebra, $C^k_{ij}$ its structure constants,
$\mathfrak{g}^*$ its dual. A linear operator $R\in\text{End}\mathfrak{g}$ \,
defines a second Lie structure on $\mathfrak{g}$ if a new bracket
\begin{equation}
\label{R}
[A,B]_R \doteq [RA,B] + [A,RB]\,, \ \ \ \ \ \
\tilde{C}^k_{ij} = R^m_i\,C^k_{mj} - R^m_j\,C^k_{mi}
\end{equation}
satisfies the Jacobi identity. This requirement is equivalent to
\begin{equation}
\label{GYBE}
[A\,,\,[RB,RC] - R\,[B,C]_R\,] + \text{cycle}\,(A,B,C) = 0 \ \ \ \ \ \ \ \
\forall \,A,B,C\in\mathfrak{g}\,.
\end{equation}
Thus, $R$ plays here the role of a classical $r$-matrix. The following two
subcases, called \emph{classical Yang-Baxter equation},
\begin{equation}
\label{CYBE}
[RA,RB] - R\,[A,B]_R = 0\,,
\end{equation}
and \emph{modified Yang-Baxter equation},
\begin{equation}
\label{MYBE}
[RA,RB] - R\,[A,B]_R \,\sim\, [A,B]\,,
\end{equation}
evidently obey the general condition (\ref{GYBE})\,.

The very existence of such $R$ results in the Lax-type dynamical
equations on $\mathfrak{g}^*$. Clearly, the two Lie structures on
$\mathfrak{g}$ induce two versions of the Kirillov brackets on
$\mathfrak{g}^*$. We are interested in the Hamiltonian dynamics on
$\mathfrak{g}^*$ generated by the second bracket. To begin with, let us
express $(\text{ad}_R)^*_A$ \, in terms of $\text{ad}^*$\,:
\begin{multline}
\label{adR}
<\!(\text{ad}_R)^*_A\,\omega\,,B\!> \ \doteq \ <\!\omega\,,[A,B]_R\!> \
= \ <\!\omega\,,[RA,B] + [A,RB]\!> \ = \ <\!\text{ad}^*_{RA}\,\omega\,,B\!> \\
  + <\!\text{ad}^*_A\,\omega\,,RB\!> \
 = \ <\!(\text{ad}^*_{RA} + R\,{}^*\!\circ\text{ad}^*_A)\,\omega\,,B\!>
\ \ \ \Longrightarrow \ \ \
(\text{ad}_R)^*_A = \text{ad}^*_{RA} + R\,{}^*\!\circ\text{ad}^*_A
\end{multline}
Our next observation is that $\text{Ad}^*$-\,invariant functions $H(\omega)$
on $\mathfrak{g}^*$ (they are defined by $H(\text{Ad}^*_g\,\omega)=H(\omega)$
and obey $\text{ad}^*_{H'(\omega)}\omega=0$) are central under the
first Kirillov bracket,
\begin{equation}
\label{main1}
\{H,F\}(\omega) \doteq \ <\!\omega\,,[H'(\omega),F'(\omega)]\!> \
 = \ <\!\text{ad}^*_{H'(\omega)}\omega\,,F'\!> \ = 0\,,
\end{equation}
and are in involution under the second:
\begin{equation}
\label{main2}
\{H_1,H_2\}_R(\omega) = \ <\!\omega\,,[H'_1,H'_2]_R\!> \
 = \ <\!\text{ad}^*_{H'_1(\omega)}\omega\,,RH'_2\!>
 - <\!\text{ad}^*_{H'_2(\omega)}\omega\,,RH'_1\!> \ = 0\,.
\end{equation}
Recalling a general formula for Hamiltonian vector fields produced by the
Kirillov bracket ($\bar{F}(\omega)=\text{ad}^*_{F'(\omega)}\omega$)
and using (\ref{adR}), we obtain the (mutually commuting) dynamical flows
\begin{equation}
\label{dyn}
\dot{\omega} \ \doteq \ (\text{ad}_R)^*_{H'(\omega)}\,\omega \
 = \ \text{ad}^*_{RH'(\omega)}\,\omega
\end{equation}
associated with $\text{Ad}^*$-\,invariant functions on $\mathfrak{g}^*$.
In the matrix group case, where both $\text{ad}$ and $\text{ad}^*$ are
commutators, this is the Lax representation.

With $\text{Ad}^*$-\,invariant $H(\omega)$\,, one can examine
$H'(\omega)$ in more detail. Let $\omega=\text{Ad}^*_g\,\omega_0$\,:
\begin{multline}
\notag
<\!\lambda\,,\,H'(\omega)\!> \ \doteq \ <\!dH\,,\,\lambda\!>_\omega \
 = (\lambda H)_\omega = \frac{d}{dt}H(\omega + t\lambda)|_{t=0}
 = \frac{d}{dt}H(\text{Ad}^*_g\,\omega_0 + t\lambda)|_{t=0} \\
= \frac{d}{dt}H(\omega_0 + t\text{Ad}^*_{g^{-1}}\lambda)|_{t=0}
 = \ <\!\text{Ad}^*_{g^{-1}}\lambda\,,\,H'(\omega_0)\!> \
 = \ <\!\lambda\,,\,\text{Ad}_{g^{-1}}H'(\omega_0)\!>\,,
\end{multline}
and we conclude that
\begin{equation}
\label{H'}
H'(\text{Ad}^*_g\,\omega) = \text{Ad}_{g^{-1}}H'(\omega)\,.
\end{equation}
In the matrix case, when \,$<\!\omega\,,A\!>\,\simeq\,\text{tr}(\omega A)\,, \
\text{Ad}_gA=gAg^{-1}, \ \text{Ad}^*_g\omega=g^{-1}\omega g$\,, one has
\begin{equation}
\label{H'2}
H'(g^{-1}\omega\,g) = g^{-1}H'(\omega)\,g\,.
\end{equation}

Surprisingly many integrable systems belong to the above scheme within the
following setting. Let $\mathfrak{g}=\mathfrak{g}_+ + \mathfrak{g}_-$ admit
a decomposition (as a linear space, not as a Lie algebra) into a direct sum
of two subalgebras, and $P_+,P_-$ be the corresponding projectors. Then
\begin{equation}
\label{RA}
R = \frac{1}{2}\,(P_+ - P_-) \ \ \ \ \ \ \ \Longrightarrow \ \ \ \ \ \
[A,B]_R = [A_+,B_+] - [A_-,B_-]
\end{equation}
obeys the modified Yang-Baxter equation (\ref{MYBE})\,:
\begin{multline}
\label{}
[RA,RB] - R\,[A,B]_R
 = \frac{1}{4}\,[A_+ - A_-\,,\,B_+ - B_-] - R\,([A_+,B_+] - [A_-,B_-]) \\
=\frac{1}{4}\,[A_+ - A_-\,,\,B_+ - B_-] - \frac{1}{2}\,([A_+,B_+]+[A_-,B_-])
 = -\frac{1}{4}\,[A,B]
\end{multline}
In this case, dynamical equations (\ref{dyn}) read
\begin{equation}
\label{dyn1}
\dot{\omega} = \text{ad}^*_{H'_+(\omega)}\,\omega
 = -\text{ad}^*_{H'_-(\omega)}\,\omega\ .
\end{equation}

The operator $R$ given by (\ref{RA}) takes on a more familiar form of the
simplest rational $r$-matrix $r(z-w)\sim(z-w)^{-1}$
when dealing with formal distributions. Let
\begin{equation}
\label{distr}
\xi(z) = \sum_n\xi_n z^n\,, \ \ \ \
P_+\xi(z) = \sum_{n\geqslant0}\xi_n z^n\,, \ \ \ \
P_-\xi(z) = \sum_{n<0}\xi_n z^n\,, \ \ \ \
\text{Res}_z\,\xi(z) = \xi_{-1}
\end{equation}
It is readily verified that
\begin{equation}
\label{}
P_-\xi(z) = \text{Res}_w\,\frac{\xi(w)}{(z-w)_w}\,, \ \ \ \ \
P_+\xi(z) = -\text{Res}_w\,\frac{\xi(w)}{(z-w)_z}\,,
\end{equation}
and, of course,
\begin{equation}
\notag
(P_+ + P_-)\,\xi(z) = \text{Res}_w\,(\frac{1}{(z-w)_w}
 - \frac{1}{(z-w)_z})\,\xi(w) = \text{Res}_w\,\delta(z-w)\,\xi(w) = \xi(z)\,.
\end{equation}
In this case,
\begin{equation}
\label{Yang}
R\,\xi(z) = -\frac{1}{2}\,\text{Res}_w\,(\frac{1}{(z-w)_z}
 + \frac{1}{(z-w)_w})\,\xi(w) \ \simeq \ \text{Res}_w\,\frac{\xi(w)}{w-z}
\end{equation}
because in the corresponding Poisson brackets, the way of expanding the
denominator usually does not matter due to the $\mathcal{O}(z-w)$ behavior
of the numerator.

\vspace{.3cm}

\begin{center}
\large\textbf{Matrix Ansatz}
\end{center}

\vspace{.1cm}

Here we illustrate the general scheme described above in terms of the matrix
Ansatz related to AKNS hierarchy. The phase space is formed by matrices $A_i$
which are allowed to depend on infinitely many variables $t_n$\,. Let
\begin{equation}
\label{Ansatz}
A = \sigma + \sum_{i=1}^{\infty}z^{-i} A_i\,, \ \ \ \ \ \ \ \
B_n \doteq (z^n A)_+ \ \ \ \ \ \ \ \ \ (n = 1,2,\ldots)
\end{equation}
Here $\sigma$ is a constant matrix, and $z$ explicitly counts the grade.
For instance, $B_1=z\sigma+A_1$\,, \
$B_2=z^2\sigma+zA_1+A_2$\,. Imposing the dynamical equations as follows,
\begin{equation}
\label{eqs}
\partial_n A = [B_n,A] \ \ \ \ \ \ \ \text{or}
\ \ \ \ \ \ \ [\partial_n - B_n\,,\,A] = 0 \ \ \ \ \ \ \ \ \ \ \ \
(\,\partial_n = \partial_{ t_n})
\end{equation}
we can prove their zero-curvature property (commutativity of the corresponding
flows)
\begin{equation}
\label{zero}
[\partial_m - B_m\,,\,\partial_n - B_n]
 = \partial_n B_m - \partial_m B_n + [B_m,B_n] = 0
\end{equation}
and thus show that (\ref{eqs}) is an integrable hierarchy:
\begin{multline}
\notag
\partial_n(z^m A) - \partial_m(z^n A) = [B_n,z^m A] - [B_m,z^n A]
 = [(z^n A)_+,z^m A] + [(z^m A)_-,z^n A] \\
 = [(z^n A)_+,(z^m A)_+] + [(z^m A)_-,(z^n A)_-]
 = [B_n,B_m] + [(z^m A)_-,(z^n A)_-]\,,
\end{multline}
and now taking the positive parts of both sides produces (\ref{zero})\,.
In addition, combining (\ref{zero}) with (\ref{eqs}) one obtains
\begin{equation}
\label{}
\partial_m B_n = [\partial_n - B_n\,,\,(z^m A)_+]
 = -[\partial_n - B_n\,,\,(z^m A)_-]\,.
\end{equation}
This implies that all flows respect grading: $B_n$ remains a polynomial
in $z$ of degree $\leqslant n$\,.

To connect this with the general geometric scheme of the preceding
section, we use
\begin{equation}
\label{id}
<\,,>\,=\text{Tr}(\ldots)\doteq\text{tr}(\ldots)_0\,, \ \ \ \
\text{Ad}_g\,\xi=g\,\xi g^{-1}\!, \ \ \ \text{Ad}^*_g\,\omega
 = g^{-1}\omega g\,, \ \ \ \text{ad}^*_\xi \omega = -[\xi,\omega]
\end{equation}
and introduce an infinite family of Hamiltonians
\begin{equation}
\label{Ham}
H_n(A) = -\frac{1}{2}\text{Tr}(A\,z^n A) \ \ \ \ \ \ \ \Longrightarrow
\ \ \ \ \ \ H'_n(A) = -z^n A\,, \ \ \ \ \ (H'_n)_+ = -B_n\,.
\end{equation}
They are invariant under $A\rightarrow g^{-1}\!Ag$\,, agree with
(\ref{H'2})\,, and do generate (\ref{eqs}) from (\ref{dyn1})\,.

In terms of $A_i$\,, the main system (\ref{eqs}) reads
\begin{equation}
\label{Ai}
\partial_n A_i = [\sigma\,,A_{n+i}] \
 + \sum_{j=1}^{\min(n,\,i-1)}[A_j\,,A_{n+i-j}]\,.
\end{equation}
Note that so far all $t_n$ (as well as the corresponding flows) are treated
on equal footing, which makes the dynamics on our phase space fairly
symmetric. However, it is a general practice to choose one `non-evolutional'
variable (say, $t_1\equiv x$) leaving all other $t_{n>1}$ as `times'.
In this case, the following equations from (\ref{Ai})\,,
\begin{equation}
\label{}
\partial_1 A_i \equiv \partial_x A_i = [\sigma\,,A_{i+1}] + [A_1\,,A_i]\,,
\end{equation}
do not introduce any time evolution, but rather express $A_{i>1}$
(actually, certain components of it, depending on the properties of $\sigma$)
in terms of $A_1$ and its $x$-derivatives. In fact, we modify the phase space
by including  $\partial^m_x A_1$ instead of $A_{i>1}$\,,
without changing eqs.\,(\ref{Ai}) themselves. In what follows, we adopt the
uniform view (with all $t_n$ treated equally) when possible, keeping, however,
in mind a possibility (or necessity) of changing the phase space.

\vspace{.3cm}

\begin{center}
\large\textbf{Dressing}
\end{center}

\vspace{.1cm}

Now we explore the origin of commuting flows (\ref{zero}) from a slightly
different point of view. Consider a family $\{b_n\} \ (n>0)$
of elements of a graded
algebra (usually, $b_n$ span the positive part of a Heisenberg subalgebra of
an affine Kac-Moody algebra) such that
\begin{equation}
\label{bn}
[b_m,b_n] = 0\,, \ \ \ \ \ \ (b_n)_+ = b_n\,, \ \ \ \ \ \
(b_n)_- = 0\,, \ \ \ \ \ \ \partial_m b_n = 0\,.
\end{equation}
Then the operators $\partial_n-b_n$ commute for any $n$, as well as their
`dressed' counterparts
\begin{equation}
\label{dress}
\partial_n - B_n \doteq W(\partial_n-b_n)\circ W^{-1} \ \ \ \ \ \
\Longleftrightarrow \ \ \ \ \ \
B_n = W b_n W^{-1} + \partial_n W\cdot W^{-1}\,.
\end{equation}
Formally, zero-curvature condition (\ref{zero}) appears here as a result of
dressing the family of trivially commuting flows.

So far these relations are useless: they do not constrain $W$ at all. But
our aim is to generalize the matrix Ansatz of the preceding section like
this: $z^n\sigma\rightarrow b_n\,, \ A\rightarrow W\sigma W^{-1}$.
It is achieved by requiring
\begin{equation}
\label{constr}
W = \mathbf{1} + (\ldots)_-\ , \ \ \ \ \ \ \ \ B_n = (W b_n W^{-1})_+
\end{equation}
that, together with (\ref{dress})\,, entails the Sato equations
\begin{equation}
\label{Sato}
\partial_n W \cdot W^{-1} = -(W b_n W^{-1})_-
\end{equation}
which can be viewed as dynamical ones if a phase space is generated by $W$.

To present a generic solution of (\ref{Sato})\,, let us introduce
\begin{equation}
\label{Gamma}
\Gamma \doteq \exp\sum_i t_i\,b_i\,, \ \ \ \ \ \ \partial_n \Gamma
 = b_n \Gamma\,, \ \ \ \ \ \ \theta = \theta_-^{-1}\theta_+
\doteq \Gamma g \Gamma^{-1}\,, \ \ \ \ \ \ \partial_n g = 0
\end{equation}
and show that $\theta_-$ obeys (\ref{Sato}) for any (constant) element $g$
of the Lie group corresponding to our original algebra. We suppose
this group to be factorizable into a product of positive and negative part,
$G=G_-^{-1}G_+$\,, in accordance with the $R$-bracket recipe (\ref{RA})\,.
Indeed,
\begin{gather*}
\partial_n\theta = \partial_n(\theta_-^{-1}\theta_+)
 = \partial_n\theta_-^{-1}\cdot\theta_+ + \theta_-^{-1}\partial_n\theta_+
 = -\theta_-^{-1}\partial_n\theta_-\cdot\theta_-^{-1}\theta_+
  + \theta_-^{-1}\partial_n\theta_+\,, \\
\partial_n\theta = \partial_n(\Gamma g \Gamma^{-1})
 = \partial_n \Gamma\cdot g\Gamma^{-1}
  - \Gamma g \Gamma^{-1}\partial_n\Gamma\cdot\Gamma^{-1}
  = b_n \theta - \theta b_n = b_n\theta_-^{-1}\theta_+
  - \theta_-^{-1}\theta_+ b_n \\
\Longrightarrow \ \ \ \ \ \ \partial_n\theta_-\cdot\theta_-^{-1}
 + \theta_- b_n\theta_-^{-1} = \partial_n\theta_+\cdot\theta_+^{-1}
 + \theta_+ b_n\theta_+^{-1}
\end{gather*}
whence the Sato relation for $\theta_-$ follows by taking a negative part.
Thus, evaluation of $(\Gamma g \Gamma^{-1})_-$ which requires solving the
factorization problem in the group $G$ provides an efficient method for
finding solutions $B_n$ of the integrable hierarchy (\ref{zero})\,, or,
equivalently, solutions $\Psi = W\,\Gamma$ of the linear problem
$(\partial_n - B_n)\Psi = 0$ (with $\Gamma$ being a `vacuum' solution)\,.
Moreover, we could start from any solution $\Psi$\,,
define $\phi=\Psi g\Psi^{-1}$ and show in complete analogy with the above
calculation that $\Phi=\phi_-\Psi$ is also a solution:
\begin{multline}
\notag
\partial_n\phi = \partial_n(\phi_-^{-1}\phi_+)
 = B_n\phi_-^{-1}\phi_+ - \phi_-^{-1}\phi_+ B_n \ \ \ \ \ \Longrightarrow
\ \ \ \ \ \partial_n\phi_-\cdot\phi_-^{-1} = -(\phi_-B_n\phi_-^{-1})_- \\
\Longrightarrow \ \ \ \phi_-(\partial_n - B_n)\circ\phi_-^{-1}
 = \partial_n - (\phi_-B_n\phi_-^{-1})_+ \ \ \ \Longrightarrow \ \ \
(\partial_n - (\phi_-B_n\phi_-^{-1})_+)\Phi = 0\,.
\end{multline}
Using \ $\phi_-\,(\theta_-b_n\theta_-^{-1})_-\,\phi_-^{-1}
=(\phi_-\,(\theta_-b_n\theta_-^{-1})_-\,\phi_-^{-1})_-$ \,we can derive
as a simple corollary that in the case $\Psi=\theta_-\Gamma$ \ a product
$\phi_-\theta_-$ satisfies (\ref{Sato})\,. This illustrates the action of the
dressing group, which can be also related with the orbit interpretation of
$\tau$ function. Assuming \,$G_+\,|0\!> \ \sim |0\!>$ we find, schematically,
\begin{equation}
\label{tau}
|\tau_g(t)\!> \ = \,\Gamma(t)\,g\,|0\!> \ = \Gamma g\Gamma^{-1}\,|0\!> \
 = \theta\,|0\!> \ \sim \ \theta_-^{-1}\,|0\!> \ = W^{-1}\,|0\!>\,.
\end{equation}

\vspace{.3cm}

\begin{center}
\large\textbf{Conservation laws}
\end{center}

\vspace{.1cm}

Coadjoint-invariant Hamiltonians of the type (\ref{Ham}) do not directly apply
within the dressing formalism because they are identically constant on
the corresponding orbits. To develop an appropriate technique,
let us adopt conventions (\ref{id}) and introduce a grading operator
$\mathcal{D}$ generalizing $z\frac{d}{dz}$\,:
\begin{equation}
\label{D}
\mathcal{D}(XY) = \mathcal{D}X\,Y + X\,\mathcal{D}Y, \ \ \ \ \
\mathcal{D}b_m = m\,b_m\,, \ \ \ \ \ \text{gr}(X) = m \ \ \
\Longrightarrow \ \ \ \mathcal{D}X = mX\,.
\end{equation}
Then the following holds:
\begin{equation}
\label{Jn}
\mathcal{J}_n \doteq \text{Tr}(\mathcal{D}W\,b_nW^{-1}) \ \ \ \ \
\Longrightarrow \ \ \ \ \
\partial_m\mathcal{J}_n = \text{Tr}(\mathcal{D}B_m\,W b_n W^{-1})
 = \partial_n\mathcal{J}_m\,.
\end{equation}
This is deduced with the use of (\ref{dress}) and
$(b_m b_n)_0=(\mathcal{D}X)_0=(X_-Y_-)_0=(\mathcal{D}X_+\,Y_+)_0=0$\,:
\begin{multline}
\notag
\partial_m\mathcal{J}_n = \partial_m \text{Tr}(\mathcal{D} W b_nW^{-1})
 = \text{Tr}(\mathcal{D}(B_m W - W b_m)\,b_n W^{-1}
 - \mathcal{D}W b_n (W^{-1}B_m - b_m W^{-1})) \\
 = \text{Tr}(\mathcal{D}B_m W b_n W^{-1} + B_m\mathcal{D}W b_n W^{-1}
  - \mathcal{D}W b_m b_n W^{-1} - m W b_m b_n W^{-1}
  - \mathcal{D}W b_n W^{-1} B_m \\
+ \mathcal{D}W b_n b_m W^{-1})
 = \text{Tr}(\mathcal{D}B_m W b_n W^{-1} - m b_m b_n)
 = \text{Tr}(\mathcal{D}B_m W b_n W^{-1}) \\
=\text{Tr}((\mathcal{D}(W b_m W^{-1})-\mathcal{D}(W b_m W^{-1})_-)W b_n W^{-1})
 = \text{Tr}(\mathcal{D}W b_m b_n W^{-1} + m b_m b_n \\
- b_m W^{-1}\mathcal{D}W b_n - \mathcal{D}(W b_m W^{-1})_-(W b_n W^{-1})_+)
 = \text{Tr}(\mathcal{D}B_n W b_m W^{-1}) = \partial_n \mathcal{J}_m\,.
\end{multline}
It is now natural to introduce a generating $\tau$-function as
\begin{equation}
\label{tau2}
\mathcal{J}_n \doteq \partial_n \log \tau \ \ \ \ \ \ \
\Longrightarrow \ \ \ \ \ \ \partial_m\mathcal{J}_n = \partial_n\mathcal{J}_m
 = \partial_m\partial_n \log \tau\,,
\end{equation}
which very likely is somehow connected with the definition (\ref{tau})\,.

To proceed further, we single out $t_1=x$ and assume (omitting details)
that our graded algebra admits a decomposition of the form
$\mathfrak{b}\!\oplus \ldots$ \,with $\mathfrak{b}$ Abelian
($b_n\in\mathfrak{b}_+$)\,, and that an equation
\begin{equation}
\label{DS}
L \equiv \partial_x - B_1 = U(\partial_x - b_1 - Q_1)\circ U^{-1}
 \equiv U L_0 U^{-1}
\end{equation}
is solved iteratively to produce $Q_1\in\mathfrak{b}_-$ and
$U=\mathbf{1}+(\ldots)_-$ as functions of $B_1$ and its
$x$-derivatives. In other words, $U$ and $Q_1$ are local.
They are also unique if we demand that the $\mathfrak{b}$-part of $U$ be zero.
Then we set $W=UT, \ \,T\in\exp(\mathfrak{b}_-)$\,, \,and find
\begin{equation}
\label{loc}
Q_1 = \partial_x T\cdot T^{-1}, \ \ \ \
T b_n T^{-1} = b_n\,, \ \ \ \ B_n = (U b_n U^{-1})_+\,, \ \ \ \
\partial_m\mathcal{J}_n = \text{Tr}(\mathcal{D}B_m U b_n U^{-1})\,.
\end{equation}
Note that $T,W$ and $\mathcal{J}_n=\partial_n\log\tau$ are nonlocal whereas
$\partial_m\mathcal{J}_n$ prove to be local.
Now, defining a set of local quantities
\begin{equation}
\label{Hn}
\mathcal{H}_n \doteq \partial_1\mathcal{J}_n = \text{Tr}(b_1 U b_n U^{-1})
\ \ \ \ \ \Longrightarrow \ \ \ \ \ \
\partial_m\mathcal{H}_n = \partial_x(\partial_m\mathcal{J}_n)
\end{equation}
($\mathcal{D}B_1=b_1$ is used) we see that $H_n=\int\!dx\mathcal{H}_n$
are conserved with respect to all flows (\ref{dress})\,.

The Hamiltonians of these flows are usually constructed as an alternative set
of conserved quantities, generated by $h\equiv Q_1$\,. Namely,
\begin{equation}
\label{Qn}
\partial_n - b_n - Q_n \doteq U^{-1}(\partial_n - B_n)\circ U
 = T(\partial_n - b_n)\circ T^{-1}
\end{equation}
implies that all $Q_n=\partial_n\log T\in\mathfrak{b}_-$\, are local and
\begin{equation}
\label{h}
\partial_m Q_n = \partial_n Q_m\,, \ \ \ \ \ \partial_m h = \partial_x Q_m\,,
\ \ \ \ \ \partial_m\int\!dx\,h = 0\,, \ \ \ \ \ \partial_m h_n = 0
\end{equation}
where $h_n\doteq\int\!dx\,\text{Tr}(b_n h)$\,. Let us express $h'_n$
in terms of the Lax operator (\ref{DS})\,. From (\ref{H'2}) we know that
$h'_n(L)=Uh'_n(L_0)U^{-1}$\,. To find $h'_n(L_0)$ we consider a variation
\begin{equation}
\label{}
L_0 \ \rightarrow \ L_0+t\lambda \doteq U(t)\,
(\partial_x-b_1-h(t))\circ U^{-1}(t)
\end{equation}
and do the following calculation:
\begin{multline}
\notag
\int\!dx\,\text{Tr}(\lambda h'_n(L_0)) \equiv \ <\!\lambda\,,\,h'_n(L_0)\!> \
 = \frac{d}{dt}\,h_n(L_0 + t\lambda)|_{t=0} \\
 = \frac{d}{dt}\int\!dx \text{Tr}(b_n h(t))|_{t=0}
= -\frac{d}{dt}\int\!dx \text{Tr}(b_n U^{-1}(t)(L_0 + t\lambda)U(t))|_{t=0} \\
 = -\frac{d}{dt}\int\!dx \text{Tr}(b_n U^{-1}(t)L_0 U(t)
  + b_n(L_0 + t\lambda))|_{t=0} = -\int\!dx\,\text{Tr}(\lambda\,b_n)\,,
\end{multline}
because of
\begin{equation}
\label{}
\frac{d}{dt}\text{tr}(b_n U^{-1}(t)L_0 U(t))|_{t=0}
 = \text{tr}(b_n[\ldots,L_0]) = \text{tr}([L_0,b_n]\ldots) = 0\,.
\end{equation}
We arrive at the desired result:
\begin{equation}
\label{h'n}
h'_n(L_0) = -b_n\,, \ \ \ \ \ \ h'_n(L) = -U b_n U^{-1}\,, \ \ \ \ \ \
(h'_n)_+ = -B_n\,.
\end{equation}

The formalism with distinguished $\partial_x$ used above is naturally
interpreted in terms of coadjoint group action underlying the whole scheme.
Let us centrally extend, via a well-known 2-cocycle, our graded algebra and
its dual as follows ($a,\alpha\in\mathbb{C}$)\,:
\begin{align}
A&\rightarrow(A,a)\,, & [(A,a),(B,b)] \
 &\doteq \ ([A,B]\,,\int\!dx\,\text{Tr}(A\,\partial_x B)\,)\,, \\
\omega&\rightarrow(\omega,\alpha)\,, & <\!(\omega,\alpha),(A,a)\!>
\ &\doteq \ \alpha a + \int\!dx\,\text{Tr}(\omega A)\,.
\end{align}
Starting from the very definition of \,$\text{ad}^*$,
\begin{equation}
\notag
<\!\text{ad}^*_{(A,a)}(\omega,\alpha),(B,b)\!> \
 \doteq \ <\!(\omega,\alpha)\,,\,[(A,a),(B,b)]\!> \
 = \int\!dx\,\text{Tr}(\text{ad}^*_A\omega\,B - \alpha\partial_x A\,B)\,,
\end{equation}
we obtain
\begin{gather}
\label{adext}
\text{ad}^*_{(A,a)}(\omega,\alpha) \ = \ (\text{ad}^*_A\omega
 - \alpha\partial_x A\,,\,0) \ =
\ (-[A\,,\,\omega - \alpha\partial_x]\,,\,0)\,, \\
\label{ident}
\text{Ad}^*_g\,(\omega\,,\alpha) \ = \ g^{-1}(\omega - \alpha\partial_x)g
 = g^{-1}\omega g - \alpha g^{-1}\partial_x g - \alpha\partial_x
 = (g^{-1}\omega g - \alpha g^{-1}\partial_x g\,,\alpha)\,,
\end{gather}
thus identifying $(\omega\,,\alpha) \ \simeq \ \omega - \alpha\partial_x$
and recognizing the Lax operator (\ref{DS}) as an element of centrally
extended phase space, with $\alpha=1$ and $\omega=B_1$\,.


\end{document}