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%%%
%%%  Corrections to first printing of GTM 107, 
%%%   "Applications of Lie Groups to Differential Equations".
%%%      Second Edition, Springer--Verlag, New York, 1993
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%%%            Last updated:   August 20, 2004
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%%%  On page 5, l. 24-27 change

Thus $T^2$ can be covered by two coordinate charts
$$U_1 = \{(\theta ,\rho ): 0 < \theta < 2 \pi ,0 < \rho < 2 \pi \},
U_2 = \{(\theta ,\rho ): \pi < \theta < 3 \pi ,\pi < \rho < 3 \pi \},$$ 
with overlap function

%%% to

Thus $T^2$ can be covered by three coordinate charts, e.g.
$$U_1 = \{(\theta ,\rho ): 0 < \theta <2 \pi , 0 < \rho < 2 \pi \},
U_2 = \{(\theta ,\rho ): \pi < \theta < 3 \pi , \pi < \rho < 3 \pi \},
U_3 = \{(\theta ,\rho ): {1\over 2} \pi < \theta < {5\over 2} \pi , 
           {1\over 2} \pi < \rho < {5\over 2} \pi \}.$$ 
The first overlap function is

%%%  On page 10, l. 6 change

$\phi \comp \tilde \phi ^{-1}: \R \to \R$

%%% to

$\phi ^{-1} \comp \tilde \phi : \R \to \R$

%%%  On page 19, l. 17 change

$x \in V_0 = \{x: |x|<{1\over 2}\}$ 

%%% to

$x \in V_0 = \{-1 < x < {1\over 3}\}$ 

%%%  On page 36, l. 9 change

for all $\varepsilon ,\theta \in \R$, $x\in M$, such 
    that both sides are defined, if and only if 

%%% to

for all $x\in M$, and $(\varepsilon ,\theta ) \in V$, 
where $V\subset \R^2$ is a connected open subset containing $(0,0)$ 
such that both sides of (1.34) are defined at all points therein, 
if and only if 

%%%  On page 37, l. 7-9 change

plane:
$$ V = \{(\theta ,\varepsilon ): 
     {\rm both sides of (1.34) are defined at} (\theta ,\varepsilon )\}$$
and
$$ U = \{(\theta ,\varepsilon ): 
 {\rm both sides of (1.34) are defined and equal at} (\theta ,\varepsilon )\}$$

%%% to

plane: first $V$ is the connected component of
$$ \widehat V = \{(\theta ,\varepsilon ): 
     {\rm both sides of (1.34) are defined at} (\theta ,\varepsilon )\}$$
containing the origin; second $U = \widehat U \cap V$, where
$$ \widehat U = \{(\theta ,\varepsilon ): 
 {\rm both sides of (1.34) are defined and equal at} (\theta ,\varepsilon )\}$$

%%%  On page 37, l. 10 delete the sentence

Note that $U\subset V$, and that $V$ is a connected subset of the 
    $(\theta ,\varepsilon )$ plane.

%%%  On page 37, l. 14 add the following sentence after the final $U=V$.

{\it Warning}: It is not, in general, true that $\widehat U = \widehat V$!

%%%***************************************************************************
%%% Remark:  The preceding corrections are because Theorem 1.34 
%%% had a subtle flaw in it, first pointed out to me by James Devlin.
%%% The following exercise gives a counterexample to the original version.
%%% More details can be found in my paper 
%%%  "Non-associative local Lie groups", J. Lie Theory; 6 (1996) 23--51
%%% Thanks also to Hans Lundmark for comments.

Exercise:

Let $M  = \set{(r,\theta )}{r>0}$.  Prove that the two vector fields 
$${\bf v} = \cos \theta \partial _r - {\sin \theta \over r} \partial _\theta ,
{\bf w} = \sin \theta \partial _r + {\cos \theta \over r} \partial _\theta ,$$
commute, $[{\bf v},{\bf w}] = 0$ on $M$, but their flows do not globally
commute.  {\it Hint}: Consider $r,\theta $ as polar coordinates.  

%%%***************************************************************************

%%%  On page 64, l. 10 delete the middle terms between the two = signs.
%%%   thus the equation should read

$$ exp(\varepsilon {\bf v}_0)^*[\omega |_{exp(\varepsilon {\bf v}_0)x}] =
\sum _{I} \alpha _I(e^\varepsilon x) e^{k\varepsilon } dx^I,$$

%%%  On page 64, l. 18 change the lower limit on the second integral sign:

$\int ^1_{log \varepsilon }$

%%% to

$\int ^1_{exp \varepsilon} $

%%%  On page 64, l. 19 change

$\lambda = log \widetilde \varepsilon $

%%% to

$\lambda = e^{ \widetilde \varepsilon}$

%%%  On page 113, l. 4 change

$\xi = u$

%%% to

$\xi = - u$

%%%  On page 197, l. -10 change

$SO(3)$-invariant solutions exist.

%%% to

$SO(3)$-invariant solutions can be constructed by this technique.

%%%  On page 207, l. -2 change

Example 2.64

%%% to

Example 2.44

%%%  On page 280, in the table, change

$I_x = xD - yA + {1\over 2} x u u_t + t M_x$

%%% to

$I_x = xD + yA + {1\over 2} x u u_t + t M_x$

%%% Thanks to Gehrt Hartjen for checking through this table in his 
%%%  Mathematics Diplomarbeit in Aachen, 2001.

%%%  On page 285, l. 5 change

4.13. (a)

%%% to

** 4.13. (a)

%%% i.e. make this exercise have two stars in front.

%%%  On page 285, l. 7 change

Prove that the reduced system 
$\Delta /G$ for the $G$-invariant solutions of $\Delta $ is 
also the Euler-Lagrange equations for some variational problem 
on the quotient manifold $M/G$.
Does this generalize to nonvariational symmetry groups?

%%% to

Is the reduced system 
$\Delta /G$ for the $G$-invariant solutions of $\Delta $ necessarily 
the Euler-Lagrange equations for some variational problem on 
the quotient manifold $M/G$?
See Anderson and Fels, Symmetry reduction of variational bicomplexes
 and the principle of symmetric criticality, 
{\it Amer. J. Math.} {\bf 119} (1997) 609--670, for details.

%%%  On page 323, l. -8 change

a third order evolution equation is integrable

%%% to

a third order evolution equation in which $u_{xxx}$ 
    occurs linearly is integrable

%%%  On page 328, l. 10 change

Bluman and Kumei, [3],

%%% to

Bluman and Kumei, [2],

%%%  On page 340, in line 4 of the table, change

$- y u_{xxx} + x u_{xyy} + u_{xy}$

%%% to

$- y u_{xxx} + x u_{xxy} + u_{xy}$

%%%  On page 340, in line 5 of the table, change

$u_{xx}(y u_{yt} + {1\over 2} u_t) - u_{yy}(x u_{xt} + {1\over 2} u_t)$

%%% to

$- u_{xx}(y u_{yt} + {1\over 2} u_t) + u_{yy}(x u_{xt} + {1\over 2} u_t)$

%%% Thanks to Gehrt Hartjen for checking through this table in his 
%%%  Mathematics Diplomarbeit in Aachen, 2001.

%%%  On page 381, Exercise 3.16a

The system does not, in fact have a recursion operator, although
there is a recursive formula for generating the higher order symmetries.
On the other hand, the related system
$$u_t = u_{xx} + v^2,  v_t = v_{xx}$$
does admit a recursion operator.  Details can be found in

Beukers, F., Sanders, J.A., and Wang, J.P., On integrability of systems of 
   evolution equations, J. Diff. Eq. 172 (2001), 396-408.

%%%  On page 381, Exercise 3.16b

A proof that the Bakirov system has only one generalized symmetry can 
now be found in

Beukers, F., Sanders, J.A., and Wang, J.P., One symmetry does not imply
   integrability, J. Diff. Eq. 146 (1998), 251-260.

%%%  On page 480, l. 2 change

Leipz. Berich. 1 (1895)

%%% to

Leipz. Berichte 47 (1895)

%%%  On page 480, l. 5 change

Leipz. Berich. 3 (1897)

%%% to

Leipz. Berichte 49 (1897)