\documentstyle[amsfonts,amssymb,amsmath,amsthm,prb,aps]{revtex}
%\usepackage[active]{srcltx} % SRC Specials for DVI Searching

\makeatletter

%run page numbers by chapter
%\def\thepage{3-\@arabic\c@page}
%these page numbers need abit more width
\def\@pnumwidth{2em}
\def\REVTeX{REV\TeX}

\makeatother

% Over-full v-boxes on even pages are due to the \v{c} in author's name
\vfuzz2pt % Don't report over-full v-boxes if over-edge is small

% THEOREM Environments ---------------------------------------------------
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{ex}[thm]{Example}
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
%\numberwithin{equation}{section}
% MATH -------------------------------------------------------------------
 \DeclareMathOperator{\RE}{Re}
 \DeclareMathOperator{\IM}{Im}
 \DeclareMathOperator{\ess}{ess}
 \newcommand{\eps}{\varepsilon}
 \newcommand{\To}{\longrightarrow}
 \newcommand{\h}{\mathcal{H}}
 \newcommand{\s}{{\mathcal{S}}}
 \newcommand{\A}{\mathcal{A}}
 \newcommand{\J}{\mathcal{J}}
 \newcommand{\M}{\mathcal{M}}
 \newcommand{\W}{\mathcal{W}}
 \newcommand{\X}{\mathcal{X}}
 \newcommand{\BOP}{\mathbf{B}}
 \newcommand{\BH}{\mathbf{B}(\mathcal{H})}
 \newcommand{\KH}{\mathcal{K}(\mathcal{H})}
 \newcommand{\Real}{\mathbb{R}}
 \newcommand{\Complex}{\mathbb{C}}
 \newcommand{\Field}{\mathbb{F}}
 \newcommand{\RPlus}{\Real^{+}}
 \newcommand{\Polar}{\mathcal{P}_{\s}}
 \newcommand{\Poly}{\mathcal{P}(E)}
 \newcommand{\EssD}{\mathcal{D}}
 \newcommand{\Lom}{\mathcal{L}}
 \newcommand{\States}{\mathcal{T}}
 \newcommand{\abs}[1]{\left\vert#1\right\vert}
 \newcommand{\set}[1]{\left\{#1\right\}}
 \newcommand{\seq}[1]{\left<#1\right>}
 \newcommand{\norm}[1]{\left\Vert#1\right\Vert}
 \newcommand{\essnorm}[1]{\norm{#1}_{\ess}}
%%% ----------------------------------------------------------------------

%%% ---------------------------------------------------------------
%miguel's-----------------------------------------------------------------
%\newcommand{\Flow}{Flow--Along}
\newcommand{\Fl}[1]{\underset{{#1}}{\mathcal{M}}}
\newcommand{\Proj}[1]{{\mathcal{P}}_{#1}}
\newcommand{\Lie}[2]{\underset{{#1}}{{\mathcal{L}}} {{#2}}}
\newcommand{\D}{{\rm D}}
%\newcommand{\ud}[3]{{#1}_{\,\,\, \mathrm{#3}}^{\mathrm{#2}}}
\newcommand{\ud}[3]{{#1}_{\,\,\, {#3}}^{{#2}}}
\newcommand{\uD}[3]{{#1}_{ \mathrm{#3}}^{\mathrm{#2}}}
\renewcommand{\d}[2]{{#1}_{\mathrm{#2}}}
\renewcommand{\u}[2]{{#1}^{\mathrm{#2}}}
\newcommand{\bnd}[1]{{\mathrm{boundary \,\,\, {#1}\textrm{--}\mathrm{form}}}}

\renewcommand{\baselinestretch}{1.37}

\begin{document}


\title{Multi--Lagrangians, Hereditary Operators and Lax Pairs for the Korteweg--de Vries Positive and Negative Hierarchies}
\author{Miguel D. Bustamante ${}^{{a,\,d}}$ \footnote[0]{${}^{{d}}$ E-mail address: miguelb@macul.ciencias.uchile.cl}
and \,\,\,Sergio A. Hojman${}^{b,\,a,\,c,\,e}$ \footnote[0]{${}^{{e}}$
E-mail address: shojman@creavirtual.org} }
\address{${}^{{a}}$ Departamento de F\'{\i}sica, Facultad
de Ciencias, Universidad de Chile, Santiago, Chile}
\address{ ${}^{{b}}$ Centro de Recursos Educativos
Avanzados, CREA, Santiago, Chile}
\address{ ${}^{{c}}$ Facultad de Educaci\'on,
Universidad Nacional Andr\'es Bello, Santiago, Chile}
\date{21 January 2003}
\maketitle
% AQU VA LA COSA QUE SE PEGAR
%(Received
\begin{abstract}
We present an approach to the construction of Action Principles for first
order (in time derivatives) differential equations (the inverse problem of
the Calculus of Variations) and generalize it to field theory in order to
construct systematically, for integrable equations which are based on the
existence of a Nijenhuis (or hereditary) operator, a ladder of Action
Principles which is complementary to the well--known multi--Hamiltonian
formulation. We work out results for the Korteweg--de Vries (KdV)
equation, which is a member of the positive hierarchy related to the
hereditary operator. Three negative hierarchies of (negative) evolution
equations are defined naturally from the hereditary operator as well, in a
concise way, suitable for field theory. We identify the Liouville,
Sinh--Gordon, Hunter--Zheng and Camassa--Holm equations as negative ones.
The same multi--Lagrangian scheme for KdV is shown to apply also to these
and other positive and negative evolution equations. The Euler--Lagrange
equations arising from the Action Principles are equivalent to
deformations of the original evolution equation, and the deformations are
obtained explicitly in terms of the positive and negative evolution
vectors, which follows from the factorized hereditary operator. For
example, several nonequivalent, nonlocal Action Principles for KdV which
are time--reparametrization invariant are constructed, and a new Action
Principle for the deformed system Sinh--Gordon $+$ spatial translation
vector is presented. The factorized hereditary operator allows for the
explicit computation of local and nonlocal Hamiltonian operators, as the
inverses of all the nonequivalent symplectic $2$--forms in the ladder.
Nonlocal constants of the motion are constructed for KdV as well as for
positive and negative vectors, in a concise way. Alternative Lax pairs for
all negative vectors are constructed. Some results are mapped to the
Krichever--Novikov and Harry--Dym equations.
\end{abstract}
{~~}\\

\noindent \textbf{Keywords}: Multi--Lagrangians, Hereditary Operators,
Korteweg--de Vries Hierarchies, Lax Pairs.

\noindent \textbf{Pacs Numbers}:
02.30.Ik,\,45.10.Db,\,11.30.-j,\,11.10.Lm\,.

\input epsf

\newpage


\section{Introduction}
Hereditary or Nijenhuis operators \cite{Fuc79,And82,Dor93} play an
important role in the description of integrable systems: in terms of these
operators, the very definition of the positive and negative hierarchies of
integrable evolution equations may be given, \cite{Fuc80} and they are
used to construct, for these equations, subsequent constants of the
motion, \cite{Boi84} alternative Lax pairs, \cite{{Bru95}}
multi--Hamiltonian \cite{Mag77} and multi--Lagrangian structures,
\cite{Nut00,{Nut01}} and symmetries. \cite{Fuc80,Boi84} For example, the
well--known Korteweg--de Vries (KdV) equation is a flow vector belonging
to the positive KdV hierarchy.

The problem of constructing multi--Lagrangian structures (i.e., an
infinite ladder of Action Principles) for the KdV equation has been
tackled recently, \cite{Nut00,{Nut01}} using the bi--Hamiltonian
formulation, with the only drawback that the explicit form of the Action
Principles associated to each symplectic $2$--form requires the
integration of the $2$--forms, a task which is increasingly difficult as
we move to the positive end of the ladder (quasi local symplectic forms),
and it gets worse as we move to the negative end of the ladder, where the
symplectic $2$--forms are increasingly nonlocal. The known way
\cite{Nut01} to get rid of the nonlocality problem is to write the Action
Principles in a ``local" coordinate system (Darboux theorem) depending on
the specific symplectic $2$--form in the ladder, a process that gets
recurrently harder as we move to the negative end.

In this work, we make use of the Galilean symmetry \cite{Sch96} and the
factorized form of the hereditary operator \cite{Gut93} for the KdV
equation, to construct explicitly the Action Principles for KdV in the
positive as well as in the negative end of the ladder. No integration of
any $2$--form is needed, nor is the search for a special coordinate
system. The factorized form of the symplectic $2$--forms allows for the
interpretation of the resulting Euler--Lagrange equations (arising from
each Action Principle) as deformed equations, with flows given by KdV $+$
vectors in the positive and negative hierarchies, which are computed
explicitly. The above construction also leads to explicit, nonrecurrent
expressions for local and nonlocal constants of the motion for KdV.

From the Action Principles obtained for the KdV equation we construct
Action Principles for flows defined by other positive and negative
vectors. In particular, new Action Principles for the Sinh--Gordon (ShG)
\cite{Rub70} equation (a negative equation) are constructed.

On the other hand, alternative Lax pairs for the KdV equation and for
positive KdV flows have been constructed from the hereditary operator.
\cite{Bru95} We do the same for all the negative KdV flows, and we
conclude that the local and nonlocal constants of the motion for KdV,
define conserved currents for the negative vectors as well.

The results here may be mapped to the following equations:
Krichever--Novikov, Harry--Dym, \cite{Gol91} Camassa--Holm, \cite{Hon99}
Hunter--Zheng, \cite{Bru02} and Liouville, all of which are essentially
flows belonging to the KdV positive and negative hierarchies. The results
are quite general and may be extended to other systems related to
hereditary operators (e.g., nonlinear Schr\"odinger equation).

This paper is organized as follows: section \ref{sec:Pre} presents a
preview and notation for the method of construction of Action Principles
for given differential evolution equations, and a brief survey of
symmetries and constants of the motion in this context. Next, the
relationship of these Principles with Hamiltonian theories, and finally
the hereditary property with the consequent construction of the positive
and negative hierarchies of integrable evolution equations. In section
\ref{sec:Ladders} we present and prove theorems on the explicit
construction of ladders of Action Principles and constants of the motion,
based on the hereditary operator and the Galilean symmetry, and we show
how these Action Principles give rise to Euler--Lagrange equations which
are deformations of the original equations, due to the fact that the
symplectic operators have a nonzero kernel. Section \ref{sec:example KdV}
is devoted to examples of the above constructions for the KdV equation. We
obtain concise expressions for the negative vectors, for the Action
Principles, and for the deformed Euler--Lagrange equations. Symplectic
operators are presented in factorized form (this allows for factorized
expressions for the Hamiltonian operators). Some known integrable
evolution equations are identified within the negative hierarchies.
Finally, nonlocal constants of the motion for KdV are concisely obtained
using the internal symmetries and the negative Action Principles. In
section \ref{sec:results negative} we work out examples of new Action
Principles for the ShG equation, which in this context is identified as a
negative vector; we construct Lax pairs for the negative equations, and
conclude that the local and nonlocal constants of the motion for KdV also
work for negative equations. Finally, some
concluding remarks are presented in section \ref{sec:conc}.\\

For simplicity, we work in finite--dimensional notation. All assertions
and theorems in sections \ref{sec:Pre} and \ref{sec:Ladders} are valid in
finite dimensions, and they can be extended to the case of field theory in
all the instances dealt with in this paper.

\section{Preview and Notation}
\label{sec:Pre} Consider the autonomous equations of motion
\begin{equation}\label{eq:motion} \dot{q}^{\textrm{a}}(t) =
V^{\textrm{a}}[q^{\textrm{b}}(t)]\,,\quad{\textrm{a}} \in A\,.
\end{equation}
$A$ is a given ordered set called ``label set": the elements of it label
the degrees of freedom of the theory. From now on, we suppress the
dependence of the coordinates $\{q^{{\textrm{a}}}\}_{{\textrm{a}} \in  A}$
on time when it is obvious.

\begin{ex} The KdV Equation for the field $u(x,t)$, \mbox{$x \in [x_-,x_+]$,}
$t\in {\mathbb{R}}$, is
\begin{equation}\label{eq:KdV}
u_t = - u_{xxx} -12\, u\, u_x\,
\end{equation}
(suffixes denote partial differentiation). The label set is identified as
${\mathrm{A}} = [x_-,x_+]$, and $x\in \mathrm{A}$ is a continuous index.

We will use standard boundary conditions for the field: $u,\,u_x,\,\ldots
\to 0\,$ as $x \to x_\pm$, although the methods may be extended for the
treatment of other boundary conditions as well (in which case the Weiss
Action Principle \cite{Sud74} and the  Witten--Zuckerman $2$--form
\cite{Wit86} come into play).
\end{ex}

\begin{rem}
Einstein summation convention over repeated indices is assumed from here
on (for the infinite--dimensional case, the summation is extended to an
integration over continuous indices).
\end{rem}

The evolution equation (\ref{eq:motion}) defines naturally a vector space,
spanned by the derivatives $\{\frac{\partial}{\partial
\u{q}{a}}\}_{{\mathrm{a}} \in {\mathrm{A}}}$ (for the
infinite--dimensional case, partial derivatives with respect to the
coordinates become functional derivatives). We call $V =
V\u{}{a}\,\frac{\partial}{\partial \u{q}{a}}$ the \textbf{flow} vector or
evolution vector for the system. It is an invariant structure, i.e., it is
independent of the coordinate system used.


\subsection{Action Principles}

The equations of motion (\ref{eq:motion}) are related to a
\textbf{Variational Principle} with \textbf{Action}
\begin{equation} \label{eq:action}
 {S}[q\u{}{a}(t),t] =
\int_{t\d{}{o}}^{t_1} dt\,\bigg(P\d{}{a}\,\left( \dot{q}\u{}{a} - V\u{}{a}
\right)+ K\bigg)\,,
\end{equation}
 where the $1$--form $P[q\u{}{b}]$ and the zero-form $K[q\u{}{b}]$
satisfy the following equation:
\begin{equation*}
   P\d{}{a,b} V\u{}{b} + P\d{}{b} V\u{}{b}{}\d{}{, a}  =
 K\d{}{ , a}\,,
\end{equation*}
with $K\d{}{ , a}\equiv \frac{\partial K}{\partial q\u{}{a}}\,.$

We rewrite the above equation in terms of invariant structures:
\begin{equation}\label{eq:pair}
 \Lie{V}{} P = \delta K\,,
\end{equation}
where $\Lie{V}{}$ is the \textbf{Lie derivative} along the vector $V$, and
$\delta$ is the \textbf{exterior differential} (see \cite{Nak92} for a
definition of these operators).

We call the pair $(P;\,K)$, $1$--form $P$ and $0$--form $K$, a
\textbf{standard Lagrangian pair} for $V$ if $K \neq 0$. In the special
case $K = 0$ we call $P$ a \textbf{non--standard Lagrangian }for $V$: the
latter case allows for the construction of constants of the motion in a
direct way \cite{Bus02} (see theorem \ref{thm:3}).

\begin{rem}The above objects should not be confused with the usual
``Lagrangian density" ${{L}}[q,\,\dot{q},\,t] = P\d{}{a}\,\left(
\dot{q}\u{}{a} - V\u{}{a} \right)+ K\,,$ which is the thing that is
integrated in time to give the action: $S = \int {{L}}\,dt\,.$ The
$1$--form $P$ is also understood as the momentum map. \cite{Nut01} In the
special case $K=0\,,$ it solves the equation for a ``conserved covariant."
\cite{Boi84}
\end{rem}

The general case of objects which depend explicitly on time is easily
worked out, \cite{Hoj81} but there is no need to do so in the applications
of this paper. Nevertheless, for symmetries and constants of the motion
the explicit time dependence will be necessarily taken into account. In
the sequel, we give the name \textbf{time--independent} to those objects
which do not depend \textbf{explicitly} on time, and viceversa.

The \textbf{Euler--Lagrange} equations which come from the Action
(\ref{eq:action}) are:
\begin{equation*}
\Sigma\d{}{a b} \left( \dot{q}\u{}{b} - V\u{}{b}\right) = 0\,,
\end{equation*}
where $\Sigma \equiv \delta P$ is the \textbf{symplectic $2$--form }or
Lagrange bracket whose components are:
\begin{equation*}
\Sigma\d{}{a b} =  P\d{}{b,a}- P\d{}{a,b}\,.
\end{equation*}
It is worth mentioning that, in \cite{Wit86}, a symplectic $2$--form is
induced by an Action Principle in essentially the same way we have derived
the above symplectic $2$--form from the Action Principle
(\ref{eq:action}).

Notice that these Euler--Lagrange equations do not imply the original
equations of motion (\ref{eq:motion}); instead they imply
\textbf{deformed} or mixed equations, where the deformation is represented
by an additive extra term which is an arbitrary linear combination of
vectors belonging to the \textbf{kernel} of the symplectic $2$--form. In
the case of KdV, we will obtain the deformations explicitly. See
\cite{Bus02} for examples in the finite dimensional case.

The symplectic $2$--form associated to this Action Principle is easily
seen to satisfy
\begin{eqnarray}
\nonumber \delta\, \Sigma &=& 0 {\textrm{\quad (closure)}}\\
\label{eq:symp2} \Lie{V}{\,} \Sigma &=& 0\,,
\end{eqnarray}
therefore the inverse process could be done: starting from a symplectic
$2$--form $\Sigma$ for the flow vector $V$, we construct the standard
Lagrangian $1$--form, from $\delta\,P = \Sigma$ and the $0$--form $K$ is
obtained by integration of equation (\ref{eq:pair}). This process suffers
from technical difficulties, which increase when the objects are infinite
dimensional and nonlocal. Fortunately, for the KdV equation there is a
constructive way of finding the Action Principles (see theorems
\ref{thm:1} and \ref{thm:2}).

\subsection{Hamiltonian theories are induced from Symplectic Structures}

The relationship of the symplectic $2$--form with the Hamiltonian
formulation is very simple: \cite{Olv93} consider the formal inverse
(i.e., except for a finite kernel that the operators may possess) of the
above $2$--form, the time--independent $(2,0)$ tensor $J$ such that
$J\cdot \Sigma = \mathbb{I}\,.$ It is possible to show that $\Sigma$ is
closed if and only if $J$ satisfies the \textbf{Jacobi identity}, which we
write in the form:
\begin{equation}\label{eq:Jac}
  \Lie{J\cdot U}{J} = J\cdot \delta U\cdot J\,,\quad \forall \,\,1\textrm{--form }\,U\,.
\end{equation}
$J$ is known as a Hamiltonian operator or Poisson bracket. Now, equation
(\ref{eq:symp2}) implies $\Lie{V}{\,J} = 0\,.$ Therefore, according to the
Jacobi Identity (\ref{eq:Jac}), a Hamiltonian theory for the flow $V$ is
induced by the symplectic $2$--form $\Sigma$: the equation $V=J\cdot U$
implies $\delta U=0$, thus $V = J\cdot \delta H\,,$ where $H$ is the
Hamiltonian, a time--independent $0$--form.

\subsection{Symmetries}

Symmetries play a crucial role in the construction of the Action
Principles. A symmetry for the system (\ref{eq:motion}) is known as a
vector with components $\u{\eta}{a}$ that takes solutions into solutions
of equation (\ref{eq:motion}), in the sense that given any solution
${q}^{\textrm{a}}(t)$ such that $\dot{q}^{\textrm{a}}(t) =
V^{\textrm{a}}[q^{\textrm{b}}(t)]\,,$ then $\u{\tilde{q}}{a} \equiv
\u{q}{a} + \epsilon\,\u{\eta}{a}[\u{q}{b},t]$ is also a solution up to
order $\epsilon^2$, i.e.: $\dot{\tilde{q}}{}^{\textrm{a}}(t) =
V^{\textrm{a}}[\tilde{q}^{\textrm{b}}(t)]\,+\,O(\epsilon^2)\,.$


It is easily seen \cite{Hoj96} that this condition leads to the equation $
\frac{\partial}{\partial t}\u{\eta}{a}
 + \ud{\eta}{\mathrm{a}}{, \mathrm{b}}\,\u{V}{b} - \ud{V}{\mathrm{a}}{, \mathrm{b}}\,\u{\eta}{b} = 0
\,$ or, in a covariant way, $\left(\frac{\partial}{\partial t} +
\Lie{V}{}\right)\eta = 0\,.$

\begin{ex}
The Galilean and the dilatation symmetries
 for the KdV equation are defined respectively by
\begin{eqnarray}
\u{\d{\eta}{G}[u,t]}{x} &=& \frac{1}{8} - \frac{3}{2} \,t \,u_x\,, \label{eq:gal} \\
\nonumber \u{\d{\eta}{D}[u,t]}{x} &=& u + \frac{1}{2}\,x\,u_x - t
\left(\frac{3}{2}\,u_{xxx} + 18\,u\,u_x\right)\,.\
\end{eqnarray}
In \cite{Sch96}, there is an open question concerning the role of the
Galilean and dilatation symmetries in the construction of constants of the
motion for KdV. An answer to this question is given in this work: these
symmetries actually lead to Action Principles for the KdV equation, as we
will see soon.
\end{ex}

\subsection{Constants of the Motion}

A constant of the motion for the system (\ref{eq:motion}) is a functional
($0$--form) $C[\u{q}{a},t]$ which is conserved in time under the
evolutionary system: $\left.\frac{\D}{\D
t}C[\u{q}{b},t]\right|_{\mathrm{on-shell}} \equiv \frac{\partial}{\partial
t}C + \d{C}{, a}\,\u{V}{a} = 0\,,$ where the partial time derivative
accounts for the explicit time dependence and $\frac{\D}{\D t}$ denotes
the convective or total derivative along the variable $t$.

This equation is best written in a covariant way: \cite{Hoj96} $\bigg(
\frac{\partial}{\partial t} + \Lie{{V}}{} \bigg) C = 0\,.$

We will usually work with time--independent constants of the motion:
$\frac{\partial}{\partial t} C = \Lie{{V}}{\,} C  =  0\,.$

\subsection{The Hereditary Property: Hierarchies of Evolution Equations}

Many integrable systems are related to a Nijenhuis or hereditary operator,
which is a time--independent $(1,1)$ tensor $R$ that solves: \cite{Olv93}
$\Lie{R \cdot \eta}{R} = R \cdot \Lie{\eta}{R}\,,\quad \forall \textrm{
vector } \eta\,.$

Out of the kernel of this operator, and of its inverse, hierarchies of
integrable evolution equations arise which are symmetries of each other
\cite{Fuc80} (this will be worked out in detail for the KdV case later
on).

According to \cite{Fuc80}, given a hereditary operator $R$ and a flow
vector field (labelled with a number) $V\d{}{1}$ such that
$\Lie{V\d{}{1}}{R}= 0\,,$ i.e., $R$ is a \textbf{recursion} operator  for
$V_1$, then a \textbf{hierarchy} is defined as a semi--infinite collection
of evolution vectors: $\{V_{j} = R^{j-1}\cdot V_1 \,,\quad
{j}=1,\,\ldots,\,\infty\}\,,$ which are symmetries of each other: $
\Lie{V_{i}}{V_{j}} = 0\,,\quad i,\,j\geq 1$, and thus every evolution
vector in the hierarchy defines an evolution equation which is integrable.
In the KdV hierarchy, the KdV equation is the second member ($V_2$). The
first vector ($V_1$) represents the translation symmetry, and is shown to
generate the kernel of the inverse of the hereditary operator, $R^{-1}$.
By convention, we refer to the above as a \textbf{positive hierarchy}.

The hereditary property for the operator $R\,$ may be used to show
formally that $R^{-1}$ is also hereditary.\cite{Gut93} Therefore, we could
conjecture that new hierarchies (referred to as \textbf{negative
hierarchies}) of evolution vectors may be constructed, which first members
generate the kernel of the operator $R$, and successive members are
defined by contraction of the first members with powers of the operator
$R^{-1}$. In the KdV case, there are three negative hierarchies. These new
negative equations include the ShG, Liouville, Camassa--Holm, and the
Hunter--Zheng equations.

\subsection{Notation: the Positive and Negative Hierarchies in terms of the Hereditary Operators}

The analysis is restricted to the KdV hierarchies, but it is easily
generalizable to other systems related to hereditary operators. A graphic
picture is given in figure $1$.

We will adopt the following notation for evolution vectors in the
hierarchies:
\[u_t = V\ud{}{(k)}{n}[u]\,,\]
where ${{k}}=1$ denotes the positive hierarchy; ${{k}} = \,-1,\,-2,\,-3$
for the three negative hierarchies, and ${{n}} = 1,\ldots,\,\infty$
denotes the place of a vector within the hierarchy, so that we have $n=1$
for the first vector of each hierarchy, i.e., the generator of the kernel
of $R^{-\mathrm{sgn}({k})}$:
\begin{equation}\label{eq:kerR}
\begin{array}{rcl}
R^{-1}[u] \cdot V\ud{}{(1)}{1}[u]& =& 0\,, \\
R[u] \cdot V\ud{}{(-1)}{1}[u] = R[u] \cdot V\ud{}{(-2)}{1}[u] &= & R[u]
\cdot V\ud{}{(-3)}{1}[u] =0\,. \
\end{array}
\end{equation}
Successive members in the hierarchies are defined by recurrence:
\[V\ud{}{(k)}{n+1}[u] = (R[u])^{\mathrm{sgn}({{k}})}\cdot V\ud{}{(k)}{n}[u]\,,\quad {{n}} \geq 1\,,\quad {{k} = 1,\, -1,\, -2,\, -3}\,.\]

In this way, the positive hierarchy begins with the vector
$V\ud{}{(1)}{1}[u] = -u_x$, continues with the KdV vector
$V\ud{}{(1)}{2}[u] = -u_{xxx}-12\,u\,u_x$, and so on (these vectors were
called $V_1$ and $V_{2}$ in the last subsection).

For the negative hierarchy, as the operator $R^{-1}$ is harder to work
with, there is a recurrent way of writing the negative vectors, in terms
of $R$:
\begin{equation}\label{eq:recNeg}
V\ud{}{(k)}{n}[u] = R[u] \cdot V\ud{}{(k)}{n+1}[u]\,,\quad {{n}} \geq
1\,,\quad {{k} = -1,\,-2,\,-3}\,.
\end{equation}


The explicit expression for negative vectors relies on the factorized form
\cite{Gut93,Vil01} of the hereditary operator, and will be realized in
section \ref{sec:example KdV} in terms of nonlocal fields which, however,
are tractable in the same scheme as the local ones. As a consequence of
this construction, we will be able to obtain concise new nonlocal
constants of the motion and nonlocal Action Principles for KdV and other
positive and negative equations.


\section{Ladders of Action Principles and Constants of the Motion}
\label{sec:Ladders} Complementary to the well--known bi--Hamiltonian
formulation,\cite{Fuc80} we may find a bi--symplectic or multi--symplectic
structure starting from the hereditary property. Assume that we have a
Nijenhuis operator $R$ along with one closed $2$--form $\Sigma^{(1)}$ such
that $\Sigma^{(2)} \equiv \Sigma^{(1)}\cdot R$ be a closed $2$--form:
then, the two semi--infinite dimensional sets (\textbf{symplectic
ladders}) of $2$--forms
\begin{equation*}
\{\Sigma^{(n)}\equiv \Sigma^{(1)}\cdot R^{\mathrm{n-1}}\,,\quad
{n}=1,\,\ldots,\,\infty\} \quad {\textrm{(positive symplectic ladder)}}\,,
\end{equation*}
and
\begin{equation*}
\{\Sigma^{(n)}\equiv \Sigma^{(1)}\cdot R^{\mathrm{n-1}}\,,\quad
{n}=0,\,-1,\,\ldots,\,-\infty\} \quad {\textrm{(negative symplectic
ladder)}}\,,
\end{equation*}
contain only closed $2$--forms. The distinction between positive and
negative ladders is somewhat arbitrary, for it depends on which hereditary
operator, $R$ or $R^{-1}$, is being used, and which symplectic $2$--form
is taken as $\Sigma^{(1)}\,.$ See figure $2$ for a schematic picture.

The symplectic ladder is completely analogous to the so--called Poisson
pencil or compatible implectic operators \cite{Fuc80} for
multi--Hamiltonian theories, if we consider the inverses of the closed
$2$--forms in the ladder.

The above result is independent of any evolution vector. When we consider
the vectors in the hierarchies, however, it is easily checked (as it holds
in the examples) that $\Lie{V\d{}{1}}{\Sigma^{(1)}} = 0$. Therefore, using
Leibnitz rule, all the $2$--forms in the ladder are symplectic operators
for the first evolution vector in the hierarchy.

Using the identity $\Lie{R\cdot \eta}{\Sigma}-\Lie{\eta}{\left(\Sigma
\cdot R\right)} = i_{R \cdot\eta}\delta\Sigma -
  i_{\eta}\delta(\Sigma \cdot R)\,,$ which holds for any vector $\eta$, $(1,1)$ tensor $R$ and $2$--form
$\Sigma$, where $i_{\eta}$ stands for interior product (contraction of the
vector $\eta$ with the left component of a $p$--form), we obtain the
important result for any hierarchy:
\begin{equation*}
  \Lie{V_{j}}{\,\Sigma^{(n)}} = 0\,,\quad
  {j}=1,\,2,\,\ldots,\,\infty,\quad
  {n}=-\infty,\,\ldots,\,\infty\,,
\end{equation*}
which means that a ladder of Action Principles may be constructed for
every evolution vector in the hierarchy (in particular, for the KdV
equation).

This fact is used in \cite{Nut01} to construct Action Principles, with the
only drawback it needs to integrate the $2$--forms (Poincar\'e lemma) in
order to get the Action Principles.

Let us assume for the rest of this section that we have a Nijenhuis
operator $R$ along with its inverse $R^{-1}$, a symplectic ladder
$\{\Sigma^{(n)}\,,\quad {n}=-\infty,\,\ldots,\,\infty\}\,,$ and a
hierarchy $\{V_{j}\,,\quad {j}=1,\,\ldots,\,\infty\}\,$ with the
corresponding properties mentioned above.

The purpose of the remaining of this section is to construct explicitly,
for every evolution vector in the hierarchy, the Action Principles
associated to every one of these symplectic $2$--forms. This will lead, in
addition, to the explicit construction of constants of the motion for the
evolution equations.

\subsection{Construction of Action Principles and Constants of the Motion out of Symmetries and Symplectic Operators}

Heuristically, if we had a symmetry for a given evolution equation we
could obtain in a direct way (by contraction of it with any symplectic
$2$--form) a Lagrangian $1$--form and therefore an Action Principle for
that equation.

We have done this procedure for any equation in the positive hierarchy,
using the Galilean symmetry, obtaining as a result a ladder of Action
Principles which are (explicitly) time--dependent (in fact, linear in
time). However, in order to make the final results simpler, the Actions
will be rewritten in such a way that they are time--independent, and the
discussion will be restricted to the second vector (which corresponds to
the KdV equation), which is from now on referred to as the vector $V_2\,.$

The following definition will be a key to the construction of the ladder
of Action Principles for the evolution vector $V_2$, and it permits a
generalization to the negative hierarchies as well as to other systems
(e.g., nonlinear Schr\"odinger equation):

\begin{defn} The
Galilean vector field $\eta\d{}{gal}$ is a time--independent vector field,
defined by three properties:
\begin{equation}\label{eq:defn:gal}
  \begin{array}{rcl}
\Lie{\eta\d{}{gal}}{R} &=& 0\,,\\
\Lie{\eta\d{}{gal}}{\Sigma^{(1)}} &=& 0\,,\\
\Lie{\eta\d{}{gal}}{V_{2}}&=&\alpha\,V_1\,,
  \end{array}
\end{equation}
where $\alpha$ is a numeric constant.
\end{defn}
As a consequence of the definition, it turns out that $\eta\d{}{gal}$ is a
Mastersymmetry \cite{Oev89} for the hierarchy $\{V_{j}\,,\quad
{j}=1,\,\ldots,\,\infty\}\,.$ Explicitly, we have $
\Lie{V_{j+1}}{\eta\d{}{gal}} = (\alpha + {j} -1)\,V_{j}\,,$ for $
{j}=1,\,\ldots,\,\infty\,. $

The aim is to construct time--independent standard Lagrangian pairs of the
form $(P;\,K)$, where $ \Lie{V_2}{P} = K\,,$ so that the Action Principles
will read $S[\u{q}{a}(t)] = \int_{t_-}^{t_+} dt\, \left(\d{P}{a}
\left(\dot{q}^{{\mathrm{a}}} - V_2^{{\mathrm{a}}}\right)+ K\right)\,, $
and the Euler--Lagrange equations involve the symplectic $2$--forms in the
ladder, i.e., $\delta P = \Sigma\,.$ See diagram in figure $3$.

\begin{thm}\label{thm:1}
The $1$--forms defined by $P^{(m)} \equiv
i_{\eta\d{}{gal}}\,\Sigma^{(m+1)}\,,$ for $
  {m}=-\infty,\,\ldots,\,\infty\,$
are ``integrals" of the symplectic $2$--forms in the ladder. That is to
say,
\begin{equation}\label{eq:lagr-Lad}
  \delta P^{(m)}=  {m}\,\Sigma^{(m)}\,,\quad
  {m}=-\infty,\,\ldots,\,\infty\,.
\end{equation}
\end{thm}

\begin{proof}
If we take the exterior derivatives of the $1$--forms, using the identity
\begin{equation}\label{eq:id-p-form}
  \Lie{\eta}{} = i_\eta\,\delta +\delta \,i_\eta\,,
\end{equation}
which holds (for every vector $\eta$) when operating on any $p$--form, we
find
\begin{equation*}
\delta P^{(m)} = \Lie{\eta\d{}{gal}}{}\Sigma^{(m+1)} =
\Lie{\eta\d{}{gal}}{}(\Sigma^{(1)}\cdot R^{m}) =  \Sigma^{(1)}\cdot
\Lie{\eta\d{}{gal}}{}(R^{m}) = {m}\,\Sigma^{(1)} \cdot R^{m-1} =
{m}\,\Sigma^{(m)}  \,,{{m}} \in {\mathbb{Z}}
\end{equation*}
after using the definition of Galilean vector and Leibnitz rule.
\end{proof}

In order to complete the Action Principles, there remains to find the
second members of the corresponding standard Lagrangian pairs.

\begin{thm}\label{thm:2}
The pair $(P^{(m)};K^{(m)})$ with $K^{(m)} \equiv \frac{\alpha}{m +
\alpha}\,i_{V_2}P^{(m)}\,,$ is a standard Lagrangian pair for the
evolution equation $\dot{q}\u{}{a} = V_2\u{}{a}\,,$ i.e.,
$\Lie{V_2}{P^{(m)}} = \delta K^{(m)}\,,$ and the Action Principle is
\begin{equation}\label{eq:action-KdV}
  S^{(m)}[\u{q}{a}(t)] = \int_{t_-}^{t_+} \,\d{P^{(m)}}{a}\,\left(
\dot{q}^{{\mathrm{a}}}- \frac{m}{m + \alpha}\,V_2\u{}{a}\right)\,dt\,.
\end{equation}

Moreover, the $0$--forms $K^{(m)}$ are constants of the motion for the
evolution equation:
\begin{equation*}
\Lie{V_2}{K^{(m)}} = 0\,.
\end{equation*}

\end{thm}

\begin{rem}The case ${m}=0$ leads to a trivial Action Principle, for the
Euler--Lagrange equations are identically zero: it is shown that this case
leads to a constant of the motion. However, the symplectic $2$--form
$\Sigma^{(0)}$ may be integrated by hand in order to get a nontrivial
Action Principle (see the end of this section).
\end{rem}

\begin{proof}[Proof: Lagrangian Pairs]
Take Lie derivatives of the $1$--forms $P^{(m)}$ along the evolution
vector $V_2$, using Leibnitz rule:
\begin{equation*}
  \Lie{V_{2}}{P^{(m)}} = \Lie{V_{2}}{}i_{\eta\d{}{gal}} \Sigma^{(m+1)}
  = -\alpha\, i_{V_1}\Sigma^{(m+1)} = -\alpha \,i_{V_{2}}\Sigma^{(m)}\,.
\end{equation*}
But, using the identity (\ref{eq:id-p-form}) and the result
(\ref{eq:lagr-Lad}) we rewrite the last expression to get
\begin{equation*}
  \Lie{V_2}{P^{(m)}} =-\alpha \left(\Lie{V_{2}}{P^{(m)}} - \delta\,
  i_{V_{2}}P^{(m)}\right)\,,
\end{equation*}
therefore
\begin{equation}\label{eq:Lagr-pair-KdV}
  \Lie{V_{2}}{}P^{(m)} = \delta \left(\frac{\alpha}{m +
\alpha}\,i_{V_{2}}P^{(m)}\right) \equiv \delta  K^{(m)}\,.
\end{equation}
\end{proof}

\begin{proof}[Proof: Constants of the Motion]
We use the above result (\ref{eq:Lagr-pair-KdV}), to find
\begin{equation*}
\Lie{V_{2}}{K^{(m)}} = \frac{\alpha}{m +
\alpha}\,i_{V_{2}}\Lie{V_{2}}{P^{(m)}} = \frac{\alpha}{m + \alpha}\,
i_{V_{2}}\delta K^{(m)} = \frac{\alpha}{m + \alpha}\, \Lie{V_{2}}{}
K^{(m)}\,,
\end{equation*}
which implies $\Lie{V_{2}}{K^{(m)}} = 0\,,\quad {m} \neq 0\,. $
\end{proof}

For the KdV equation, when ${m}>0$ these correspond to the usual
denumerably infinite set of constants of the motion.\cite{Eil81} Notice
that this theorem represents also a constructive method to obtain such
constants.
\begin{proof}[Proof: Action Principles]
The Action Principles (\ref{eq:action-KdV}) arise directly from equation
(\ref{eq:action}), using the definition of $K^{(m)}$.
\end{proof}


From the point of view of theorem \ref{thm:2}, the case ${m}=0$ actually
leads to a time--dependent constant of the motion. From equation
(\ref{eq:lagr-Lad}), it follows that $P^{(0)} = \delta C^{(0)}\,,$ and
thus $\Lie{V_{2}}{}C^{(0)} = K^{(0)}\,,$ where $K^{(0)} =
i_{V_{2}}P^{(0)}\,.$ Although we cannot prove that $K^{(0)}$ is a constant
of the motion, it holds in the examples, so we may take it as an extra
assumption, obtaining the following time--dependent constant of the motion
for the evolution vector $V_{2}$:
\begin{equation}\label{eq:t--d const}
  C[q\u{}{a}(t),\,t] = C^{(0)} - t\,K^{(0)}\,.
\end{equation}

Finally, for the case ${m}=0\,,$ a special (``missing") Action Principle
is constructed by hand from integration of the $2$--form $\Sigma^{(0)}\,,$
which leads to the $1$--form $P^{(M)}\,,$ such that $\delta P^{(M)} =
\Sigma^{(0)}\,.$ We will have $\Lie{V_{2}}{P^{(M)}} = \delta K^{(M)}\,,$
and the Action is
\begin{equation*}
  S^{(0)}[q\u{}{a}(t)] = \int_{t_-}^{t_+} \, \left(\d{P^{(M)}}{a}\,\left(
\dot{q}^{{\mathrm{a}}}- V_{2}\u{}{a}\right) + K^{(M)}\right)\,dt\,.
\end{equation*}


\subsection{The Euler--Lagrange Equations as Deformed Evolution Equations}

The Euler--Lagrange equations that arise from variation of each Action
$S^{(m)}\,,\quad {m} \in \mathbb{Z}\,$ are, apart from nonzero numeric
factors,
\begin{equation*}
  \Sigma^{(m)}\d{}{a b}(\dot{q}\u{}{b} - V_{2}\u{}{b}) = 0\,,\quad {m} \in \mathbb{Z}\,.
\end{equation*}
The kernel of the symplectic operators, ${\mathrm{Ker}} \Sigma^{(m)}\,,$
is of importance here. For each Action Principle we obtain an equivalent,
\textbf{deformed}, evolution equation:
\begin{equation*}
  \dot{q}\u{}{a} = V_{2}\u{}{a} + \sum_{j=1}^{N_{m}}
  \theta_j\,\eta_{j;{m}}\u{}{a}\,,
\end{equation*}
where $N_{m} = {\mathrm{dim}}({\mathrm{Ker}} \Sigma^{(m)})\,,$ the vectors
$\{\eta_{j;{m}}\}_{j=1}^{N_{m}}$ generate the kernel of $\Sigma^{(m)}\,,$
and $\theta_j$ are arbitrary $0$--forms: it can be said that these
Euler--Lagrange equations and the Action Principles acquire extra
symmetries (as compared to the symmetries of the original equations).

As the $2$--forms here are formed by contraction of powers of the
hereditary operators $R$ and $R^{-1}$ with $\Sigma^{(1)}$, it is clear
that the kernel of the $2$--forms are computed essentially from vectors in
the kernel of the operators $R^m$ and $R^{-m}\,,\,$ for $m>0:$ as we have
mentioned, these are the positive and negative evolution vectors. In the
next section, we will find explicitly the deformed equations for the KdV
equation in terms of the positive and negative vectors.

\subsection{Construction of local and nonlocal Constants of the Motion from Symmetries and non--standard Lagrangian $1$--forms}
Let us assume, as it will be demonstrated in subsection \ref{subsec:neg
lag KdV} for the KdV case (under usual boundary conditions), that the
constants of the motion from theorem \ref{thm:2} are $K^{(m)} = 0$ or a
numeric constant for $m \leq -2\,.$ This implies that $\Lie{V_2}{}P^{(m)}
= 0\,,$ i.e., $P^{(m)}$ is a non--standard Lagrangian $1$--form for the
flow $V_2$. Assume also that the evolution equation defined by the flow
$V_2$ possesses a symmetry $\eta\,.$ Then
\begin{thm}\label{thm:3}The $0$--forms defined by $Q^{(m)} \equiv i_\eta P^{(m)}\,,$
for $m \leq -2\,,$ are constants of the motion for the flow $V_2\,,$ i.e.,
$\left(\partial_t + \Lie{V_2}{}\right){Q^{(m)}} = 0\,$ for $m \leq -2\,.$
\end{thm}
\begin{proof}The proof follows directly from Leibnitz rule.
\end{proof}
In subsection \ref{subsec:neg lag KdV} we find a concise expression of
nonlocal constants of the motion for the KdV equation, obtained with
$\eta$ equal to a nonlocal internal symmetry for KdV.

\section{Example: The KdV Equation}
\label{sec:example KdV}
\subsection{Known Objects}
We begin by presenting the Nijenhuis operator relevant for the KdV
hierarchy: as an operator,\cite{Fuc80} $R[u] = \D^2 + 8\, u + 4\, u_x
\,\D^{-1}\,,$ where $\D$ and $\D^{-1}$ are, respectively, the derivative
and the anti--derivative operators: $\D f(x)  \equiv  \frac{\partial
f(x)}{\partial x}\,,$ $\,\,\D^{-1}g(x)  \equiv
\int_{x_-}^{x_+}\epsilon(x-x')\,g(x')\,dx'\,,$ with $\epsilon(x-x') =
1/2\,{\mathrm{sign}}(x-x')$.

Next, the  \textbf{positive} hierarchy begins with the vector
$V\ud{}{(1)}{1}[u] = -u_x\,.$ The second vector in the positive hierarchy
is obtained after application of the Nijenhuis operator on the latter
vector: $V\ud{}{(1)}{2}[u] = R[u]\cdot V\ud{}{(1)}{1}[u] =
  -u_{xxx}-12\,u\,u_x \equiv V_2[u]\,.$ We see it represents the KdV equation (\ref{eq:KdV}).

Next, we write the first symplectic $2$--form: as an operator,\cite{Fuc80}
$\Sigma^{(1)}[u] = \D^{-1}\,.$ The second symplectic operator is
constructed just by contracting the latter operator with the Nijenhuis
operator: $ \Sigma^{(2)}[u] =   \Sigma^{(1)}[u] \cdot R[u] = \D +
4\,u\,\D^{-1} + 4\,\D^{-1}\,u\,.\,$ These operators are closed under usual
boundary conditions for the vector fields: the ladder, then, contains only
closed $2$--forms. Now, it is easy to show that these operators are
symplectic for the flow defined by $V\ud{}{(1)}{1}$, therefore all
operators in the ladder are symplectic for the KdV flow $V_{2}$ (see first
paragraphs of section \ref{sec:Ladders}).

Finally, the Galilean vector is just the time--independent part of the
Galilean symmetry (\ref{eq:gal}): $ \eta\d{}{gal}[u] = \frac{1}{8}\,, $
and the constant in the last of the defining equations (\ref{eq:defn:gal})
is $\alpha = 3/2\,.$

\subsection{Explicit form of the KdV Negative Hierarchies: Linear
generalization and factorization of the Hereditary Operator}

In order to find an explicit and concise expression for the KdV
\textbf{negative} hierarchies, we factorize a generalization of the
hereditary operator $R$, which is obtained by addition of a multiple of
the identity tensor ${\mathbb{I}}$:
\begin{equation*}
  R(\lambda)[u] \equiv R[u] + 4\, \lambda\,{\mathbb{I}}\,.
\end{equation*}
This is also a hereditary operator, for fixed $\lambda$, which is taken as
an arbitrary real number.

The idea behind this generalization, is that the kernel of $R(\lambda)[u]$
contains all the negative hierarchies in its Taylor expansion around
$\lambda= 0\,,$ so we will write the negative hierarchies in a compact
way.

\begin{lem}\label{lemLeft}
The vectors
$V\ud{}{(-1)}{1}(\lambda)[u],\,V\ud{}{(-2)}{1}(\lambda)[u],\,V\ud{}{(-3)}{1}(\lambda)[u]$,
defined by
\[V\ud{}{(k)}{1}(\lambda)[u] \equiv \sum_{{{n}}=0}^{\infty}(-4\,\lambda)^{{n}}
\,V\ud{}{(k)}{n+1}[u]\,,\quad {{k}} = -1,\,-2\,-3\,\]
 generate the vectorial kernel of $R(\lambda)[u]$.

Conversely, all vectors in the negative hierarchies may be obtained from
the vectorial kernel of $R(\lambda)[u]$:
\begin{equation*}
V\ud{}{(k)}{n+1}[u] = \frac{1}{{n}!}(-4)^{-{n}}
\left.\frac{\partial^{{n}}}{\partial\lambda^{{n}}}
V\ud{}{(k)}{1}(\lambda)[u]\right|_{\lambda = 0}\,,\quad {{k}} =
-1,\,-2\,-3 \,,\quad {{n}} = 0,\,1,\ldots,\,\infty\,.
\end{equation*}
\end{lem}

\begin{proof}Consider the action of $R[u]$ on the vector
$V\ud{}{(k)}{1}(\lambda)[u]\,.$ Using equations (\ref{eq:kerR}) and
(\ref{eq:recNeg}), we get
\begin{equation*}
  R[u] \cdot V\ud{}{(k)}{1}(\lambda)[u] = \sum_{{{n}}=0}^{\infty}(-4\,\lambda)^{{n}}
\,  R[u] \cdot V\ud{}{(k)}{n+1}[u] =
\sum_{{{n}}=1}^{\infty}(-4\,\lambda)^{{n}} \, V\ud{}{(k)}{n}[u] = -4
\,\lambda\, V\ud{}{(k)}{1}(\lambda)[u]\,,
\end{equation*}
therefore $R(\lambda)[u]\cdot V\ud{}{(k)}{1}(\lambda)[u] = (R(\lambda)[u]
+ 4 \,\lambda )\cdot V\ud{}{(k)}{1}(\lambda)[u] = 0\,.$
\end{proof}

The factorization process \cite{Vil01} implies the definition of auxiliary
fields, which are directly related to nonlocal pre--potentials found in
the literature \cite{Gut93} and to the associated isospectral linear
eigenvalue problem: \cite{Lax68}
\begin{equation}\label{eq:lin-eig}
\psi_{xx} + 2 \, u \, \psi = -\lambda\,\psi\,,
\end{equation}
where $\psi = \psi(x,t;\lambda)\,.$ As usual, we assume $\lambda_t = 0\,,$
and $u = u(x,t)$ is independent of $\lambda$.

Alternatively, we write the above equation as: $L(\lambda)\cdot \psi = 0$,
where $L(\lambda) = D^2+2\, u +\lambda$, so that elements in the kernel of
the \textbf{Lax operator} $L(\lambda)$ are solutions of the linear problem
(\ref{eq:lin-eig}). This operator is easily factorizable: $L(\lambda) =
\frac{1}{\psi}\,\D\,\psi^2\,\D\,\frac{1}{\psi}\,.$ From this
factorization, we construct directly two l.i. solutions of the linear
problem: $\psi(\lambda) \equiv \psi(x,t;\lambda)$ and
\begin{equation}\label{eq:2ndSol}
 \bar{\psi}(\lambda) \equiv {\psi}(\lambda)
\D^{-1}(1/{\psi}(\lambda)^2)\,.
\end{equation}
The above eigenvalue problem may be understood as an extended coordinate
system labelled by $\psi(x,t;\lambda)$, with $\lambda$ as an additional
variable (just like $x$), and which however must solve an extra equation,
$L(\lambda)\cdot \psi(\lambda) = 0$, which we call \textbf{constraint}.
This constraint let us write derivatives of the field $\psi(\lambda)$ with
respect to $\lambda$ in terms of the field itself, in a nonlocal way. This
will be useful in the next section, when we write the negative ladder of
Action Principles. We obtain, apart from integration constants,
\begin{equation}\label{eq:local--nonlocal}
\begin{array}{rcl}
\frac{\partial^n}{\partial \lambda^{n}} \psi
&=&(-1)^n\,n!\,L(\lambda)^{-n}\,\psi\,,\\
\frac{\partial^n}{\partial \lambda^{n}} \bar{\psi}
&=&(-1)^n\,n!\,L(\lambda)^{-n}\,\bar{\psi}\,,\quad n\geq 1\,,
\end{array}
\end{equation}
where $L(\lambda)^{-1} =
{\psi}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,{\psi}\,,$  $\psi =
\psi(\lambda)$ and $\bar{\psi} = \bar{\psi}(\lambda)\,.$

Now, the factorization of $R(\lambda)[u]$ is found to be
\[R(\lambda)[u] =
\frac{1}{\psi(\lambda)^2}\,\D\,\psi(\lambda)^2\,\D\,\psi(\lambda)^2\,\D\,\frac{1}{\psi(\lambda)^2}\,\D^{-1}\,
.\] It is remarkable that the above operator is \emph{linear} in
$\lambda$, which is a consequence of the eigenvalue equation
(\ref{eq:lin-eig}).

The kernel of the operator $R(\lambda)[u]$ is easily found to be composed
by three nonlocal vectors:
\begin{equation}\label{eq:negKdV}
\begin{array}{rcl}
u_t =     {V}\ud{}{(-1)}{1}(\lambda)[u] & = & \left({\psi(\lambda)^2}\right)_x \,,\\
u_t =     {V}\ud{}{(-2)}{1}(\lambda)[u]& = & \left(\psi(\lambda)\,\bar{\psi}(\lambda)\right)_x \,,\\
u_t ={V}\ud{}{(-3)}{1}(\lambda)[u] & =
&\left(\bar{\psi}(\lambda)^2\right)_x\,, \
\end{array}
\end{equation}
where $\psi(\lambda),\,\bar{\psi}(\lambda)$ (see equation
(\ref{eq:2ndSol})) are two l.i. solutions of the eigenvalue equation
(\ref{eq:lin-eig}). These vectors contain the whole negative hierarchies
if $\lambda$ is left arbitrary, as it is stated in lemma \ref{lemLeft}.
These negative vectors have been constructed as nonlocal symmetries of the
KdV equation in \cite{Gut93} using a discrete infinite dimensional
representation. In the continuous representation (\ref{eq:negKdV}) of
these vectors, more suitable for field theory, we consider them also as
possible evolution equations, which will be identified as known integrable
equations in subsection \ref{subsec:examples neg}.

\subsection{Explicit form and Kernel of the Inverse Hereditary Operator $R^{-1}$}

The inverse hereditary operator, $R^{-1}(\lambda)$, is found easily after
inverting every factor. Assuming appropriate boundary conditions on the
field $\psi$, we get:
\[R^{-1}(\lambda)[u] =
\D\,{\psi(\lambda)^2}\,\D^{-1}\,\frac{1}{\psi(\lambda)^2}\,\D^{-1}\,\frac{1}{\psi(\lambda)^2}\,\D^{-1}\,{\psi(\lambda)^2}\,\,
.\]
 It is easy to show that the kernel of this operator is generated by the
 vector $V\ud{}{(1)}{1} = -u_x\,.$

\subsection{Factorization of Positive and Negative Symplectic Operators}

Using the factorized form of the hereditary operators, we get easily the
symplectic operators in factorized form. For the positive ones, we have
\begin{equation*}
\begin{array}{rcl}
\Sigma^{(2)}[u] &=&
\D^{-1}\,\frac{1}{\psi^2}\,\D\,\psi^2\,\D\,\psi^2\,\D\,\frac{1}{\psi^2}\,\D^{-1}\,,
\\
\Sigma^{(3)}[u] &=&
\D^{-1}\,\frac{1}{\psi^2}\,\D\,\psi^2\,\D\,\psi^2\,\D\,\frac{1}{\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D\,\psi^2\,\D\,\psi^2\,\D\,\frac{1}{\psi^2}\,\D^{-1}\,,
\end{array}
\end{equation*}
and so on, where $\psi=\psi(\lambda=0)\,.$ Notice that the inverses of
these operators give new nonlocal Hamiltonian operators for KdV.

For the negative ones, on the other hand, we have
\begin{equation}\label{eq:neg sym}
\begin{array}{rcl}
 \Sigma^{(0)}[u] &=&
 {\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,{\psi^2}\,,
\\
\Sigma^{(-1)}[u] &=&
{\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,{\psi^2}\,\D\,{\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,{\psi^2}\,,
\end{array}
\end{equation}
and so on. By the way, the above expression for $\Sigma^{(0)}[u]$ turns
out to solve a puzzle in the recent literature, \cite{Nut00} for it is the
inverse of Magri's Hamiltonian operator. As we see, we have got all
inverses of all Hamiltonian operators within the multi--Hamiltonian
structure.

There is another, concise way to write these negative operators, which
resembles the way we wrote the negative vectors in terms of the
$\lambda$--dependent first one. We state the lemma without proof:
\begin{lem}\label{lem:symp}
The negative symplectic operators for KdV are written in terms of
$\Sigma^{(0)}(\lambda)[u] \equiv \Sigma^{(1)}\cdot R^{-1}(\lambda)[u]$ in
the following way:
\begin{equation*}
\Sigma^{(-n)}[u] = \left.\frac{1}{{n}!}(-4)^{-{n}}
\frac{\partial^{{n}}}{\partial\lambda^{{n}}}
\Sigma^{(0)}(\lambda)[u]\right|_{\lambda=0}\,,\quad {n} \geq 1\,.
\end{equation*}
\end{lem}


\subsection{Negative Vectors as Kernel of Positive Symplectic Operators}

The kernel spaces ${\mathrm{Ker}} \Sigma^{(n)}[u]\,,$ for $
{n}=1,\,\ldots,\,\infty\,,$ are easily computed in terms of the kernel of
positive powers of $R$, from the fact that ${\mathrm{Ker}}
\Sigma^{(1)}[u]$ is null. We get
\begin{equation*}
{\mathrm{Ker}} \Sigma^{(n+1)} = {\mathrm{span}}\{V\ud{}{(k)}{m}[u]\,,\quad
{k} = -1,\,-2,\,-3,\quad {m} = 1,\ldots,{n}\, \}\,,\quad {n} \geq 0 \,.
\end{equation*}

\subsection{Positive Vectors as Kernel of Negative Symplectic Operators}

Finally we compute the kernel spaces ${\mathrm{Ker}} \Sigma^{(n)}[u]\,,$
for ${n}=0,\,-1,\,\ldots,\,-\infty$. It is easily seen that the operator
$\Sigma^{(0)}[u]$ has a null kernel. This time we have to evaluate the
kernel of negative powers of $R$. We get
\begin{equation*}
{\mathrm{Ker}} \Sigma^{(-n)}[u] =
{\mathrm{span}}\{V\ud{}{(1)}{m}[u]\,,\quad {m} = 1,\ldots,{n}\, \}\,,\quad
{n} > 0 \,,
\end{equation*}
so that, in particular, the Action Principle  for the KdV equation
associated to $\Sigma^{(-2)}[u]$ has the translation vector as well as the
KdV vector as generators of its kernel, therefore the action has to be
time--reparametrization invariant (we will see an example soon).



\subsection{Action Principles for KdV: Positive Lagrangian Ladders}

\begin{rem}If $P\d{}{x}$ denotes a $1$--form, and $\mathrm{x}$ is a continuous index,
we will write the $1$--form as ${\mathcal{P}}(x,t)$ (which looks more like
a density) when dealing with it inside an integral sign.
\end{rem}

Following theorems \ref{thm:1} and \ref{thm:2}, we write the Action
Principles from equation (\ref{eq:action-KdV}):
\begin{equation}\label{eq:action ladder KdV}
S^{(m)}[u(x,t)] = \int_{t_-}^{t_+} \, \int_{x_-}^{x_+}
\,{\mathcal{P}}^{(m)}[u](x,t)\,\left(u_t + \frac{m}{m + 3/2}\,
(u_{xxx}+12\,u\,u_x))\right)\,dx\,dt\,,
\end{equation}
for ${m} > 0\,,$ where
\begin{equation*}
{\mathcal{P}}^{(1)}[u](x,t) = i_{\eta\d{}{gal}}\Sigma^{(2)}[u] = -(\D +
4\,u\,\D^{-1} + 4\,\D^{-1}\,u)\frac{1}{8} = -\frac{1}{2}\left(x\,u +
\D^{-1}(u)\right)\,,
\end{equation*}
and successive $1$--forms are defined by recurrence:
${\mathcal{P}}^{(m+1)}[u](x,t) = R^\dag[u] \cdot
{\mathcal{P}}^{(m)}[u](x,t)$, where ${R}^\dag[u]$ is the transpose
Nijenhuis operator.

The Action Principle $S^{(1)}[u(x,t)]$ gives rise to the following
Euler--Lagrange equations:
\begin{equation*}
  \D^{-1}(u_t + u_{xxx}+12\,u\,u_x) = 0\,,
\end{equation*}
which are equivalent to KdV. The associated constant of the motion is:
\begin{equation*}
  H^{(1)}[u] \equiv i_{V_{2}} P^{(1)} = \frac{1}{2}\int dx\,\left(x\,u
+ \D^{-1}(u)\right)\left(u_{xxx}+12\,u\,u_x\right) = \frac{5}{4}\,\int
dx\,\left(u_x^2-4\,u^3\right)\,,
\end{equation*}
which is a member of the known set.

The next Action Principle is written as equation (\ref{eq:action ladder
KdV}), with
\begin{equation*}
{\mathcal{P}}^{(2)}[u](x,t) = R^\dag[u]\cdot{\mathcal{P}}^{(1)}[u](x,t) =
-\frac{1}{2}\left(3\,u_x + x\,u_{xx} +6\,x\,u^2 + 4\,u\,\D^{-1}(u) +
6\,\D^{-1}(u^2) \right)\,.
\end{equation*}
The Euler--Lagrange equations are: $2\,\left(\D + 4\,u\,\D^{-1} +
4\,\D^{-1}\,u\right)(u_t + u_{xxx}+12\,u\,u_x) = 0\, $ or, in factorized
form,
 $\,2\,\D^{-1}\,\frac{1}{\psi^2}\,\D\,\psi^2\,\D\,\psi^2\,\D\,\frac{1}{\psi^2}\,\D^{-1}\,(u_t
+ u_{xxx}+12\,u\,u_x) = 0\,.\,$ These equations are not equivalent to the
original equations: instead, they are equivalent to the deformed equations
\begin{equation*}
  u_t =- u_{xxx} - 12\,u\,u_x + \theta_1\,\left(\psi^2\right)_x+ \theta_2\,\left(\psi\,\bar{\psi}\right)_x+ \theta_3\,\left(\bar{\psi}^2\right)_x\,,
\end{equation*}
where $\theta_j$ are arbitrary $0$--forms that multiply the generators of
${\mathrm{Ker}} \Sigma^{(2)}[u]\,,$ and $\psi = \psi(\lambda=0)\,.$

We could continue this process \textbf{constructively}, obtaining
explicitly the constants of the motion and the Action Principles as well
as the Euler--Lagrange equations: the original KdV equation gets deformed
with vectors in the negative hierarchies, as it was mentioned before.

\subsection{Action Principles for KdV: Negative Lagrangian Ladders}
\label{subsec:neg lag KdV} Now we turn to the construction of the negative
Action Principles for KdV, whose action functionals are defined by
\begin{equation*}
S^{(m)}[u(x,t)] = \int_{t_-}^{t_+} \, \int_{x_-}^{x_+}
\,{\mathcal{P}}^{(m)}[u](x,t)\,\left(u_t + \frac{m}{m + 3/2}\,
(u_{xxx}+12\,u\,u_x))\right)\,dx\,dt\,,
\end{equation*}
for ${m}<0\,.$ We only need to evaluate ${\mathcal{P}}^{(-m)}[u](x,t) =
(R^\dag[u])^{-m} \cdot {\mathcal{P}}^{(0)}[u](x,t)$, for $m<0\,.$ This is
done easily after stating the following corollary (from lemma
\ref{lem:symp}):

\begin{cor}\label{cor:neg1form}
The negative $1$--forms are obtained from the first negative one as
follows:
\begin{equation*}
{\mathcal{P}}^{(-n-1)}[u](x,t) = \left.\frac{1}{{n}!}(-4)^{-{n}}
\frac{\partial^{{n}}}{\partial\lambda^{{n}}}
{\mathcal{P}}^{(-1)}(\lambda)[u](x,t)\right|_{\lambda=0}\,,\quad {n} \geq
1\,,
\end{equation*}
where ${\mathcal{P}}^{(-1)}(\lambda)[u](x,t) =
-\frac{1}{8}\,\psi^2\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\psi^2$
or, using equation (\ref{eq:local--nonlocal}),
\begin{equation*}
{\mathcal{P}}^{(-1)}(\lambda)[u](x,t) = \frac{1}{16}\,\left(\psi_\lambda
\,\bar{\psi}-\psi\,\bar{\psi}_\lambda \right)\,.
\end{equation*}
\end{cor}
The first negative Action Principle from equation (\ref{eq:action ladder
KdV}) is thus:
\begin{equation*}
S^{(-1)}[u(x,t)] = \int_{t_-}^{t_+} \, \int_{x_-}^{x_+}
\,\frac{1}{16}\,\left( \psi_\lambda \,\bar{\psi}-\psi\,\bar{\psi}_\lambda
\right)\,\left(u_t + \frac{-1}{-1 + 3/2}\,
(u_{xxx}+12\,u\,u_x))\right)\,dx\,dt\,,
\end{equation*}
or, after some manipulations, $S^{(-1)}[u(x,t)] = \int_{t_-}^{t_+} \,
\int_{x_-}^{x_+} \,\left(\frac{1}{16}\,\left(\psi_\lambda \,\bar{\psi}
-\psi\,\bar{\psi}_\lambda \right)\,u_t -\frac{1}{4}\,u\right)\,dx\,dt\,, $
where we have to evaluate the fields $\psi,\,\bar{\psi}$ at $\lambda =
0\,.$ This Action Principle is highly nonlocal, even in terms of the
auxiliary fields (see equation (\ref{eq:local--nonlocal})). However, the
Euler--Lagrange equations are obtained as usual, varying the Action with
respect to the field $u$, and using the appropriate transformation
matrices. We obtain $-\Sigma^{(-1)}[u]\cdot(u_t + u_{xxx}+ 12\,u\,u_x) =
0\,,\,$ or explicitly, using the fact that the kernel of this operator is
generated by $V\ud{}{(1)}{1}\,,$
\begin{equation*}
  u_t =- u_{xxx}- 12\,u\,u_x + \theta_1 \,u_x\,,
\end{equation*}
where $\theta_1$ is arbitrary.

We stress there is no need to hesitate about the inclusion of auxiliary
fields in the negative Action Principles, for they are not varied
independently. Alternatively, we may map the above Action into a mixed
Action Principle, in which the fields $\psi$, $u$ and a Lagrange
multiplier $\rho$ are varied independently:
\begin{equation}\label{eq:action -1 Lax}
S^{(-1)}[\psi(x,t),\, u(x,t),\, \rho(x,t)] = \int_{t_-}^{t_+} \,
\int_{x_-}^{x_+} \,\left(\frac{1}{16}\,\frac{\psi_t}{\psi^3}\left(\D^{-1}
\psi^2\right) + \frac{1}{8}\,\frac{\psi_{xx}}{\psi} + \rho \left(u +
\frac{\psi_{xx}}{2\,\psi}\right) \right)\,dx\,dt\,.
\end{equation}
See \cite{Nut01} for a general discussion.


The next negative Action Principles are quite simple. Recall that the
evolution vector itself $V_{2}$ is in the kernel of the symplectic
operators $\Sigma^{(m)}\,,$ for ${m} = -2,\ldots, -\infty\,,$ so that the
Action Principles should be time--reparametrization invariant.\cite{Bus02}
From equation (\ref{eq:action-KdV}), the only chance is $K^{(m)}\propto
i_{V_{2}}P^{(m)} = 0$ or a numeric constant (which would not change the
Action Principle), for ${m}\leq -2\,.$ This is easily shown, from the fact
that the interior product $I(\lambda) =
i_{V\ud{}{(1)}{1}}P^{(-1)}(\lambda)[u] = -\frac{1}{16}\,\int_{x_-}^{x_+}
dx\,\left(\psi_\lambda \,\bar{\psi} -\psi\,\bar{\psi}_\lambda
\right)\,u_x\,, $ where $\psi=\psi(\lambda)\,,$ is a numeric constant for
all $\lambda$: $I(\lambda) = -\frac{1}{32}(x_+-x_-)\,.$ We get, then,
manifestly time--reparametrization invariant Actions:
\begin{equation*}
S^{(-n-1)}[u(x,t)] = \frac{1}{16\,({n}!)}\,(-4)^{-{n}}\,\int_{t_-}^{t_+}
\, \int_{x_-}^{x_+}
\,\frac{\partial^{{n}}}{\partial\lambda^{{n}}}\left.\left( \psi_\lambda
\,\bar{\psi}-\psi\,\bar{\psi}_\lambda
\right)\right|_{\lambda=0}\,u_t\,dx\,dt\,,\quad {n} \geq 1\,,
\end{equation*}
and we may use equation (\ref{eq:local--nonlocal}) in order to write the
$\lambda$--derivatives in terms of nonlocal expressions. For example, the
second negative Action Principle is
\begin{equation*}
S^{(-2)}[u(x,t)] = -\frac{1}{64}\,\int_{t_-}^{t_+} \, \int_{x_-}^{x_+}
\,\left.\left( \psi_{\lambda \lambda}
\,\bar{\psi}-\psi\,\bar{\psi}_{\lambda \lambda}
\right)\right|_{\lambda=0}\,u_t\,dx\,dt\,,
\end{equation*}
where $\psi_{\lambda\lambda} = 2
\,{\psi}\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,{\psi}^2\,\D^{-1}\,\frac{1}{\psi^2}\,\D^{-1}\,{\psi}^2$
and $\bar{\psi} = \psi\,\D^{-1}(1/\psi^2)\,;$ the Euler--Lagrange
equations are equivalent to
\begin{equation*}
u_t = \theta_1\,\left(u_{xxx} + 12\,u\,u_x\right) + \theta_2\,u_x\,
\end{equation*}
where, as usual, $\theta_j$ are arbitrary functionals. The invariance $t
\to \tau(t)$ is evident.

\subsection{The Missing Action Principle for KdV, a time--dependent Constant of the Motion and the Internal Vectors}

So far we have obtained two ladders of Action Principles for the KdV
equation: the positive (quasi local) and the negative ladders (highly
nonlocal). However, there is a missing Action Principle: this is the case
${m} = 0$, which is actually twofold: first, the $1$--form $P^{(0)}[u]
\equiv i_{\eta\d{}{gal}}\Sigma^{(1)}[u] = -x/8 $ is closed: $P^{(0)}[u] =
\delta C^{(0)}[u]\,,$ where $C^{(0)}[u] = -\int_{x_-}^{x_+} dx\,x\,u/8\,.$
From equation (\ref{eq:t--d const}), we obtain a known \cite{Eil81}
time--dependent constant of the motion for KdV:
$C[u,t]=\frac{1}{8}\,\int_{x_-}^{x_+} (6\,t\,u^2-x\,u)\,dx\,.$

Second, the Action Principle for the symplectic $2$--form
$\Sigma^{(0)}[u]$ (see equation (\ref{eq:neg sym})) has to be evaluated by
hand. After some hard but straightforward calculations, we find that the
$1$--form $\,{\mathcal{P}}^{(M)}[u](x,t) \equiv \frac{\psi^2}{4}
\D^{-1}\left(\frac{1}{\psi^2}\,\ln \psi\right)\,$ is a solution of
$\,\delta {P}^{(M)} = \Sigma^{(0)}\,.$

In this case, we map to the $\psi$--coordinate system for simplicity. We
get the Action Principle
\begin{equation*}
S^{(0)}[\psi,u,\rho] =
-\frac{1}{8}\,\int_{x_-}^{x_+}\left(\frac{\psi_x\,\psi_t}{\psi^2} -
\frac{\psi_{xx}^2}{\psi^2} + \rho\,\left(u +
\frac{\psi_{xx}}{2\,\psi}\right)\right)\,dx\,dt\,.
\end{equation*}
The Euler--Lagrange equations we obtain are: for the field $\psi$,
\begin{equation*}
\frac{1}{\psi}\,\D\,\frac{1}{\psi} \left(\psi_t + \psi_{xxx} -
3\frac{\psi_x\psi_{xx}}{\psi}\right) = 0\,,
\end{equation*}
for the field $u$: $u =-\frac{\psi_{xx}}{2\,\psi} \,\Rightarrow\,u_t =
  -u_{xxx}-12\,u\,u_x\,,$
and for the Lagrange multiplier:  $\rho = 0\,.$

Notice that the Euler--Lagrange equations for $\psi$ are equivalent to
\begin{equation*}
  \psi_t =- \psi_{xxx} +
3\frac{\psi_x\psi_{xx}}{\psi} + \theta_1\,\psi\,,
\end{equation*}
where $\theta_1$ is arbitrary. This symmetry is one of the three
known\cite{Gut93} internal symmetries (i.e., those which do not affect the
field $u$) of the eigenvalue problem (\ref{eq:lin-eig}). In the
$\psi$--coordinate system we write it as $\psi_t =  V\ud{}{(-2)}{0}[\psi]
= -\frac{1}{2}\,\psi\,$ (the numeric factor is only for simplicity). In
the cited reference it is shown that all negative vectors and the internal
symmetries span a loop algebra over $SL(2,{\mathbb{R}})$.

\subsection{Nonlocal Constants of the Motion for KdV}

As a final result, we will construct explicitly three new sets of
constants of the motion for the KdV equation, starting from the nonlocal
objects we have obtained. We denote $\psi=\psi(x,t;\lambda=0)$ for
simplicity from here on, unless explicitly stated. Consider the Action
Principle (\ref{eq:action -1 Lax}): the term which multiplies the velocity
$\psi_t$ is the mapping of the Lagrangian $1$--form $P^{(-1)}[u]$ into
$\psi$--coordinates:
\begin{equation*}
{\mathcal{P}}^{(-1)}[\psi](x,t) = \frac{1}{16\,\psi^3}\left(\D^{-1}
\psi^2\right)\,.
\end{equation*}
On the other hand, from theorem \ref{thm:2} we have $\Lie{V_2}{}{P}^{(-1)}
= \delta K^{(-1)}\,,$ where $K^{(-1)}
=\frac{3}{16}\,\int_{x_-}^{x_+}dx\,\frac{\psi_x^2}{\psi^2}\,.$ Consider
now the $0$--form
\begin{equation*}
  H^{(-2)}[\psi] \equiv -16\,i_{V\ud{}{(-2)}{0}}P^{(-1)} =
 - \int_{x_-}^{x_+}dx\,\frac{1}{\psi^2}\,\left(\D^{-1}\,\psi^2\right)\,
\end{equation*}
or, more concisely, $H^{(-2)}[\psi] =
\int_{x_-}^{x_+}dx\,\psi\,\bar{\psi}\,. $ We use Leibnitz rule to show
that this is a constant of motion for the KdV equation, which in
$\psi$--coordinates reads $V_2[\psi] = - \psi_{xxx} +
3\frac{\psi_x\psi_{xx}}{\psi}\,.$

As $\Lie{V_2}{V\ud{}{(-2)}{0}}[\psi] = 0\,,$ we find:
$\Lie{V_2}{}H^{(-2)}[\psi] =  \Lie{V\ud{}{(-2)}{0}}{K^{(-1)}}[\psi] =
  0\,,$ where the last equality comes from the fact that $K^{(-1)}[\psi]$ is
invariant under scaling of $\psi$.

Thus $H^{(-2)}[\psi]$ is a nonlocal constant of the motion for the KdV
flow. But if we recall that the fields $\psi,\,\bar{\psi}$ are solutions
of the linear problem (\ref{eq:lin-eig}), and that $u$ does not change if
these fields are replaced by other arbitrary linear combinations, we get
indeed three constants of the motion: $H^{(-1)}[\psi]  =
\int_{x_-}^{x_+}dx\,\psi^2\,,\, $
 $H^{(-2)}[\psi]  = \int_{x_-}^{x_+}dx\,\psi\,\bar{\psi}\,,\,$
 $H^{(-3)}[\psi]  = \int_{x_-}^{x_+}dx\,\bar{\psi}^2\,. $

There is a reference that supports this construction: in \cite{Lax68}, in
the context of the eigenvalue ``Schr\"odinger" problem (\ref{eq:lin-eig}),
the author assumes that the total probability (here denoted by
$H^{(-1)}[\psi]$) is equal to $1$. But it is indeed a constant of motion
of its own! Moreover, these are indeed special cases ($\lambda=0$) of more
general constants of motion. Along the same lines, we get three families,
parametrized by the eigenvalue $\lambda$:
\begin{equation}\label{eq:nonlocal constants}
 \begin{array}{rcl}
    H^{(-1)}(\lambda)[\psi] & = & \int_{x_-}^{x_+}dx\,\psi(\lambda)^2\,, \\
    H^{(-2)}(\lambda)[\psi] & =& \int_{x_-}^{x_+}dx\,\psi(\lambda)\,\bar{\psi}(\lambda)\,, \\
    H^{(-3)}(\lambda)[\psi] & =& \int_{x_-}^{x_+}dx\,\bar{\psi}(\lambda)^2\,. \
  \end{array}
\end{equation}
These are real new constants (indeed they contain, in their Taylor series
around $\lambda=0$, the constants of the motion from theorem \ref{thm:3}).
In order to evaluate them explicitly, take, for example, successive
derivatives of the first one with respect to $\lambda$, evaluate at
$\lambda=0$ and use equation (\ref{eq:local--nonlocal}). We get:
\begin{equation*}
Q^{(-1;n)}[\psi] \equiv \int_{x_-}^{x_+}dx\,\psi\,L^{-n}\,\psi\,,\quad n=
0,\ldots,\,\infty\,,
\end{equation*}
and we see they are increasingly nonlocal constants of the motion.

It is worth to mention that these nonlocal constants, when mapped to the
coordinate system in which the KdV equation maps into the Harry--Dym
equation (see \cite{Gol91}), reproduce the results obtained independently
in a recent work \cite{Bru02}, and add three more constants to the
Harry--Dym equation: the mappings of the nonlocal constants of motion
(\ref{eq:nonlocal constants}) for $\lambda=0\,$ into the Harry--Dym
equation $\omega_t = (\omega^{-1/2})_{sss}$ for the field $\omega(s,t)\,,$
are $H^{(-1)}[\omega]
 =  \int_{s_-}^{s_+}ds\,\omega\,,$
 $H^{(-2)}[\omega]  =
\int_{s_-}^{s_+}ds\,s\,\omega\,,$
 $ H^{(-3)}[\omega]  =
\int_{s_-}^{s_+}ds\,s^2\,\omega\,. $

\section{Results for other KdV Positive and Negative Equations}
\label{sec:results negative}
\subsection{Some Positive, Negative and Internal Vectors as known
Integrable Equations} \label{subsec:examples neg} We write the internal
vectors after transformation to Schwartzian coordinates, defined by $
\zeta_x(x,t;\lambda) =  {\psi(x,t;\lambda)^{-2}}\,. $ The internal vectors
in $\zeta$--coordinates are $V\ud{}{(-1)}{0}[\zeta] =  1\,,$
 $V\ud{}{(-2)}{0}[\zeta]  =  \zeta\,,$
 $V\ud{}{(-3)}{0}[\zeta]  = \zeta^2\,.$

The vectors $V\ud{}{(-3)}{0}\,,\,(V\ud{}{(-3)}{0} - V\ud{}{(-1)}{1})/2\,,$
and $V_2$ give the evolution equations $ \zeta_t =  \zeta^2\,,$ $\zeta_t =
\frac{1}{2}\,\zeta^2 - \frac{1}{8}\frac{\zeta_{x
  \lambda}}{\zeta_x}\,,$ and $\zeta_t = 6\,\lambda\,\zeta_x +
   \frac{3\,{\zeta_{xx}}^2}{2\,
      \zeta_x} -
   \zeta_{xxx}\,,$ where the fields are evaluated at $\lambda=0\,.$ The last of these
equations is the Krichever--Novikov \cite{Bus03} equation, and the first
and the second equations, via the transformation $z =
\ln\left(2\,\zeta_x\right)\,,$ may be mapped to the Liouville equation
$z_{x t} = \exp {z}\,,$ and the ShG equation $z_{x t}  = \sinh z\,.$ For
completeness, we just mention that the associated Camassa--Holm equation
\cite{Cam93} and the Hunter--Zheng equation \cite{Bru02} are obtainable
from the negative vectors $V\ud{}{(-2)}{1}$ and $V\ud{}{(-1)}{1}$,
respectively, via suitable coordinate transformations, and that the
Harry--Dym equation, \cite{Gol91} just like the Krichever--Novikov
equation, is a mapping of the KdV equation.


\subsection{Action Principles for the Sinh--Gordon Equation}
As a representative of the extension of the results on Action Principles
for equations in the negative hierarchies, we work out some examples for
the ShG equation. The results in this subsection are new up to our
knowledge, except when it is explicitly stated. We will work in the
$z$--coordinate system, where the ShG equation is $z_t = \d{V}{ShG}[z] =
\D^{-1}\,\sinh z\,.$

\subsubsection{Pure ShG equation: Symplectic matrix $\Sigma^{(0)}$\,}

We look for a standard Lagrangian pair for the ShG equation of the form
(${P}^{(M)}[z]$;\,${K}^{(M;ShG)}[z]$), where ${\mathcal{P}}^{(M)}[z] =
-\frac{1}{32}  z_x\,$ is the mapping of ${\mathcal{P}}^{(M)}[u]$ to
$z$--coordinates. The symplectic $2$-form $\u{\Sigma}{(0)}[z] =
\frac{1}{16}\, \D $ has only one vector in the kernel, namely
${V}\ud{}{(-2)}{0}[z] = 4\,.$ On the other hand, the standard Lagrangian
$0$--form solves $\nonumber \frac{\delta}{\delta z} {K}^{(M;ShG)}[z]  =
\Lie{\d{V}{ShG}}{{P}^{(M)}[z]}= -\frac{1}{32} \left(\sinh z - z\,\cosh z
\right) \,.$ We get after integration the usual Action Principle for the
ShG equation: \cite{Eil81}
\[S[z(x,t)] = \frac{1}{32}\int_{t_-}^{t_+} dt\,dx\,\left(-z_x\,z_t - 2\,\cosh z\right)\,,\]
and the Euler--Lagrange equations are simply $z_t  = \D^{-1}\,\sinh z +
\theta_1 \,,\,$ where $\theta_1$ is arbitrary.

\subsubsection{ShG equation deformed with first Positive Vector: Symplectic
matrix $\Sigma^{(-1)}$\,}

The next negative $1$--form, $\u{P}{(-1)}$, reads
$\u{\mathcal{P}}{(-1)}[z](x,t) = -\frac{1}{32} \,\,
{\mathrm{e}}^z\,\left(\D^{-1}\,{\mathrm{e}}^{-z}\right)\,.$ The associated
symplectic $2$--form is $\u{\Sigma}{(-1)}[z] = -\frac{1}{32}
\left({\mathrm{e}}^{z} \,\D^{-1}\,{\mathrm{e}}^{-z} + {\mathrm{e}}^{-z}
\,\D^{-1}\,{\mathrm{e}}^{z}\right)\,,$ which inherits the kernel
(generated from $\ud{V}{(1)}{1}[z] = - z_x $) from that in the
$u$--coordinate system only for special boundary conditions: defining the
boundary terms $\overline{f} \equiv f_+ +f_-,$ and $f_\pm = f(x_\pm),$ the
expression
\[\u{\Sigma}{(-1)}[z] \cdot \ud{V}{(1)}{1}[z] = \frac{1}{64} \, \left({\mathrm{e}}^z \,\overline{{\mathrm{e}}^{-z}} - {\mathrm{e}}^{-z} \,\overline{{\mathrm{e}}^{z}}\right)\,\]
is zero only for boundary conditions $z_+ = z_- + i \pi (2\,{{n}} +
1)\,,\quad {{n}} \in {\mathbb{Z}}$.

For other boundary conditions, however, this Lagrange bracket has no
kernel, which will show up in the variational principle for the ShG vector
by the fact that the Euler--Lagrange equations get deformed by a factor of
the vector $\ud{V}{(1)}{1}[z]$, which is not arbitrary: it depends on the
boundary conditions used for the $z$--coordinates.

In the generic case when $\overline{{\mathrm{e}}^{z}}\neq 0$ (invertible
symplectic $2$--form $\Sigma^{(-1)}$), the Action Principle is explicitly
\[S[z(x,t)] = \int_{t_-}^{t_+} dt\,\left[-\frac{1}{32}\int_{x_-}^{x_+} dx\, {\mathrm{e}}^z(\D^{-1}{\mathrm{e}}^{-z})(z_t - \D^{-1}\sinh z) + \u{K}{(-1;ShG)}[z]  \right]\,,\]
 and the Euler--Lagrange equations are
\[-\frac{1}{32}\left({\mathrm{e}}^{z} \,\D^{-1}\,{\mathrm{e}}^{-z} + {\mathrm{e}}^{-z} \,\D^{-1}\,{\mathrm{e}}^{z}\right)\left(z_t - \D^{-1} \sinh z + \alpha[A_+,\,A_-]\,z_x\right) = 0\,\]
or, equivalently,
\[z_t - \D^{-1} \sinh z + \alpha[A_+,\,A_-]\,z_x = 0\,,\]
where $\u{K}{(-1;ShG)}[z]  = \int_{x_-}^{x_+} dx\,{\mathrm{e}}^z
(\D^{-1}{\mathrm{e}}^{-z})^2/128 + F[A_+,\,A_-]\,,$ $\quad A_{\pm} \equiv
\int_{x_-}^{x_+}dx\,{\mathrm{e}}^{\pm z}\,,$ and $\alpha ,\,F$ solve the
equation:
\begin{equation}\label{Fa}
\delta F[A_+,\,A_-]  =
-\alpha[A_+,\,A_-]\,(\overline{{\mathrm{e}}^{-z}}\,\delta A_+ +
\overline{{\mathrm{e}}^{z}}\,\delta A_-) + \frac{1}{8} \,A_-\,(A_- - 2\,
A_+)\,\delta A_+\,.
\end{equation}

There are many solutions of the above equation for a given set of boundary
conditions on the limiting values of $z_\pm$, so we discuss, as examples,
only two representative, nonintersecting cases of boundary conditions, for
which the symplectic $2$--form $\Sigma^{(-1)}$ is invertible:



\begin{enumerate}
\item[(i)]{$z_+ = -z_- + i \pi (2\,{{n}})\,,\quad {{n}} \in
{\mathbb{Z}}\,; \quad \cosh z_+ \neq 0\,.$}

A solution of equation (\ref{Fa}) is $\alpha = - A_+\,A_- /{4
\,\overline{{\mathrm{e}}^z}}\,,\quad F= A_+\,A_-^2/512 \,,$
 which is well defined because of the boundary conditions
used.

\item[(ii)]{$z_+ = -z_- + i \pi (2\,{{n}} + 1)\,,\quad {{n}} \in
{\mathbb{Z}}\,; \quad \sinh z_+ \neq 0\,.$}

In this case, a solution of equation (\ref{Fa}) is
\[\alpha = - \left(A_+^2 - A_-^2 \right)/{8
\,\overline{{\mathrm{e}}^z}}\,,\qquad F= - \left(A_+^2\,A_- - \frac{1}{3}
(A_+^3 + A_-^3) \right)/{512}\,.\]

The usual constant of the motion for the ShG equation, $H[z] =
\int_{x_-}^{x_+} dx\,\cosh z\,,$ works in this case also: under the
boundary conditions used, we get
\[\dot{H}[z] = \int_{x_-}^{x_+} dx\,\sinh z \,(\D^{-1}\, \sinh z - \alpha\, z_x)
= \alpha \,\overline{\cosh z} = 0\,.\]
\end{enumerate}

\subsection{Alternative Lax Pairs and Constants of the Motion for Negative
Equations} In reference \cite{Bru95}, the authors find alternative Lax
pairs for the KdV equation (as well as for every evolution equation in the
KdV positive hierarchy) by making no ansatz: they just use the evolution
equation and the hereditary operator.

We present a similar construction, this time for the negative vectors. By
so doing we are answering an open question in reference \cite{Bru02}.

As it is shown in \cite{Bru95}, given an evolution equation $u_t = V[u]$,
and a recursion operator $R$ for $V$ (i.e., $\Lie{V}{\,R}= 0\,$), it
follows that
\begin{equation}\label{eq:Lax pair}
  \frac{\D}{\D t}R = \left[V',\,R\right]\,,
\end{equation}
where $V'$ denotes the Frechet derivative, and the square bracket is the
commutator. The above equation defines the alternative Lax Pair
$(R,\,V')$.

Now, take $R$ as the hereditary operator $R(\mu)[u]$, and $V$ as the
negative vector $V\ud{}{(-1)}{1}(\lambda)[u] = (\psi(\lambda)^2)_x$, for
arbitrary $\mu,\,\lambda\,.$ We need to evaluate the Frechet derivative of
this vector with respect to the field $u$. Using the transformation matrix
$\frac{\delta \psi}{\delta u} =
-2\,\psi\,\D^{-1}\,\frac{1}{\psi^2}\D^{-1}\,\psi^2\,,$ where $\psi$ stands
for $\psi(\lambda)$ from here on, we get $V\ud{}{(-1)}{1}(\lambda)'[u] =
-4\,\D\,\psi^2\,\D^{-1}\,\frac{1}{\psi^2}\D^{-1}\,\psi^2\,.$
 On the other hand, the hereditary operator is
\begin{equation*}
R(\mu)[u] = R(\lambda)[u] + 4\,(\mu-\lambda)\,{\mathbb{I}}=
\frac{1}{\psi^2}\,\D\,\psi^2\,\D\,\psi^2\,\D\,\frac{1}{\psi^2}\,\D^{-1} +
4\,(\mu-\lambda)\,{\mathbb{I}}\,,
\end{equation*}
and it is also written as $R(\mu)[u] = \D^{2} + 8\,u + 4\,u_x\,\D^{-1}\,.$
Now we apply the Lax pair equation (\ref{eq:Lax pair}), getting after some
rearrangements the operator equation $8\,u_t + 4\,u_{x t}\,\D^{-1} =
8\,(\psi^2)_x + 4\,(\psi^2)_{xx}\,\D^{-1}\,,$ which implies $u_t =
(\psi(\lambda)^2)_x\,.$ Recall this equation contains all the negative
vectors in the corresponding negative hierarchy, so that the Lax pair we
have presented indeed works for all vectors in that hierarchy. Similarly,
for the other two negative hierarchies we get the Lax pairs $(R,\,B_2)$
and $(R,\,B_3)$, with
\begin{equation*}
\begin{array}{rcl}
B_2 = V\ud{}{(-2)}{1}(\lambda)'[u]& = & -4\,\D\,\psi^2\,\D^{-1}\,{\bar{\psi}}\,{\psi}^{-1}\,\D\,{\bar{\psi}}\,{\psi}^{-1}\,\D^{-1}\,\left(\bar{{\psi}}\right)^{-2}\,\D^{-1}\,\psi\,\bar{\psi}\,, \\
B_3 = V\ud{}{(-3)}{1}(\lambda)'[u]& = &
-4\,\D\,\bar{\psi}^2\,\D^{-1}\,{\left(\bar{\psi}\right)^{-2}}\,\D^{-1}\,\bar{\psi}^2\,.\
\end{array}
\end{equation*}

In this way, constants of the motion for the negative vectors are
constructed from Adler traces of positive, semi--integer powers of the
Nijenhuis operator $R$: these are the usual (local) constants of the
motion for the KdV equation. \cite{Bru95} A natural conjecture is that
Adler traces of positive, semi--integer powers of $R^{-1}$ will give our
nonlocal constants for KdV defined in equation (\ref{eq:nonlocal
constants}). If that is true, we could infer that the nonlocal constants
of the motion for KdV should also work for the negative vectors, and they
do. We present the results only for the hierarchy
$V\ud{}{(-1)}{1}(\mu)\,,$ because Lie derivatives of the results along the
internal vector $V\ud{}{(-3)}{0}$ map the objects into similar ones for
the other two negative hierarchies.

\begin{enumerate}

\item[(i)]{}Conserved currents: defining the boundary term $\overbrace{f} \equiv
f(x_-)-f(x_+)\,,$ the integral $H^{(-1)}(\lambda)[\psi]  =
\int_{x_-}^{x_+}dx\,\psi(\lambda)^2\, $ solves
\begin{equation*}
\Lie{V\ud{}{(-1)}{1}(\mu)}{H^{(-1)}(\lambda)[\psi]} = -
  \overbrace{E(\lambda,\mu)[\psi]}\,,
\end{equation*}
where $E(\lambda,\mu)[\psi] \equiv
\frac{\psi(\mu)\,\psi_x(\lambda)-\psi(\lambda)\,\psi_x(\mu)}{\mu-\lambda}\,.$

\item[(ii)]{}Constants of the motion: the expression
\begin{equation*}
G^{(-1)}(\lambda,\mu)[\psi,t] = \int_{x_-}^{x_+}dx\,\psi(\lambda)^2 +
t\,\overbrace{\frac{E(\lambda,\mu)[\psi]^2}{1-t\,E(\mu,\mu)[\psi]^2}}\,
\end{equation*}
is a constant of the motion for the flow $V\ud{}{(-1)}{1}(\mu)\,.$

These constants and currents are infinite in number and work for every
vector in the respective negative hierarchy, because $\lambda$ and $\mu$
are arbitrary.
\end{enumerate}
\section{Conclusion}
\label{sec:conc}The Lagrangian point of view determines a unifying scheme
for the study of integrable equations belonging to hierarchies related to
hereditary operators. Nonlocal symmetries, Lax pairs, constants of the
motion and Action Principles all come out in a constructive, explicit way
from the same structure. Moreover, new equations, which are mixed or
deformed versions of known integrable equations, arise as the
Euler--Lagrange equations of the Action Principles obtained. We have
obtained results for the KdV equation, and other related equations in the
positive as well as in the negative KdV hierarchies (e.g., Harry--Dym and
Sinh--Gordon equation). The construction of alternative Lax pairs for
negative equations arises naturally, without any ansatz, from this scheme,
and it is shown that negative equations essentially share the constants of
the motion (local as well as nonlocal) of the KdV equation.

\section{Acknowledgements}
M.B. wishes to mention that this work was done in part during a research
stay at the Relativity Center, The University of Austin, Texas, U.S.A.,
where M.B. was invited by Prof. L. C. Shepley; M.B. is deeply grateful to
him, the professors and students in the Center for their support and the
following people whose comments and suggestions were fundamental for his
understanding of the topics involved in the present work: P. J. Morrison,
C. deWitt--Morette, C. Valls, J. Zanelli, C. Teitelboim and R. Troncoso.

Finally, M.B. acknowledges the financial support from a Fundaci\'on Andes
Grant for Doctoral Studies, and a Conicyt Grant for Thesis Completion.


\bibliographystyle{unsrt}
\bibliography{finalbib-kdv}




\newpage


 \epsfbox[110 436 576 833]{KdV015-fig1.ps}

\newpage
\epsfbox[110 316 576 713]{KdV015-figs2-3.ps}





\end{document}

**********************
\newpage

\begin{figure}[h]
\begin{center} \hspace{-2cm}
 \begin{tabular*}{145truemm}{lr}
    \cline{1-2}
\multicolumn{2}{c}{
 $\left.\begin{array}{cccccc}
%                    &                   &                             &                       &       &\\
\vspace{-3truemm}   &{\Ss R^{-1}(\lambda)}&    &{\Ss R^{-1}(\lambda)}&         &   \\
\vspace{-4truemm} \ldots &{\leftrightarrows }& V\ud{}{(-1)}{2}(\lambda)&{\leftrightarrows }& V\ud{}{(-1)}{1}(\lambda)&\\
    &{\Ss R(\lambda)}         &      &{\Ss R(\lambda)}&                   &  \\
                    &                   &                             &                       &
                    &\\
\vspace{-3truemm}   &{\Ss R^{-1}(\lambda)}&    &{\Ss R^{-1}(\lambda)}&         &   \\
\vspace{-4truemm}\ldots &{\leftrightarrows }& V\ud{}{(-2)}{2}(\lambda)&{\leftrightarrows }& V\ud{}{(-2)}{2}(\lambda)&\\
    &{\Ss R(\lambda)}         &      &{\Ss R(\lambda)}&                   &  \\
                    &                   &                             &                       &
                    &\\
\vspace{-3truemm}    &{\Ss R^{-1}(\lambda)}&    &{\Ss R^{-1}(\lambda)}&         &   \\
\vspace{-4truemm} \ldots &{\leftrightarrows }& V\ud{}{(-3)}{2}(\lambda)&{\leftrightarrows }& V\ud{}{(-3)}{1}(\lambda)&\\
\vspace{-3truemm}    &{\Ss R(\lambda)}         &      &{\Ss R(\lambda)}&                   &  \\
                    &                   &                             &                       &       &\\
                    &                   &                             &                       &
 \vline \hfill \textrm{ Kernel of } \hfill \vline                          &\\
                    &                   &                             &                       &\vline \hfill R(\lambda) \hfill \vline       &\
\end{array}\right.$
 $\left.\begin{array}{ccccc}
%                   &                             &                       &       &\\
                   &                             &                       &       &\\
                   &                             &                       &       &\\
                   &                             &                       &       &\\
\vspace{-3truemm}   &{\Ss R^{-1}}&    &{\Ss R^{-1}}&            \\
 \vspace{-4truemm} V\ud{}{(1)}{1}&{\leftrightarrows }& \;\;\;V\ud{}{(1)}{2}\;\;\;&{\leftrightarrows }&\ldots \\
\vspace{-4truemm}    &{\Ss R}         &      &{\Ss R}&                \\
\vspace{-2truemm}           &         & (\textrm{KdV}                &       &\\
    &             &      \textrm{Equation})          &       &\\
    &             &                &       &\\
    &             &                &       &\\
\vspace{-4.5truemm}     &             &                &       &\\
\vline \hfill \textrm{ Kernel of } \hfill \vline&
&                       & &\\
\vline \hfill R^{-1} \hfill \vline       & &                       & &\
\end{array}\right.$}\vspace{-6.7truemm} \\
$   \dashleftarrow \textrm{ Negative Hierarchies }\dashleftarrow $&$
\quad\dashrightarrow \textrm{ Positive Hierarchy}
\dashrightarrow  $ \\
 \cline{1-2}
\end{tabular*}

\end{center}
\caption{ Negative and Positive KdV Hierarchies of flow vectors. They
begin with vectors in the Kernel of the Hereditary Operators $R(\lambda)$
and $R^{-1}$. Successive members are generated in general by contraction
with the respective operators, or by derivation with respect to $\lambda$
(negative hierarchies). Lie derivation along the internal vector
$V\ud{}{(-3)}{0}$ moves down across different negative hierarchies. Lie
derivation along Galilean vector for $R$ moves to the negative end and
viceversa with the Galilean vector for $R^{-1}\,.$ Sinh--Gordon (as well
as Hunter--Zheng) and Camassa--Holm equations arise as evolution equations
using the negative vectors $V\ud{}{(-1)}{1}(\lambda)$ and
$V\ud{}{(-2)}{1}(\lambda)\,,$ respectively.}
 \end{figure}


\newpage
%\thispagestyle{empty}

 \begin{figure}[t]
\begin{center}
 \begin{tabular*}{145truemm}{|c|}
    \cline{1-1}
$\begin{array}{ccccccccccc}
 &&        &         &                          &                    &                   &                             &                       &       &\\
\vspace{-3truemm}&{\Ss R^{-1}(\lambda)}&       &{\Ss R^{-1}(\lambda)}&         &  {\Ss R^{-1}(\lambda)}              &                     &       {\Ss R^{-1}}                   &        &{\Ss R^{-1}}&\\
\vspace{-4truemm} \ldots &{\leftrightarrows }&  \Sigma^{(-1)}(\lambda)&\leftrightarrows                  & \Sigma^{(0)}(\lambda)&\leftrightarrows                  &   \;\;\;\Sigma^{(1)}\;\;\;      &\leftrightarrows                         &\;\;\;\Sigma^{(2)}\;\;\; & \leftrightarrows &  \ldots \\
       &{\Ss R(\lambda)}&                         &{\Ss R(\lambda)}&                   &  {\Ss R(\lambda)}                 &                     &       {\Ss R}                          &       & {\Ss R}&\\
&&        &         &                           &                    & & &
&       &\
 \end{array}$\\
$\begin{array}{ccccc}
\dashleftarrow      &\textrm{Negative Ladder}&   \dashleftarrow\parallel\dashrightarrow &    \textrm{Positive Ladder}   &    \dashrightarrow   \\
 \end{array}$\\
    \cline{1-1}
 ${\Sigma}^{(-n)}(\lambda)[u] =
\frac{1}{{{n}}!}{\left(-\frac{1}{4}\frac{\partial}{\partial
\lambda}\right)}^{n}
\Sigma^{(0)}(\lambda)[u]\,, \quad {{n}} = 1,\ldots,\infty.\ $\\
    \cline{1-1}
\end{tabular*}
\end{center}
\caption{Negative and Positive Ladders of closed Symplectic Operators.
They are generated from successive contractions of $\Sigma^{(1)}$ with the
Hereditary Operators. Symplectic operators in the negative ladder may be
generated by derivation of $\Sigma^{(0)}(\lambda)$ with respect to
$\lambda$. Lie derivation along the Galilean vector for $R$ moves to the
negative end and viceversa with the Galilean vector for $R^{-1}\,.$}
 \end{figure}

%\newpage


\begin{figure}[b]
\begin{center}


 \begin{tabular*}{143truemm}{|c|c|c|}
    \cline{1-3}
\multicolumn{3}{|c|}{
 $\begin{array}{ccccccccccc}
   &&     &         &                          &                    &                   &                             &                       &       &\\
\vspace{-3truemm}   &{\Ss R^{-1}(\lambda)}&    &{\Ss R^{-1}(\lambda)}&         &  {\Ss R^{-1}}              &                     &       {\Ss R^{-1}}                   &        &{\Ss R^{-1}}&\\
\vspace{-4truemm}\ldots &{\leftrightarrows }& P\u{}{(-1)}(\lambda)&{\leftrightarrows }& \;\;\;P\u{}{(0)}\;\;\;&\leftrightarrows                  &   \;\;\;P\u{}{(1)}\;\;\;      &\leftrightarrows                         &\;\;\;P\u{}{(2)}\;\;\; &\leftrightarrows&\ldots\\
    &{\Ss R(\lambda)}         &      &{\Ss R(\lambda)}&    \textrm{\scriptsize(closed)}               &  {\Ss R}                 &                     &       {\Ss R}                          &       & {\Ss R}&\\
    &                      &        &          &                           &                    &                   &                             &                       &       &\
 \end{array}$}\\
\multicolumn{3}{|c|}{ $\begin{array}{ccccc} \dashleftarrow
&\textrm{Negative Ladder}& \dashleftarrow\parallel\dashrightarrow &
\textrm{Positive Ladder} & \dashrightarrow   \
 \end{array}$}\\
    \cline{1-3}
  $\delta P^{(n)}(\lambda) = {n}\,\Sigma^{(n)}(\lambda)\,,\quad {n} \leq 0
$& $ \delta P^{\Ss(M)}(\lambda)=\Sigma\u{}{(0)}(\lambda)$ & $ \delta P^{(n)} = {n}\,\Sigma^{(n)}\,,\quad {n} > 0 $\\
 \cline{1-3}
\end{tabular*}

\end{center}
\caption{Negative and Positive Ladders of Standard Lagrangian
\mbox{$1$--forms}. Their exterior derivatives are proportional to the
Symplectic Operators in the Ladders and thus define nontrivial Action
Principles, except for $P\u{}{(0)}$, which is closed and leads to a
constant of the motion. In its place, the ``missing" \mbox{$1$--form}
$P^{\Ss(M)}$ is obtained by direct integration of $\Sigma^{(0)}\,.$ For
the KdV hierarchies, the missing $1$--form gives Action Principles for the
following equations: Krichever--Novikov, Harry--Dym (both are maps of
KdV), Sinh--Gordon, Hunter--Zheng, Camassa--Holm, and Liouville.}
 \end{figure}

