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\begin{document}

\title{Lecture\\ Generalized Symmetries} \author{Paul H.M. Kersten \\
Department of Applied Mathematics\\ University of Twente\\
P.O. Box 217\\ 7500 AE Enschede\\ The Netherlands}


\date{}
\maketitle

\begin{abstract}
\mbox{\ }
\end{abstract}

\section{Introduction.}
The classical notion of symmetry of a system of differential equations
was based on transformations in the space of independent en dependent
variables, transforming solutions into solutions. These symmetries are
called {\it point} symmetries. The first generalization of this
concept is to consider transformations of independent, dependent
variables and first order partial derivatives, and transforming
solutions into solutions. This leads to the socalled {\it contact}
symmetries. Generalized symmetries, the subject of this lecture, can
be understood as transformations in the space of independent,
dependent variables and {\it all} partial derivatives \cite{O,V}.\\

Notations will be as follows.\\
$X$ is the space of independent variables, local coordinates being
\begin{displaymath}
 (x_1,...,x_p)
\end{displaymath}
$U$ is the space of dependent variables where local coordinates are
\begin{displaymath}
 (u^1,...,u^q)
\end{displaymath}
The $k^{th}$ order jetbundle $J^k(x,u)$ has local coordinates
\begin{equation}
\label{1.1a}
 (x_i,u^{\alpha},u^{\alpha}_I) \qquad (|I| \leq k,i=1,...,p ; \alpha = 1,...,q)
\end{equation}
while the infinite jetbundle $J(x,u) = J^{\infty}(x,u)$ has local
coordinates
\begin{equation}
\label{1.1b}
 (x_i,u^{\alpha},u_I ^{\alpha}) \; \; |I| < \infty
\end{equation}
In (\ref{1.1a}),(\ref{1.1b}) we used the multiindex notation
$I=(i_1,...,i_p) \; |I| = \sum\limits_{k=1}^p i_r$\\

Throughout we shall use summation convention in case an index occurs
twice; latin indices run from $1$ to $p$ while greek indices run from
$1$ to $q$.\\

Functions $f:J^k(x,u) \rightarrow \Bbb R$ are supposed to be
$C^{\infty}$, while functions $g:J(x,u) \rightarrow \Bbb R$ are just
those dependent on a {\it finite} number of variables, so in effect
\begin{displaymath}
 g = \pi_k^* f \mbox{ for some } f \mbox{ and } k,
\end{displaymath}
(see previous lectures).
notation $f = f[u], g = g[u]$.\\
A system of $k-th$ order differential equations is denoted by
\begin{equation}
\label{1.1c}
 \Delta_j[u] = 0 \; (j=1,...,\l)
\end{equation}
where $\Delta_j$ is defined on $J^k(x,u)$.\\
The total partial derivative operators $D_i$ are given by
\begin{equation}
\label{1.1d}
 D_i = \frac{\partial}{\partial x _ {i}} + u^\alpha_{I,i}
\frac{\partial}{\partial u_I^\alpha} \; \; (i=1,...,q) 
\end{equation}
and they re-create in an algebraic way, what is realized classically
by partial differentiation, using chain-rule.\\

In section 2 we give a short recapitulation of the notion of
infinitesimal symmetry. In section 3 the concept of generalized
symmetry is given, some theorems are proved and an explicit example is
given.\\
In section 4 the notion of nonlocal symmetry, \cite{KV,KV2} being a
generalization of generalized symmetry, is introduced and an illustration
through the famous Korteweg-de Vries equation (KdV) is
discussed. In the conclusions we point out that even generalizations
of this concept are very interesting.\\ 

Applications of symmetries to construct explicit solutions,
conservation laws etc are beyond the scope of this lecture, and are
dealt with in p.e. ref \cite{O}, \cite{KB}.

\setcounter{equation}{0}
\section{Classical Symmetries.}
We give a short review of classical (infinitesimal) symmetries of
differential equations.\\
We start at a $k$-th order system of differential equations
\begin{equation}
\label{1.1}
 \Delta_j[u] = 0 \; \; j = 1,\ldots,\l.
\end{equation}
A vector field $V \epsilon T(J^0(x,u))$ is given by
\begin{equation}
\label{1.2}
 V = \xi^i(x,u) \frac{\partial}{\partial x ^ {i}} +
\varphi_{\alpha}(x,u) \frac{\partial}{\partial u ^ {\alpha}}
\end{equation}
The $k^{th}$ prolongation of the vector field $V$ defined in
$T(J^k(x,u))$ and denoted $pr^{k}(V)$ is given by
\begin{equation}
\label{1.3}
 pr^{k}(V) = \xi^i (x,u) \frac{\partial}{\partial x ^ {i}} +
\Phi_{\alpha}^I [u] \frac{\partial}{\partial u
_{I} ^ {\alpha}}
\end{equation}
where
\begin{equation}
\label{1.4}
 \Phi_{\alpha}^I [u] = D^I (\varphi_{\alpha}(x,u) - u_i^{\alpha}
\xi^i (x,u)) + u_{I,i} ^{\alpha} \xi^i (x,u)
\end{equation}
and
\begin{equation}
\label{1.4a}
 D^I = D_1^{i _{1}} o D_2^{i _ {2}} ... o D_p^{i _ {p}}
\end{equation}

Formula (\ref{1.4}) can be obtained by the conditions that the
prolongation of the vector field $V$ leaves the contact structure
\begin{equation}
\label{1.5}
 \omega_J^{\alpha} = du_J^{\alpha} - u_{J,i}^\alpha dx^i \; \;
(|J| \leq k-1)
\end{equation}
invariant \cite{O}.\\
We now arrive at the following definition.

\begin{df}
A vector field $V$ (\ref{1.2}) is a (infinitesimal) symmetry of the
system of differential equations (\ref{1.1}) if
\begin{equation}
\label{1.6}
 \hspace{5cm} pr^{(k)}(V)(\Delta_j) = 0 \; \; \mbox{ on } \Delta = 0
\end{equation}
We shall not compute symmetries here; but postpone it to the next
section.\\
Computerprograms to construct solutions of the symmetry condition
(\ref{1.6}) are discussed in p.e. \cite{K}.
\end{df}

\setcounter{equation}{0}
\section{Generalized Symmetries.}

In this section we generalize the classical notion of infinitesimal
symmetries to generalized symmetries, sometimes called
Lie-B\"{a}cklund transformations: not te be confused with B\"{a}cklund
transformations which are of a completely different nature.\\
Remind that classically a vector field $V \epsilon T(J^0(x,u))$ is
given by
\begin{equation}
\label{2.1}
 V = \xi^i(x,u) \frac{\partial}{\partial x^{i}} + \varphi_\alpha(x,u)
\frac{\partial}{\partial u_{\alpha}}
\end{equation}
We now pass to the infinite jetbundle $J(x,u)$ where local coordinates
are given by
\begin{displaymath}
 (x^i,u^{\alpha},u^{\alpha}_I) \; \; I = (i_1,...,i_p) \; i_k \geq 0(k=1,...,p)
\end{displaymath}
functions $F:J(x,u) \rightarrow \Bbb R$ are to be understood to depend
on an arbitrary but finite number of variables, $F=F[u]$.

\begin{df}
A (formal) generalized vector field is given by the following
expression
\begin{equation}
\label{2.2}
 V = \xi^i[u] \frac{\partial}{\partial x ^ {i}} +
\varphi_{\alpha}[u] \frac{\partial}{\partial u ^ {\alpha}}
\end{equation}
The formal prolongation of $V$ to the infinite jetbundle is defined by
\begin{equation}
\label{2.3}
 pr(V) = \xi^i[u] \frac{\partial}{\partial x ^{i}} +
\Phi^J_{\alpha}[u] \frac{\partial}{\partial u^{\alpha}_{J}}
\end{equation}
whereas in the second term summation runs over $\alpha$ and all
possible multiindices $J$ and
\begin{equation}
\label{2.4}
 \Phi^J_{\alpha} = D^J(\varphi_{\alpha}[u] - \xi^i [u] u^{\alpha}_i) +
\xi^i [u] u^{\alpha}_{J,i},
\end{equation}
compare this with formula (\ref{1.4}).\\

{\bf Note} there arise no convergence problems in defining the action
of an (infinitely) prolonged vector field on a function $F[u]$
since the latter only depends on a finite number of variables.\\

We now arrive at the definition of generalized symmetry. 
\end{df}

\begin{df}
A generalized vector field $V$ is a generalized symmetry of a system of
differential equations
\begin{displaymath}
 \Delta_j[u] = 0 \; \; (j=1,...,\l)
\end{displaymath}
if and only if
\begin{equation}
\label{2.5}
 pr(V)(\Delta_j) = 0 \; \; (j=1,...,\l)
\end{equation}
for solutions $u = f(x)$.\\

{\bf Note} it can be proved that for applications one has in mind
that condition (\ref{2.5}) results in
\begin{equation}
\label{2.6}
 pr(V)(\Delta_j) = \sum P_{k,j}^J [u]D^J (\Delta_k) \; \;
j=k=1,\ldots l \qquad |J|< \infty, \; \; P^J_{k,j}[u]\in C^\infty(J(x,u)) 
\end{equation}
or $pr(V)(\Delta_j) = 0$ when restricted to the manifold $Y \subset
J(x,u)$ defined by the system of differential equations and all its
differential consequences.\\

The concept of evolutionary or vertical vector field is a great
advantage in the computation of generalized symmetries.
\end{df}

\begin{df}
 A generalized vector field
\begin{equation}
\label{2.7}
 V = V_F = F_{\alpha}[u] \frac{\partial}{\partial u^\alpha}
\end{equation}
is called an evolutionary or vertical vector field.\\
The set of functions $(F_{\alpha})$ is called the {\em characteristic} of the
vector field $V$.\\

Note that for evolutionary vector fields we have a very elegant way for
the expression of the infinite prolongation (\ref{2.4}) i.e.
\begin{equation}
\label{2.8}
 pr(V_F) = D^J(F_{\alpha} [u]) \frac{\partial}{\partial u ^{\alpha} _ {J}}
\end{equation}
because $\xi^i [u] \equiv 0 \; \; (i=1,\ldots,p)$.\\
\end{df}

Moreover every infinitely prolonged vector field $V$ (\ref{2.2},\ref{2.3})
can be written as a sum of an evolutionary vector field and total
partial derivative vector fields i.e.
\begin{equation}
\label{2.9}
 pr(V) = pr(V_F) + \xi^i[u]D_i
\end{equation}
where the characteristic $F$ of the evolutionary vector field is given
by
\begin{equation}
\label{2.10}
 F_{\alpha}[u] = \varphi_{\alpha}[u] - u^{\alpha}_i \xi^i [u] \; \;
(\alpha=1,\ldots,q)
\end{equation}
Since vector fields $\xi^i[u]D_i$ satisfy the symmetry condition
(\ref{2.5}),(\ref{2.6}) in a trivial way; we can restrict the search for
generalized symmetries to the search for {\it evolutionary} vector fields.\\

To show the complexity of the computations involved in constructing
generalized symmetries we compute {\it third} order symmetries of the
potential form of Burgers' equation.

\begin{ex}
Burgers' equation is the following partial differential equation
\begin{equation}
\label{2.11}
 u_t = u_1^2 + u_2 \; \; (u_1=u_x,u_2=u_{xx})
\end{equation}
Note that differential consequences are given by
\begin{eqnarray}
\label{2.11b}
 u_{1t} & = & 2u_1u_2 + u_3 \qquad (u_{1t} = u_{xt})\\\nonumber
 u_{2t} & = & 2u_2^2 + 2u_1u_3 + u_4\\
 u_{3t} & = & 6u_2u_3 + 2u_1u_4 + u_5\nonumber
\end{eqnarray}
The characteristic of the evolutionary vector field $V_F$ is
\begin{equation}
\label{2.12}
 F[u] = F(x,t,u,u_1,u_2,u_3)
\end{equation}
Since we restrict to the solution manifold $u_t,u_{1t},..$ can be
eliminated by (\ref{2.11}),(\ref{2.11b})
\end{ex}
Now due to (\ref{2.8}) the symmetry condition (\ref{2.5}),(\ref{2.6})
reduces to
\begin{equation}
\label{2.13}
 D_tF - 2u_1 D_xF - D_x^2 F = 0
\end{equation}

\begin{equation}
\begin{array}{ll}
\label{2.14}
 \mbox{i.e. }& F_t + F_u(u_1^2 + u_2) + F_{u_1} (2u_1u_2 + u_3) +
  F_{u_2} (2u_2^2 + 2u_1u_3 + u_4)\\
 &+ F_{u_3} (6u_2u_3 + 2u_1u_4 + u_5)\\
 &- 2u_1(F_x + F_u u_1 + F_{u_1} u_2 + F_{u_2} u_3 + F_{u_3} u_4)\\
 &- \{F_{xx} + F_{xu} u_1 + F_{xu_1} u_2 + F_{xu_2} + F_{xu_3} u_4\\
 &+ u_1(F_{xu} + F_{uu} u_1 + F_{uu_1} u_2 + F_{uu_2} u_3 + F_{uu3} u_4)\\
 &+ u_2(F_{xu_1} + F_{uu_1} u_1 + F_{u_1u_1} u_2 + F_{u_1u_2}
   u_3 + F_{u_1u_3} u_4)\\
 &+ u_3(F_{xu_2} + F_{uu_2} u_1 + F_{u_1u_2} u_2 + F_{u_2u_2} u_3 +
   F_{u_2u_3} u_4)\\
 &+ u_4(F_{xu_3} + F_{uu_3} u_1 + F_{u_1u_3} u_2 + F_{u_2u_3} u_3 +
   F_{u_3u_3} u_4)\\
 &+ F_u u_2 + F_{u_1} u_3 + F_{u_2} u_4 + F_{u_3} u_5 \} = 0
\end{array}
\end{equation}

>From (\ref{2.14}) we see that the coefficient of $u_5$ vanishes
identically. The vanishing of the coefficients of $u_4,u_4^2$ lead to
\begin{eqnarray}
\label{2.15}
 u_4^2 : F_{u_3u_3} = 0
\end{eqnarray}
\begin{eqnarray}
\label{2.16}
 u_4 : -F_{xu_3} - u_1 F_{uu_3} - u_2 F_{u_1u_3} - u_3 F_{u_2u_3} = u_4
\end{eqnarray}

The first equation leads to the fact that $F_3$ is a polynomial of
degree $\leq 1$ in $u_3$ while the second equation results in

\begin{equation}
\label{2.17}
 F = \alpha(t)u_3 + \bar F(x,t,u,u_1,u_2)
 \end{equation}

Substitution of (\ref{2.17}) into (\ref{2.14}) leads to a polynomial
of degree 2 in $u_3$, the coefficients of which have to vanish i.e.
\begin{eqnarray}
\label{2.18}
 u_3^2 : \bar F_{u_2u_2} = 0
\end{eqnarray}
\begin{eqnarray}
\label{2.19}
 u_3 : \alpha'(t) + 6u_2\alpha(t) = 2 \bar F_{xu_2} + 2 \bar F{_uu_2} u_1 +
 2u_2 \bar F_{u_1u_2}
\end{eqnarray}
which results in

\begin{equation}
\label{2.20}
 \bar F(x,t,u,u_1,u_2) = 3\alpha u_1u_2 + (\frac{1}{2} \alpha'x +
\beta(t))u_2 + \tilde F(x,t,u,u_1)
\end{equation}
proceeding in this way we finally arrive at the fact that the solution
of (\ref{2.13}) is a linear combination of 10 vector fields whose
characteristics are given by

\begin{equation}
\label{2.21a}
 \begin{array}{rcl}
  F_0 &=& 1\\
  F_1 &=& u_1\\
  F_2 &=& tu_1 + \frac{1}{2} x\\
  F_3 &=& u_2 + u_1^2\\
  F_4 &=& t(u_2 + u_1^2) + \frac{1}{2} xu_1
 \end{array}
\end{equation}
\begin{equation}
\label{2.21b}
 \begin{array}{rcl}
  F_5 &=& t^2(u_2 + u_1^2) + txu_1 + (\frac{1}{2} t + \frac{1}{4} x^2)\\
  F_6 &=& u_3 + 3u_1u_2 + u_1^3\\
  F_7 &=& tF_6 + \frac{1}{2} x F_3\\
  F_8 &=& t^2F_6 + txF_3 + (\frac{1}{2} t + \frac{1}{4} x^2)F_1\\
  F_9 &=& t^3F_6 + \frac{3}{2} t^2 xF_3 + (\frac{3}{2} t^2 + \frac{3}{4}
  tx^2)F_1 + \frac{3}{4} tx + \frac{1}{8} x^3\\
 \end{array}
\end{equation}
  and\\
\begin{eqnarray*}
  F_{10} &=& \rho (x,t)e^{-u}
\end{eqnarray*}
whereas in (\ref{2.21b}) $\rho(x,t)$ is an arbitrary solution of the
heat equation $\rho_t = \rho_{xx}$.   

The existence of a symmetry (\ref{2.21b}) reflects the fact that the
equation at hand (\ref{2.11}) is in 1-1 correspondence with the heat
equation. The general theorem concerning this was proved by Kumei \&
Bluman \cite{KB}.\\
At the moment a number of computerprograms is available in
REDUCE,...,to handle the computations for symmetries p.e. \cite{K}.\\
In order to introduce the Lie bracket of generalized vector fields we
first prove the following lemma.

\begin{lemma}
 If $V_F$ is an evolutionary vector field then

\begin{equation}
\label{2.22}
 [pr(V_F),D_i] = 0
\end{equation}
interpreted as componentwise.
\end{lemma}

\noindent{\bf Proof.} First of all $\frac{\partial}{\partial
 u_j^\alpha} (D_iP) = \frac{\partial P}{\partial u_{J\backslash
 i}^\alpha} + D_i(\frac{\partial}{\partial u_J^\alpha}P)$\\
where $J\backslash i = (j_1,\ldots,j_{i-1},\ldots,j_p)$.\\
This implies that

\begin{equation}
\label{2.23}
 pr(V_F)(D_iP) = (D^J F_\alpha)\cdot \frac{\partial}{\partial
 u_J^\alpha}(D_iP) = (D^J F_\alpha)D_i(\frac{\partial}{\partial
 u_J^\alpha}P) + D^J F_\alpha \cdot \frac{\partial P}{\partial
 u^\alpha_{J\backslash i}} 
\end{equation}
We know that

\begin{equation}
\label{2.24}
 D_i(pr(V_F)P) = D_i((D^J F_\alpha) \cdot \frac{\partial P}{\partial
 u_J^\alpha}) = (D^J F\alpha)D_i(\frac{\partial
 P}{\partial u_J^\alpha}) + (D_iD_JF_\alpha) \frac{\partial P}{\partial
 u_J^\alpha}.   
\end{equation}

By changing summation index $J$ tot $J\backslash i$ we see that the
right hand sides in (\ref{2.23},\ref{2.24}) are equal, which proves
the Lemma.\\
As a corollary to this lemma we have
\begin{equation}
\label{2.25}
 pr(V_F)(D^JP) = D^J(pr(V_F)P) 
\end{equation}

\begin{th}
Let $V_Q,V_R$ be two evolutionary vector fields and $pr(V_Q)$,
$pr(V_R)$ their prolongations to $J(x,u)$ then the formal commutator is

\begin{equation}
\label{2.26}
 [pr(V_Q),pr(V_R)] = \tilde S
\end{equation}
where $\tilde S$ is the prolongation of an evolutionary vector field

\begin{equation}
\label{2.27}
 \tilde S = pr(V_S)
\end{equation}
and $S$ is defined by

\begin{equation}
\label{2.28}
 S_\alpha = pr(V_Q)(R_\alpha) - pr(V_R)(Q_\alpha) \qquad \alpha=1,\ldots,q
\end{equation}
\end{th}

\noindent{\bf Proof.} The definition of $S$ in (\ref{2.28}) is just
the computation of the $\frac{\partial}{\partial u^\alpha}$ component in
(\ref{2.26}).\\

The component of $\partial_{u_J^\alpha}$ in $\tilde S$ (\ref{2.26}) is
obtained from

\begin{displaymath}
 \tilde S_{u_J^\alpha} = pr(V_Q)D^J(R_\alpha) - pr(V_R)D^JQ_\alpha
\end{displaymath}
and by Lemma 2.1
\begin{displaymath}
 \tilde S_{u_J^\alpha} = D^J\{pr(V_Q)R_\alpha - pr(V_R)Q_\alpha\} =
D^J S_\alpha 
\end{displaymath}
stating that $\tilde S$ is just the prolongation of $V_S$ (cf.\ref{2.8}).\\
>From theorem 6 and the symmetry condition (3.5,6), we now have the
following

\begin{th}
the evolutionary generalized symmetries of a system of differential
equations
\begin{displaymath}
 \Delta_J[u]=0 \qquad (j=1,\ldots,\ell
\end{displaymath}
constitute a Lie algebra by the Lie bracket (\ref{2.26}).
\end{th}

\begin{ex} (Burgers' equations)
We compute some Lie brackets of evolutionary symmetries of example
3.4, (\ref{2.21a}),(\ref{2.21b}).\\
Take
\begin{eqnarray*}
 &Z_1=F_6=t(u_3+3u_1u_2+u^3_1) + \frac{1}{2}
          x(u_2+u^2_1)\\
 &X_1=u_1\\
 &X_2=u_2+u^2_1\\
 &X_3=u_3+3u_1u_2+u^3_1
\end{eqnarray*}
We now have the following result
\begin{eqnarray*}
 \left[V_{Z_1},V_{X_1}\right] &=& \frac{1}{2}V_{X_2}\\
 \left[V_{Z_1},V_{X_2}\right] &=& V_{X_3}\\
 \left[V_{Z_1},V_{X_3}\right] &=& \frac{3}{2}V_{X_4}
\end{eqnarray*}
where $V_{X_4}$ is a fourth-order generalized symmetry and
\begin{displaymath}
 X_4=u_4+3u^2_2+4u_1u_3+6u^2_1u_2+u^4_1
\end{displaymath}
In effect the generalized symmetries of example 3.4
(\ref{2.21a}),(\ref{2.21b}) constitute on {\em infinite dimensional
Lie algebra}.
\end{ex}

\setcounter{equation}{0}
\section{Nonlocal symmetries.}

Here we shall discuss special types of nonlocal symmetries as they
arise in certain special types of coverings. The notion of covering
has been introduced in \cite{KV} and \cite{KV2} (also called
Wahlquist-Estabrook prolongation) and has been discussed by
P. Gragert in his lecture \cite {G}.\\ 
For simplicity we restrict to two independent variables
$(x,t)(p=2)$.\\

In the discussion of coverings or prolongation one starts at the
infinite prolongation $Y$ of a $k$-th-order system of partial
differential equations, i.e. the original system together with all of
its differential consequences, defined on the infinite jet bundle
$J((x,t),u)$ i.e.

\begin{equation}
\label{3.1}
 D^J(\Delta_j[u]) = 0 \qquad j=1,\ldots,\ell,|J|<\infty
\end{equation}
 
An $s$-dimensional covering of (\ref{3.1}), with $(y_1,\ldots,y_s)$ as
local coordinates in the fibres, requires the existence of functions
\begin{displaymath}
 X_r([u],y_1,\ldots,y_s),T_r([u],y_1,\ldots,y_s) \qquad r=1,\ldots,s
\end{displaymath}
such that the extended or generalized total partial derivative
operators

\begin{eqnarray}
\label{3.2}
 \tilde D_x & = & D_x + X_r \frac{\partial}{\partial_{y_r}}\nonumber\\
            &   & \hspace{5cm} \mbox{(summation $r=1,\ldots,s$)}\\
 \tilde D_t & = & D_t + T_r \frac{\partial}{\partial_{y_r}}\nonumber
\end{eqnarray}
commute, i.e.

\begin{equation}
\label{3.3}
 [\tilde D_x,\tilde D_t] = 0
\end{equation}
which yields besides (\ref{3.1}) the covering condition

\begin{equation}
\label{3.4}
 \tilde D_xT_r - \tilde D_tX_r = 0 \; \; \; \mbox{ on (\ref{3.1}) }
\end{equation}
i.e.

\begin{equation}
\label{3.4a}
 D_xT - D_tX + [X,T] = 0
\end{equation}
where $X=(X_1,\ldots,X_s) T=(T_1,\ldots,T_s)$ and the bracket in
(\ref{3.4a}) is taken with respect to the fibre coordinates
$y=(y_1,\ldots,y_s)$.\\

As a special case we now consider coverings (\ref{3.2}),(\ref{3.4})
where $X_r,T_r$ are independent of $y=(y_1,\dots,y_s)$; (\ref{3.4})
then reduces to

\begin{equation}
\label{3.5}
 D_x(T_r) - D_t(X_r) = 0 \; \; \; \mbox{ on (\ref{3.1}) } (r=1,\ldots,s)
\end{equation}
i.e. $X_r,T_r$ determines a conservation law for (\ref{3.1})
and $Y_r=D_x^{-1}(X_r)$, as formal integral.\\

Analogously to (\ref{2.6}) we now introduce a {\em nonlocal} vertical
(generalized) vector field

\begin{equation}
\label{3.6}
 V_F = F_\alpha([u],y_1,\ldots,y_s) \frac{\partial}{\partial u_\alpha}
\end{equation}
and its prolongation to the infinite jetbundle

\begin{equation}
\label{3.7}
 pr(V_F) = \tilde D^J(F_\alpha([u],y_1,\ldots,y_s))
\frac{\partial}{\partial u_J^\alpha}
\end{equation}

We now define the notion of nonlocal symmetry.

\begin{df}

A nonlocal vector field $V_F$ (\ref{3.6}) determines a nonlocal
symmetry of (\ref{3.1}) if and only if

\begin{equation}
\label{3.8}
 pr(V_F)(\Delta_j) = 0 \; \; \; \mbox{ on (\ref{3.1}) },
j=1,\ldots,\ell
\end{equation}
where $pr(V_F)$ is defined by (\ref{3.7}).\\

{\bf Note:} The interested reader, comparing this definition with the
one given in Vinogradov \& Krasilshchik's work \cite{KV2}, might notice a
difference; in order to keep things simple and to outline the ideas we
just use this simplified definition.\\

We apply the notion of nonlocal symmetries to the construction of
nonlocal symmetries of the famous Korteweg-de Vries equation
(KdV-equation).
\end{df}

\begin{ex}
We start at the infinite prolongation of the KdV-equation i.e.,

\begin{equation}
\label{3.9}
 u_t = uu_1 + u_3 \qquad (u_1=u_x,u_3=u_{xxx})
\end{equation}
and its differential consequences.\\

If we apply the technique of the preceding section and search for
generalized symmetries of (\ref{3.9}) with characteristic
$F=F(u,u_1,\ldots,u_5)$, we arrive at the existence of

\begin{equation}
\label{3.10}
 \begin{array}{rclrcl}
 F_1 &=& u_1      &F_4 &=& 2u+xu_1+3t(uu_1+u_3)\nonumber\\
 F_2 &=& uu_1+u_3 &F_5 &=& 1+tu_1\\
 F_3 &=& \frac{5}{6} u_1u^2+\frac{10}{3}u_1u_2+\frac{5}{3}uu_3+u_5\nonumber
 \end{array}
\end{equation}
being the characteristics of $5$ generalized symmetries $V_{F_i} \;
(i=1,\ldots,5)$.\\
\end{ex}

Note that (\ref{2.3}),(\ref{2.4})

\begin{equation}
\label{3.11}
 \begin{array}{rcll}
 V_{F_1} &\doteq& \frac{\partial}{\partial x} , \qquad V_{F_2}
 \doteq \frac{\partial}{\partial t} &(x,t \mbox{-translation
 )}\\  
 V_{F_3} &\doteq& -x \frac{\partial}{\partial x} - 3t
 \frac{\partial}{\partial t} + 2u \frac{\partial}{\partial u}
 &(\mbox{scale transformation )}\\
 V_{F_4} &\doteq& t \frac{\partial}{\partial x} +
 \frac{\partial}{\partial u} &(\mbox{Gallilean Boost)}
\end{array}
\end{equation} 

It is an easy observation that $X_1=u, \; \; T_1=\frac{1}{2}u^2+u_2$
yield a conservation law for KdV-equation (\ref{3.9})

\begin{equation}
\label{3.12}
 D_xT_1-D_tX_1=0 \qquad \mbox{on (\ref{3.9}) }
\end{equation}

We introduce the {\em 1-dimensional covering} of (\ref{3.9}) with
$y=D_x^{-1}(u)$, and we are interested in the existence of a nonlocal
symmetry of (\ref{3.9}) i.e. solution of (\ref{3.8}) where

\begin{equation}
\label{3.13}
 H=H(x,t,u,\ldots,u_s,y)
\end{equation}
i.e.
\begin{equation}
\label{3.14}
 \tilde D_tH - u_1H - u \tilde D_xH - \tilde D_x^3H = 0
\end{equation}
\begin{eqnarray*}
 \tilde D_x &=& D_x + u\frac{\partial}{\partial y}\\
 \tilde D_t &=& D_t + (\frac{1}{2}u^2+u_2)\frac{\partial}{\partial y}
\end{eqnarray*}

Using an integration package the solution can be constructed in a
straightforward way, but since this would be very lengthy we proceed
in a more convenient way.\\
First of all, note that KdV-equation is graded due to the scale
transformation (\ref{3.11}).\\
i.e.
\begin{displaymath}
 [u]=2 \; ; \; [x]=-1 \; ; \; [t]=-3 \; ; \; [D_x]=1 \; ; \;
 [D_t]=3
\end{displaymath}
which implies

\begin{equation}
\label{3.15}
 [F_1]=3 \; ; \; [F_2]=5 \; ; \; [F_3]=7 \; ; \; [F_4]=2 \; ; \; [F_5]=0.
\end{equation}

We now search for a nonlocal symmetry whose characteristic is of
degree 4 and which is of polynomial degree 1 in $x,t,y$.\\

>From this we arrive at the Ansatz, based on the grading (\ref{3.15})

\begin{equation}
\label{3.16}
 H=t(F_3) + \alpha xF_2 + \beta {\bf y}u_1 + \gamma u_2 + \delta u^2u_1
\end{equation}
where $F_2,F_3$ are defined by (\ref{3.10}) and
$\alpha,\beta,\gamma,\delta$ constants to be determined. The symmetry
condition, due to the fact that $F_3,F_2$ satisfy (\ref{3.14})
themselves, reduces to

\begin{equation}
\label{3.17}
 \begin{array}{ll}
 &F_3 - \alpha uF_2 - 3\alpha D_x^2F_2 + \beta(\frac{1}{2}u^2+\beta
  {\bf y}(u_1^2+uu_2+u_4)\\
 &+ \gamma(3u_1u_2+uu_3+u_5) + 2\delta u(uu_1+u_3)-u_1(\beta
  {\bf y}u_1+\gamma u_2+\delta u^2)\\
 &-u(\beta uu_1+\beta {\bf y}u_2+\gamma u_3+2\delta uu_1)\\
 &-[4\beta u_1u_2+3\beta uu_3+\beta {\bf y}u_4+\gamma u_5+2\delta
  uu_3+6\delta u_1u_2] = 0
 \end{array}
\end{equation}

This condition leads to the following conditions for
$\alpha,\beta,\gamma,\delta$

\begin{equation}
\label{3.18}
 \begin{array}{rcl}
 u_5    &:& 1-3\alpha + \gamma-\gamma = 0\\
 uu_3   &:&
 \frac{5}{3}-\alpha-3\alpha+\gamma+2\delta-\gamma-3\beta-2\delta=0\\
 u_1u_2 &:&
 \frac{10}{3}-9\alpha+\beta+3\gamma-\gamma-4\beta-6\delta=0\\
 u^2u_1 &:&
 \frac{5}{6}-\alpha+\frac{1}{2}\beta+2\delta-\delta-\beta-2\delta=0 
 \end{array}
\end{equation}
or equivalently

\begin{equation}
\label{3.19}
 \begin{array}{l}
 1-3\alpha=0\\
 \frac{5}{3}-4\alpha-3\beta=0\\
 \frac{10}{3}-9\alpha-3\beta+2\gamma-6\delta=0\\
 \frac{5}{6}-\alpha-\frac{1}{2}\beta-\delta=0
 \end{array}
\end{equation}
solving (\ref{3.19}) we arrive at
\begin{displaymath}
 \alpha=\frac{1}{3} \; ,\; \beta=\frac{1}{9} \; ,\; \gamma=\frac{4}{3}
 \; ,\; \delta=\frac{4}{9}
\end{displaymath}
which leads to the characteristic ({\em nonlocal}) of a symmetry of
KdV-equation 

\begin{equation}
\label{3.20}
 H=tF_3+\frac{1}{3}xF_2+\frac{1}{9}yu_1+\frac{4}{3}u_2+\frac{4}{9}u^2u_1
\end{equation}

\setcounter{equation}{0}
\section{Recursion Operators and Nonlocal Symmetries.}

In this section we indicate the importance of nonlocal symmetries in
connection with the existence of recursion operators.\\
For simplicity we restrict to the case of (\ref{3.9}) two independent and one
dependent variable, keeping the KdV-equation as principal example in
mind.\\

Let us take a deeper look at the (generalized) symmetry condition
(\ref{2.5}),(\ref{3.8}) i.e.

\begin{equation}
\label{4.1}
 pr(V)(\Delta)=0 \mbox{ on } Y.
\end{equation}

If we use the prolongation formula (\ref{2.8}),(\ref{3.7}) it is a
straightforward procedure to see that the symmetry condition can be
rewritten as

\begin{equation}
\label{4.2}
 \sum(\frac{\partial\Delta}{\partial u_J}) D^J(F)=0
\end{equation}
which is reflected in (\ref{2.13}),(\ref{3.14}).\\
This observation urges us to introduce the socalled {\it linearization
operator} \cite{KV}

\begin{equation}
\label{4.3}
 \ell_\Delta = \sum(\frac{\partial\Delta}{\partial u_J}) D^J
\end{equation}
while (\ref{4.1}),(\ref{2.5}) can be written as

\begin{equation}
\label{4.4}
 \ell_\Delta F=0
\end{equation}

Suppose there exists a differential or integro- differential operator
$\cal R$, and associated to it some $\cal S$ such that the following relation
for operators, $\cal R,\cal S,$ hold

\begin{equation}
\label{4.5}
 \ell_\Delta {\cal R}={\cal S}\ell_\Delta
\end{equation}

Now assume that $F_0$ is a characteristic of a generalized symmetry of
$\Delta$ i.e.

\begin{equation}
\label{4.6}\
 \ell_\Delta F_0=0
\end{equation}

We then have

\begin{equation}
\label{4.7}
 \ell_\Delta({\cal R}F_0) = {\cal S}(\ell_\Delta F_0)=0
\end{equation}
i.e. ${\cal R}(F_0)$ is a characteristic of a generalized symmetry.\\

More generally, if an {\bf operator $\cal R$ satisfying (\ref{4.5})}
for some $\cal S$ exists then starting from a {\bf characteristic
$F_0$} of a symmetry we obtain a {\bf infinite hierarchy} (if not
zero) of generalized symmetrics whose {\bf characteristics} are
defined by 

\begin{equation}
\label{4.8}
 {\cal F}_n={\cal R}^n(F_0) \qquad n=0,\ldots
\end{equation}
Such an operator $\cal R$ is called a {\bf recursion operator} for
generalized symmetries

\begin{ex}
The KdV-equation

\begin{equation}
\label{4.9}
 \Delta(u)=u_t-uu_1-u_3=0
\end{equation}
admits a recursion operator for symmetries

\begin{equation}
\label{4.10}
 {\cal R}=D_x^2 + \frac{2}{3}u + \frac{1}{3}u_1D_x^{-1}
\end{equation}
where $D^{-1}_x$ has to be understood as a formal integral [1].
It is a somewhat tedious calculation to show that

\begin{equation}
\label{4.11}
 \ell_\Delta {\cal R} = {\cal R}\ell_\Delta
\end{equation}
i.e. ${\cal S}={\cal R}$.\\

If we start with $F_1=u_1$ then

\begin{equation}
\label{4.12}
\begin{array}{rcccccl}
 F_2 &=& {\cal R}F_1   &=& {\cal R}u_1 &=& uu_1 + u_3\\
 F_3 &=& {\cal R}^2F_1 &=& {\cal R}F_2 &=&
 \frac{5}{6}u_1u^2+\frac{10}{3}u_1u_2+\frac{5}{3}uu_3+u_5
\end{array}
\end{equation}
and so on.
\end{ex}

We are now in a position underlign the importance of the notion of
nonlocal symmetry.\\
First of all, if we would apply the recursion formula (\ref{4.8})
starting at $F_4$ or $F_5$ (in effect $F_4={\cal R}F_5$) then in order to
compute ${\cal R}F_4$ we would have to allow nonlocal variables $y$ to come
in.\\
Moreover the nonlocal characteristic $H$ (\ref{3.20}) is just nothing
else but

\begin{equation}
\label{4.13}
 H=3{\cal R}F_4
\end{equation}

Secondly, if we compute the generalized Lie-Bracket for generalized
vector field (\ref{2.28}) and compute Lie-Brackets with the non local
vector field $V_H$ we arrive at

\begin{equation}
\label{4.14}
 \begin{array}{rcl}
 \left[V_H,V_{F_1}\right] &=& c_1F_2\\
 \left[V_H,V_{F_2}\right] &=& c_2F_3 
 \end{array}
\end{equation}
$c_1,c_2$ being some nonzero constants.\\

In effect the nonlocal generalized symmetry $V_H$ acts as {\bf
recursion symmetry}.\\

{\bf Final Remarks.}\\
In this lecture I have tried to give you an introduction to and an
impression of the beautiful world of symmetries of differential
equations, where a lot of research is needed to explore the beautiful
structures in this field of applied mathematics.\\

For the interested reader I would recommend the book of Bluman-Kumei
\cite{KB} as a starting point, the book by Olver as a rigorous and
deep discussion of all the mathematics involved, and the work of my
Russian friends Vinagradov, Krasil'shchik for the beautiful and rich
geometrical structures underlying all the notions.

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\bibitem{KV} Krasil'shchik I.S. \& Vinagradov A.M., Nonlocal
symmetries and the theory of coverings. Acta Applicandae Mathematicae
Vol 3 (1984), pp. 79-96.
\bibitem{KV2} Krasil'shchik I.S. \& Vinagradov A.M., Nonlocal Trends
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Laws, and B\"{a}cklund Transformations Acta Applicandae Mathematicae
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\bibitem{K} Kersten P.H.M., Infinitesimal Symmetries: a Computational
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\bibitem{KB} Kumei S. \& Bluman G., Symmetries and Differential
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\end{thebibliography}

\end{document}
