Composite $S$-brane solutions in multidimensional gravity with scalar fields and fields of forms related to Toda-like systems are presented. These solutions are defined on a product manifold $\R_{*} \times M_1 \times \ldots \times M_n$, where $\R_{*}$ is a time manifold, $M_1$ is an Einstein manifold and $M_i$ ($i > 1$) are Ricci-flat manifolds. Certain examples of $S$-brane solutions related to $A_1 + ... + A_1$, $A_m$ Toda chains and those with "block-orthogonal" intersections (e.g. $SM$-brane solutions) are singled out. Under certain restrictions imposed a Kasner-like asymptotical behaviour of the solutions is shown. \vspace{10cm}
hep-th0208101.tex
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\begin{document}


\begin{flushright}        IGC-PFUR-03/2002  \end{flushright}

\vspace{15pt}


\begin{center}
\large\bf
%\\
COMPOSITE S-BRANE SOLUTIONS \\
RELATED TO TODA-TYPE SYSTEMS
\\[15pt]

\normalsize\bf V.D. Ivashchuk\fnm[1]\fnt[1]{ivas@rgs.phys.msu.su},

\it Center for Gravitation and Fundamental Metrology,
VNIIMS, 3/1 M. Ulyanovoy Str.,
Moscow 119313, Russia

\it Institute of Gravitation and Cosmology,
Peoples' Friendship University of Russia,
6 Miklukho-Maklaya St., Moscow 117198, Russia


\end{center}
\vspace{15pt}

\small\noi

\begin{abstract}

Composite $S$-brane solutions
in multidimensional gravity with scalar fields and
fields of forms related to Toda-like systems
are presented. These solutions are defined on a product
manifold $\R_{*} \times M_1 \times  \ldots  \times M_n$,
where $\R_{*}$ is a time manifold, $M_1$ is an Einstein manifold
and $M_i$ ($i > 1$) are Ricci-flat manifolds.
Certain examples of $S$-brane solutions  related to
$A_1 + ... + A_1$, $A_m$  Toda
chains and those with "block-orthogonal" intersections
(e.g.  $SM$-brane solutions) are singled out.
Under certain restrictions imposed
a  Kasner-like asymptotical behaviour of the
solutions  is shown.

\vspace{10cm}

\end{abstract}


\pagebreak

\normalsize


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Currently, there is a  certain interest  to  so-called
$S$-brane solutions \cite{S1}, i.e. space-like analogues of $D$-branes,
see also \cite{S2,S3,S4,S5} and references therein.

We remind that in  perturbative string theory
$D$-branes \cite{Polc} are  hypersurfaces where open strings end
and the Dirichlet boundary conditions along transverse spacelike directions
are imposed. Alternatively,  $D$-branes may be also described  as classical
solutions in  supergravity theories. Open strings can also obey Dirichlet
boundary conditions along time-like  directions that gives   rise to
space-like  analogues of $D$-branes, i.e. $S$-branes.
(For Euclidean brane solutions in
type $II$ string theories see also \cite{Hull1,Hull2}.)

It is well-known that $D$-branes play an important role
in studying non-perturbative aspects of string/M theory and the AdS/CFT
duality \cite{Ads1,Ads2,Ads3}. Analogously
$S$-branes are expected to play the role of $D$-branes in realizing $dS/CFT$
duality \cite{Str} in string/M theory.

Nevertheless, from pure mathematical point of view $S$-brane solutions
\cite{S1,S2,S3,S4,S5} are not new ones but were considered (mostly)
in some earlier publications on string cosmology,
see \cite{BF,LOW,LMPX,LMP,LP1} etc.
\fnm[2]\fnt[2]{The solutions from
\cite{LMPX} contain the spacelike brane
solutions subsequently obtained by
Strominger and  Gutperle \cite{S1} and some other authors.}
They are also special cases of more general
exact solutions found in certain publications
of Moscow group, see \cite{GrIM,BGIM,IMJ,GM,IK,IMtop}
and references therein.
These publications
contain a lot of exact solutions defined on product of several Ricci-flat or
Einstein spaces of arbitrary signatures. Among them
generalized $S$-brane solutions governed by harmonic function
with brane  intersections corresponding to hyperbolic algebras were
considered \cite{IMBl}. In \cite{IMb1}
a general class of cosmological solutions with composite
$S$-branes exhibiting an oscillating behaviour near the singularity
was described, e.g. using the billiard representation near
the singularity.

The aim of this paper is twofold. Firstly, we single out a family of
cosmological solutions with composite $S$-branes from
general Toda-like cosmological-type  solutions suggested in \cite{IK}.
Secondly, for special  intersections (e.g. "orthogonal" ones) we get
a subfamily of $S$-brane solutions, containing
the main part of solutions from
\cite{S1,S2,S3,S4,S5} as special cases.

The paper is organized as follows. In Section 2
cosmological-type solutions related to Toda-type systems from \cite{IK}
are presented (in a more or less condensed manner).
In Section 3 we single out  composite  $S$-brane solutions of general
(Toda-like) form. Here we also consider special subclasses
of $S$-brane solutions, e.g. with "orthogonal", "block-ortogonal"
and $A_m$ intersection rules.
In Section 4  certain examples
of solutions in $D= 11$ supergravity  (describing $SM$-branes)
are singled out.
Section 5 is devoted to asymptotical Kasner-like behaviour
of $S$-brane solutions.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Cosmological-type solutions related to Toda-type systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The model}


We consider a  model governed by the action
\beq{1.1}
S_g=\int d^Dx \sqrt{|g|}\biggl\{R[g]-h_{\alpha\beta}g^{MN}\p_M\varphi^\alpha
\p_N\varphi^\beta-\sum_{a\in\tri}\frac{\theta_a}{n_a!}
\exp[2\lambda_a(\varphi)](F^a)^2\biggr\}
\eeq
where $g=g_{MN}(x)dx^M\otimes dx^N$ is a metric,
$\varphi=(\varphi^\alpha)\in\R^l$ is a vector of scalar fields,
$(h_{\alpha\beta})$ is a  constant symmetric
non-degenerate $l\times l$ matrix $(l\in \N)$,
$\theta_a=\pm1$,
\beq{1.2a}
F^a =    dA^a
=  \frac{1}{n_a!} F^a_{M_1 \ldots M_{n_a}}
dz^{M_1} \wedge \ldots \wedge dz^{M_{n_a}}
\eeq
is a $n_a$-form ($n_a\ge1$), $\lambda_a$ is a
1-form on $\R^l$: $\lambda_a(\varphi)=\lambda_{\alpha a}\varphi^\alpha$,
$a\in\tri$, $\alpha=1,\dots,l$.
In (\ref{1.1})
we denote $|g| =   |\det (g_{MN})|$,
\beq{1.3a}
(F^a)^2_g  =
F^a_{M_1 \ldots M_{n_a}} F^a_{N_1 \ldots N_{n_a}}
g^{M_1 N_1} \ldots g^{M_{n_a} N_{n_a}},
\eeq
$a \in \tri$. Here $\tri$ is some finite set.
For pseudo-Euclidean metric of signature $(-,+, \ldots,+)$
all $\theta_a = 1$.

\subsection{Solutions with $n$ Ricci-flat spaces}

Let us consider a family of
solutions to field equations corresponding to the action
(\ref{1.1}) and depending upon one variable $u$
\cite{IK}. These solutions are defined on the manifold
\beq{1.2}
M =    (u_{-}, u_{+})  \times
M_1  \times M_2 \times  \ldots \times M_{n},
\eeq
where $(u_{-}, u_{+})$  is  an interval belonging to $\R$.
The solutions read \cite{IK}
\bear{1.3}
g= \biggl(\prod_{s \in S} [f_s(u)]^{2 d(I_s) h_s/(D-2)} \biggr)
\biggr\{ \exp(2c^0 u + 2 \bar c^0) w du \otimes du  + \\ \nn
\sum_{i = 1}^{n} \Bigl(\prod_{s\in S}
[f_s(u)]^{- 2 h_s  \delta_{i I_s} } \Bigr)
\exp(2c^i u+ 2 \bar c^i) g^i \biggr\}, \\ \label{1.4}
\exp(\varphi^\alpha) =
\left( \prod_{s\in S} f_s^{h_s \chi_s \lambda_{a_s}^\alpha} \right)
\exp(c^\alpha u + \bar c^\alpha), \\ \label{1.5}
F^a= \sum_{s \in S} \delta^a_{a_s} {\cal F}^{s},
\ear
$\alpha=1,\dots,l$; .
In  (\ref{1.3})  $w = \pm 1$,
$g^i=g
_{m_i n_i}^i(y_i) dy_i^{m_i}\otimes dy_i^{n_i}$
is a Ricci-flat  metric on $M_{i}$, $i=  1,\ldots,n$,
\beq{1.11}
\delta_{iI}=  \sum_{j\in I} \delta_{ij}
\eeq
is the indicator of $i$ belonging
to $I$: $\delta_{iI}=  1$ for $i\in I$ and $\delta_{iI}=  0$ otherwise.

The  $p$-brane  set  $S$ is by definition
\beq{1.6}
S=  S_e \sqcup S_m, \quad
S_v=  \sqcup_{a\in\tri}\{a\}\times\{v\}\times\Omega_{a,v},
\eeq
$v=  e,m$ and $\Omega_{a,e}, \Omega_{a,m} \subset \Omega$,
where $\Omega =   \Omega(n)$  is the set of all non-empty
subsets of $\{ 1, \ldots,n \}$.
Here and in what follows $\sqcup$ means the union
of non-intersecting sets. Any $p$-brane index $s \in S$ has the form
\beq{1.7}
s =   (a_s,v_s, I_s),
\eeq
where
$a_s \in \tri$ is colour index, $v_s =  e,m$ is electro-magnetic
index and the set $I_s \in \Omega_{a_s,v_s}$ describes
the location of $p$-brane worldvolume.

The sets $S_e$ and $S_m$ define electric and magnetic
$p$-branes, correspondingly. In (\ref{1.4})
\beq{1.8}
\chi_s  =   +1, -1
\eeq
for $s \in S_e, S_m$, respectively.
In (\ref{1.5})  forms
\beq{1.9}
{\cal F}^s= Q_s
\left( \prod_{s' \in S}  f_{s'}^{- A_{s s'}} \right) du \wedge\tau(I_s),
\eeq
$s\in S_e$, correspond to electric $p$-branes and
forms
\beq{1.10}
{\cal F}^s= Q_s \tau(\bar I_s),
\eeq
$s \in S_m$,
correspond to magnetic $p$-branes; $Q_s \neq 0$, $s \in S$.
Here  and in what follows
\beq{1.13a}
\bar I \equiv I_0 \setminus I, \qquad I_0 = \{1,\ldots,n \}.
\eeq

All manifolds $M_{i}$ are assumed to be oriented and
connected and  the volume $d_i$-forms
\beq{1.12}
\tau_i  \equiv \sqrt{|g^i(y_i)|}
\ dy_i^{1} \wedge \ldots \wedge dy_i^{d_i},
\eeq
and parameters
\beq{1.12a}
 \varepsilon(i)  \equiv {\rm sign}( \det (g^i_{m_i n_i})) = \pm 1
\eeq
are well-defined for all $i=  1,\ldots,n$.
Here $d_{i} =   {\rm dim} M_{i}$, $i =   1, \ldots, n$,
$D =   1 + \sum_{i =   1}^{n} d_{i}$. For any
 set $I =   \{ i_1, \ldots, i_k \} \in \Omega$, $i_1 < \ldots < i_k$,
we denote
\bear{1.13}
\tau(I) \equiv \tau_{i_1}  \wedge \ldots \wedge \tau_{i_k},
\\ \label{1.14}
M(I) \equiv M_{i_1}  \times  \ldots \times M_{i_k}, \\
\label{1.15}
d(I) \equiv {\rm dim }(M(I)) =  \sum_{i \in I} d_i, \\
\label{1.15a}
\varepsilon(I) \equiv \varepsilon(i_1) \ldots \varepsilon(i_k).
\ear

$M(I_s)$ is isomorphic to $p$-brane worldvolume manifold
(see (\ref{1.7})).

The parameters  $h_s$ appearing in the solution
satisfy the relations
\beq{1.16}
h_s = (B_{s s})^{-1},
\eeq
where
\beq{1.17}
B_{ss'} \equiv
d(I_s\cap I_{s'})+\frac{d(I_s)d(I_{s'})}{2-D}+
\chi_s\chi_{s'}\lambda_{\alpha a_s}\lambda_{\beta a_{s'}}
h^{\alpha\beta},
\eeq
$s, s' \in S$, with $(h^{\alpha\beta})=(h_{\alpha\beta})^{-1}$.

Here we assume that
\beq{1.17a}
({\bf i}) \qquad B_{ss} \neq 0,
\eeq
for all $s \in S$, and
\beq{1.18b}
({\bf ii}) \qquad {\rm det}(B_{s s'}) \neq 0,
\eeq
i.e. the matrix $(B_{ss'})$ is a non-degenerate one. In (\ref{1.9})
another non-degenerate matrix (a so-called "quasi-Cartan" matrix)
appears
\beq{1.18}
(A_{ss'}) = \left( 2 B_{s s'}/B_{s' s'} \right).
\eeq
Here  some ordering in the set $S$ is assumed.

In (\ref{1.3}), (\ref{1.4})
\beq{1.19}
f_s = \exp( - q^s),
\eeq
where $(q^s) = (q^s(u))$ is a solution to Toda-type equations
\beq{1.20}
\ddot{q^s} = -  \eps_s B_{s s} Q_s^2 \exp( \sum_{s' \in S} A_{s s'}
q^{s'} ),  \eeq
$s \in S$. Here
\beq{1.22}
\eps_s=(-\eps[g])^{(1-\chi_s)/2}\eps(I_s) \theta_{a_s},
\eeq
$s\in S$, $\eps[g]\equiv\sign\det(g_{MN})$. More explicitly
(\ref{1.22}) reads: $\eps_s=\eps(I_s) \theta_{a_s}$ for
$v_s = e$ and $\eps_s=-\eps[g] \eps(I_s) \theta_{a_s}$, for
$v_s = m$.

Vectors $c=(c^A)= (c^i, c^\alpha)$ and
$\bar c=(\bar c^A)$ obey the following constraints
\beq{1.27}
\sum_{i \in I_s}d_ic^i-\chi_s\lambda_{a_s\alpha}c^\alpha=0,
\qquad
\sum_{i\in I_s}d_i\bar c^i-
\chi_s\lambda_{a_s\alpha}\bar c^\alpha=0, \quad s \in S,
\eeq
\bear{1.30aa}
c^0 = \sum_{j=1}^n d_j c^j,
\qquad
\bar  c^0 = \sum_{j=1}^n d_j \bar c^j,
\\  \label{1.30a}
2E = 2E_{TL} +
 h_{\alpha\beta}c^\alpha c^\beta+ \sum_{i=1}^n d_i(c^i)^2
- \left(\sum_{i=1}^nd_ic^i\right)^2 = 0,
\ear
where
\beq{1.31}
E_{TL} = \frac{1}{4}  \sum_{s,s' \in S} h_s
A_{s s'} \dot{q^s} \dot{q^{s'}}
  + \frac{1}{2} \sum_{s \in S} \eps_s Q_s^2
  \exp( \sum_{s' \in S} A_{s s'} q^{s'} )
\eeq
is an integration constant (energy) for the solutions from
(\ref{1.20}).

Eqs. (\ref{1.20}) correspond to the
Toda-type Lagrangian
\beq{1.31a}
L_{TL} = \frac{1}{4}  \sum_{s,s' \in S}
h_s  A_{s s'} \dot{q^s}\dot{q^{s'}}
-  \frac{1}{2} \sum_{s \in S}  \eps_s Q_s^2 \exp( \sum_{s' \in S} A_{s s'}
q^{s'} ).  \eeq

The reduction of multidimensional cosmology
with composite $p$-branes (and block-diagonal metric)
to Toda-like system was performed earlier in \cite{IMJ}.

Here we identify notations  for $g^{i}$  and  $\hat{g}^{i}$, where
$\hat{g}^{i} = p_{i}^{*} g^{i}$ is the
pullback of the metric $g^{i}$  to the manifold  $M$ by the
canonical projection: $p_{i} : M \rightarrow  M_{i}$, $i = 1,
\ldots, n$. An analogous agreement will be also kept for volume forms etc.

Due to (\ref{1.9}) and  (\ref{1.10}), the dimension of
$p$-brane worldvolume $d(I_s)$ is defined by
\beq{1.16a}
d(I_s)=  n_{a_s}-1, \quad d(I_s)=   D- n_{a_s} -1,
\eeq
for $s \in S_e, S_m$, respectively.
For a $p$-brane we have $p =   p_s =   d(I_s)-1$.


\subsection{Solutions with one curved Einstein space and $n-1$
Ricci-flat spaces}


The cosmological solution with Ricci-flat spaces
may be also  modified to the following case:
\beq{1.2.2}
{\rm Ric}[g^1] = \xi_1 g^1, \  \xi_1 \ne0, \qquad {\rm Ric}[g^i] = 0, \ i
>1,
\eeq
i.e. the first space $(M_1,g^1)$ is  Einstein space of non-zero
scalar curvature and
other  spaces $(M_i,g^i)$ are Ricci-flat and
 \beq{1.2.3}
 1 \notin I_s,  \quad  \forall s  \in S,
\eeq
i.e. all ``brane'' submanifolds  do not  contain $M_1$.

In this case the exact solution may be obtained by a little modifications
of the solutions from the previous subsection \cite{IK}.

The metric reads as follows
\bear{1.2.4}
g= \biggl(\prod_{s \in S} [f_s(u)]^{2 d(I_s) h_s/(D-2)} \biggr)
\biggl\{[f_1(u-u_1)]^{2d_1/(1-d_1)}\exp(2c^1u + 2 \bar c^1) \quad
\\ \nn
\times[w du \otimes du+ f_1^2(u-u_1)g^1] +
\sum_{i = 2}^{n} \Bigl(\prod_{s\in S}
[f_s(u)]^{- 2 h_s  \delta_{i I_s} } \Bigr)\exp(2c^i u+ 2 \bar c^i)
g^i\biggr\}.
\ear
where
\bear{1.2.5}
f_1(\tau) =R \sh(\sqrt{C_1}\tau), \ C_1>0, \ \xi_1 w>0;
\\ \label{1.2.6}
R \sin(\sqrt{|C_1|}\tau), \ C_1<0, \  \xi_1 w>0;   \\ \label{1.2.7}
R \ch(\sqrt{C_1}\tau),  \ C_1>0, \ \xi_1w <0; \\ \label{1.2.8}
\left|\xi_1(d_1-1)\right|^{1/2} \tau, \ C_1=0,  \ \xi_1w>0,
\ear
$u_1$ and $C_1$ are constants, $R =  |\xi_1(d_1-1)/C_1|^{1/2}$, and
\beq{1.2.9}
C_1\frac{d_1}{d_1-1}= 2E,
\eeq
where $E$ is defined in  (\ref{1.30a}).

Now, vectors $c=(c^A)$ and $\bar c=(\bar c^A)$ satisfy
also additional constraints
\bear{1.2.10}
c^1 = \sum_{j=1}^nd_jc^j = c^0, \qquad
\bar c^1= \sum_{j=1}^nd_j\bar c^j = \bar c^0.
\ear

All other  relations from the previous subsection are unchanged.


\subsection{Restrictions on $p$-brane configurations.}

The solutions  presented above are valid if two
restrictions on the sets of $p$-branes are satisfied \cite{IK}.
These restrictions
guarantee  the block-diagonal form of the  energy-momentum tensor
and the existence of the sigma-model representation (without additional
constraints) \cite{IMC}.

Let us denote
$w_1\equiv\{i|i\in\{1,\dots,n\},\quad d_i=1\}$, and
$n_1=|w_1|$ (i.e. $n_1$ is the number of 1-dimensional spaces among
$M_i$, $i=1,\dots,n$).

{\bf Restriction 1.} {\em For any $a\in\tri$ and $v= e,m$ there are no
$I,J \in\Omega_{a,v}$ such that
$ I= \{i\} \sqcup (I \cap J)$, and $J= (I \cap J) \sqcup \{ j \}$
for some $i,j \in w_1$, $i \neq j$.}

{\bf Restriction 2.}.
{\em For any $a\in\tri$ there are no
$I\in\Omega_{a,e}$ and $J\in\Omega_{a,m}$ such that
$\bar J=\{i\}\sqcup I$.}

Restriction 1  is satisfied for $n_1 \leq 1$ and also in
the non-composite case: $|\Omega_{a,e}|+ |\Omega_{a,m}| = 1$ for all
$a\in\tri$.  For $n_1\ge2$ it forbids the following
pairs of two electric or two magnetic $p$-branes,
corresponding to the same form $F^a, a \in \tri$:

\begin{center}
\begin{tabular}{cccc}
\cline{1-2}
\multicolumn{1}{|c|}{$i$} &
\multicolumn{1}{|c|}{\hspace*{1cm}} & & $\quad I$ \\
\cline{1-2}
 & & & \\
\cline{2-3}
 & \multicolumn{1}{|c|}{\hspace*{1cm}} &
\multicolumn{1}{|c|}{$j$} & $\quad J$ \\
\cline{2-3}
\end{tabular}
\end{center}

\begin{center}

{\bf \small Figure 1.
A forbidden by Restriction 1 pair of two electric or two
magnetic p-branes.
}

\end{center}

Here $d_i = d_j =1$, $i \neq j$, $i,j =1,\dots,n$. Restriction 1
may be also rewritten in terms of intersections
\beq{1.3.1a}
{\bf (R1)} \quad d(I \cap J) \leq d(I)  - 2,
\eeq
for any $I,J \in\Omega_{a,v}$, $a\in\tri$, $v= e,m$ (here $d(I) = d(J)$).

Restriction 2 is satisfied for $n_1=0$. For
$n_1\ge1$ it forbids the following electro-magnetic pairs,
corresponding to the same form $F^a, a \in \tri$:

\begin{center}
\begin{tabular}{ccc}
\cline{1-2}
\multicolumn{1}{|c|}{\hspace*{1cm}} &
\multicolumn{1}{|c|}{$i$}  & $\quad\bar J$ \\
\cline{1-2}
 & & \\
\cline{1-1}
\multicolumn{1}{|c|}{\hspace*{1cm}} & & $\quad I$
\\
\cline{1-1}
\end{tabular}
\end{center}

\begin{center}

{\bf \small Figure 2.
Forbidden by Restriction 2 electromagnetic pair of
p-branes}

\end{center}

Here $d_i =1$, $i =1,\dots,n$. In terms of intersections
Restriction 2 reads
\beq{1.3.1b}
{\bf (R2)} \quad d(I \cap J) \neq 0,
\eeq
for $I\in\Omega_{a,e}$ and $J\in\Omega_{a,m}$, $a \in \tri$.


{\bf Intersection rules.}
>From  (\ref{1.16}), (\ref{1.17}) and (\ref{1.18})  we get
the  $p$-brane intersection rules  corresponding
to the quasi-Cartan matrix $(A_{s s'})$ \cite{IMJ}
\beq{1.3.4}
d(I_s \cap I_{s'})= \frac{d(I_s)d(I_{s'})}{D-2}-
\chi_s\chi_{s'}\lambda_{a_s}\cdot\lambda_{a_{s'}} + \frac12 B_{s's'}
A_{ss'}, \eeq
where $\lambda_{a_s}\cdot\lambda_{a_{s'}} =
\lambda_{\alpha a_s}\lambda_{\beta a_{s'}} h^{\alpha\beta}$;
$s, s' \in S$.


\section{Composite S-brane solutions}

In this section we single out special
cosmological solutions called as $S$-branes.

\subsection{Toda-type solutions}

In what follows we suppose
that all  metrics $g^i$ have Euclidean signatures $(+,...,+)$
and  $w = -1$.
Thus the total metric $g$ has the pseudo-Euclidean signature  $(-,+,...,+)$.
We put in  action (\ref{1.1})  $\theta_a = 1$.

Then, from definitions (\ref{1.15a}) and (\ref{1.22}) we get
\beq{2.2}
\eps_s=\eps(I_s) =1
\eeq
for all $s$.

We also put
\beq{2.3}
d_1 > 1
\eeq
and $1 \notin I_s$, i.e. all branes do not contain $M_1$-submanifold.

Let us assume that
the (brane) matrix $(B_{ss'})$ is a positive definite
and hence
\beq{2.5}
 B_{ss} > 0
\eeq
for all $s \in S$.
This suggestion and  (\ref{2.2}) imply
\beq{2.6}
E_{TL} > 0.
\eeq

Now we make  the following choice of parameters in the solutions
>from subsections 2.2 and 2.3
\beq{2.7}
c^i = 0, \ i >1, \qquad c^\alpha=0,
\eeq
for all $\alpha$ and
\beq {2.7a}
\bar c^i = \bar c^\alpha = 0,
\eeq
for all $i, \alpha$.
With this choice the brane constraints (\ref{1.27}) are satisfied
identically due to  (\ref{1.2.3}).

With the adopted choice of parameters both
 solutions from
subsections 2.2 and 2.3 may be unified by the following relations
\bear{2.8a}
g= \biggl(\prod_{s \in S} H_s^{2 d(I_s) h_s/(D-2)} \biggr)
\biggl\{H^{2d_1/(1-d_1)} [ - dt \otimes dt+ H^2 g^1] \\ \nn
+ \sum_{i = 2}^{n} \Bigl(\prod_{s\in S}
H_s^{- 2 h_s  \delta_{i I_s} } \Bigr) g^i\biggr\},
\\ \label{2.8b}
\exp(\varphi^\alpha) =
\prod_{s\in S} H_s^{h_s \chi_s \lambda_{a_s}^\alpha},
\\ \label{2.8c}
F^a= \sum_{s \in S_e} \delta^a_{a_s} Q_s
\Bigl( \prod_{s' \in S}  H_{s'}^{- A_{s s'}} \Bigr) dt \wedge\tau(I_s)
+ \sum_{s \in S_m}\delta^a_{a_s} Q_s \tau(\bar I_s),
\ear
$a \in \tri$; $\alpha = 1, \ldots, l$;
where
\bear{2.9a}
H = |\xi_1(d_1-1)|^{1/2}  \frac{\sh(M t)}{M},  \ \xi_1 > 0;
\\ \label{2.9b}
|\xi_1(d_1-1)|^{1/2} \frac{\ch(M t)}{M},  \ \xi_1 < 0;  \\
 \label{2.9c}
\exp( M t),  \ \xi_1  = 0;
\ear
$M > 0$, and
\beq{2.10}
\frac{M^2 d_1}{d_1-1}= 2E_{TL} = \frac{1}{2}  \sum_{s,s' \in S} h_s
A_{s s'} \dot{q^s} \dot{q^{s'}}
  +  \sum_{s \in S}  Q_s^2 \exp( \sum_{s' \in S} A_{s s'} q^{s'} ).
\eeq

Here
\beq{2.11}
H_s  = \exp( - q^s(t)),
\eeq
with $q^s(t)$ obeying Toda-type equations
\beq{2.12}
\ddot{q^s} = -   B_{s s} Q_s^2 \exp( \sum_{s' \in S} A_{s s'}
q^{s'} ), \quad s \in S.
\eeq

The metric  $g^1$ is an Einstein metric on $M_1$:
${\rm Ric}[g^1] = \xi_1 g^1$, and all other metrics are Ricci-flat, i.e.
${\rm Ric }[g^i] = 0$ for $i > 1$.

In combining of two subfamilies of solutions from the previous section
we used the following definitions and identifications: $u = t$,
$f_1 = H$, $f_s = H_s$, $M = \sqrt{C_1}$ ($C_1 > 0$) for $\xi_1 \neq 0$ and
$(d_1 -1) c^1 = M$ for $\xi_1 = 0$.

\subsection{Solutions with "orthogonal" intersections}

Let us consider "orthogonal" (or $A_1 \oplus ...\oplus A_1$) intersection
 rules
 \beq{2.13} B_{s s'}=0, \quad s \neq s', \eeq
 (remind that
 $B_{s s'} = (U^{s},U^{s'})$
 are scalar products of brane vectors \cite{IMC,IMtop}).

Then using relations from Appendix we get
 \beq{2.14}
H_s = |Q_s||h_{s}|^{-1/2}  \frac{\ch(M_s (t-t_s))}{M_s},
\eeq
where all $M_s > 0$  and $t_s$ are constants and
\beq{2.15}
2 E_{TL} =  \sum_{s \in S} M_s^2 h_s.
\eeq

Here we used the notations $M_s =\sqrt{C_s}$, $C_s > 0$ (see Appendix).
For $\xi_1 \neq 0$ these "orthogonal" $S$-brane solutions are special cases
of more general ones from \cite{IMJ}.


\subsection{Solutions with "block-orthogonal" intersections}

Let us suppose that
\beq{2.13a}
B_{s s'}=0, \quad  s \in S_i, \ s'\in S_j, \ i \neq j,
\eeq
where
\beq{2.13b}
S=S_1\sqcup\dots\sqcup S_k,
\eeq
$S_i\ne\emptyset$, $i,j=1,\dots,k$.
Relation (\ref{2.13b}) means that the set $S$ splits into  $k$
mutually non-intersecting subsets (blocks) $S_1,\dots,S_k$.

Then using relations from Appendix we get
 \beq{2.14b}
H_s = \left[ |Q_s||h_s b_s|^{-1/2}  \frac{\ch(M_s (t-t_s))}{M_s}
\right]^{b_s},
 \eeq
where all $M_s > 0$ and $t_s$ are constants
coinciding inside "blocks":
\beq{2.14c}
t_s = t_{s'}, \quad M_s = M_{s'},
\eeq
$s,s' \in S_i$, $i = 1, \ldots, k$.
The charges should be proportional to each other inside blocks
\bear{1.4.10b}
\frac{Q_s^2}{b_s h_s} = \frac{Q_{s'}^2}{b_{s'} h_{s'}},
\ear
$s,s' \in S_i$, $i = 1, \ldots, k$.
Here
\beq{1.4.4}
b_s = 2 \sum_{s' \in S} A^{s s'}, \quad (A^{ss'})= (A_{ss'})^{-1}.
\eeq

For the Toda-like part of energy we get from Appendix
\beq{2.15b}
2 E_{TL} =  \sum_{s \in S} M_s^2 b_s h_s.
\eeq

More general "block-orthogonal" solutions where considered in
\cite{Br1,IMJ1,IMJ2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Solutions related to $A_m$  Toda  chain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Here we  consider exact solutions to
Toda-chain  (\ref{2.12}) equations \cite{T}
corresponding to the Lie algebra
 $A_m= sl(m+1, \C)$
with the Cartan matrix
\beq{B.1a}
\left(A_{ss'}\right)=
\left( \begin{array}{*{6}{c}}
2&-1&0&\ldots&0&0\\
-1&2&-1&\ldots&0&0\\
0&-1&2&\ldots&0&0\\
\multicolumn{6}{c}{\dotfill}\\
0&0&0&\ldots&2&-1\\
0&0&0&\ldots&-1&2
\end{array}
\right)\quad
\eeq
$s,s' \in S =  \{1, \ldots, m \}$.


The Toda chain equations have the following solutions \cite{And,GM,IK}
\beq{B.3}
H_s =
C_s^{-1} \sum_{r_1< \dots <r_s}^{m+1} v_{r_1}\cdots v_{r_s}
\Delta^2( w_{r_1}, \ldots, w_{r_s}) \exp[(w_{r_1}+\ldots +w_{r_s}) t],
\eeq
$s = 1, \ldots, m$, where
\beq{B.4a}
\Delta( w_{r_1}, \ldots, w_{r_s})  =
\prod_{i<j}^{s} \left(w_{r_i}-w_{r_j}\right); \quad
\Delta(w_{r_1}) \equiv 1,
\eeq
denotes the  Vandermonde determinant.
The real constants $v_r$ and $w_r$, $r = 1, \ldots, m + 1$, obey the
relations
\beq{B.5}
\prod_{r=1}^{m+1} v_r= \Delta^{-2}(w_1,\ldots, w_{m+1}), \qquad
\sum_{r=1}^{m+1}w_r=0.
\eeq
In (\ref{B.3})
\beq{B.6}
C_s = \prod_{s'=1}^{m} [B_{s's'} Q_{s'}^2]^{-A^{s s'}},
\eeq
where
\beq{B.7}
A^{s s'}= \frac1{m+1}\min(s,s')[m+1-\max(s,s')],
\eeq
$s, s' = 1, \ldots, m$,  \cite{FS}.
Here $v_r \neq 0$ and $w_r \neq w_{r'}$, $r \neq r'$;
$r, r' = 1, \ldots, m +1$.
Due to  $B_{ss} > 0$, $s \in S$,  all $w_r, v_r$ are real, and, moreover,
all  $v_r > 0$, $r =1, \ldots, m+1$ \cite{IK}.

For the (Toda) energy  we get
\beq{B.10}
E_{TL} =  \frac{h}{4} \sum_{r=1}^{m+1} w^2_r,
\eeq
where $h = (B_{ss})^{-1}$,
$s \in S$ (here all $B_{ss}$ are equal).

It should be noted that pioneering
cosmological solutions with $p$-branes related to
Toda chains were considered earlier in  \cite{LPX,LMPX,LP1}.


\section{Examples}

Here we present well-known $SM$-brane solutions in $D =11$ supergravity
\cite{CJS}.

\subsection{Solutions for algebra $A_1$}

We start with single $S$-branes.


\subsubsection{$SM2$-brane}

Let $n =3$, $d_2 = 3$.
The $SM2$-brane solution reads
\bear{4.1}
g=  H^{1/3}_1  \biggl\{H^{2d_1/(1-d_1)}[ - dt \otimes dt+ H^2 g^1]
+  H_1^{-1} g^2 + g^3   \biggr\},
\\  \label{4.1a}
F= Q_1 H^{- 2}_1 dt  \wedge \tau_2,
\ear
where
$H$ is defined in (\ref{2.9a})-(\ref{2.9c}),
$H_1$ is defined in  (\ref{2.14}) ($h_{1} =1/2$)
and $2M^2 d_1/(d_1-1)=  M_1^2$.

\subsubsection{$SM5$-brane}

Let us consider the magnetic solution dual to $SM2$.
We put $n =3$, $d_2 = 6$.
The solution reads
\bear{4.2b}
g=  H^{2/3}_1  \biggl\{H^{2d_1/(1-d_1)}[ - dt \otimes dt+ H^2
g^1]
+  H_1^{-1} g^2 + g^3   \biggr\},
\\  \label{4.2ab}
F= Q_1 \tau_1 \wedge \tau_3 ,
\ear
where $H$ and $H_1$ are defined in (\ref{2.9a})-(\ref{2.9c})
and (\ref{2.14}),
respectively,  ($h_{1} =1/2$) and $2M^2 d_1/(d_1-1)=  M_1^2$.

\subsection{$SM2 \cap SM5$-branes with $A_1 \oplus A_1$ intersection}

Here we present a "superposition" of $SM2$ and $SM5$
solutions corresponding to "ortogonal intersection":
$d(I_1 \cap I_2) =2$.
We put $n =5$, $d_1 =2$, $d_2 = 1$, $d_3 = 2$, $d_4 = 4$, $d_5 = 1$
and $I_1 = \{ 2,3 \}$, $I_2 = \{ 3, 4 \}$.
The solution reads
\bear{4.3}
g= H_1^{1/3}  H_2^{2/3}  \biggl\{H^{-4}[ - dt \otimes dt+ H^2 g^1]
+  H_1^{-1} g^2 +
\\ \nn H_1^{-1} H_2^{-1} g^3 + H_2^{-1} g^4 +   g^5   \biggr\},
\\  \label{4.3a}
F= Q_1 H_1^{- 2}dt  \wedge \tau_2 \wedge \tau_3 + Q_2 \tau_1 \wedge \tau_2
\wedge \tau_5,  \ear
where $H$ and $H_s$ are  defined in  (\ref{2.9a})-(\ref{2.9c}) and
(\ref{2.14}), respectively (all $h_{s} =1/2$) and $4M^2 =  M_1^2 + M_2^2$.

\subsection{$SM2 \cap SM5$-branes with $A_2$ intersection}

Now we consider a (dyonic) solution consisting of  $SM2$ and
$SM5$ branes and with $A_2$ intersection: $d(I_1 \cap I_2) =1$ \cite{IMtop}.
We put $n =4$, $d_1 = d_2 = 2$, $d_3 = 1$, $d_4 = 5$
and $I_1 = \{ 2,3 \}$, $I_2 = \{ 3, 4 \}$.
The  solution reads (see subsection 3.3)
\bear{4.4}
g= H_1  \biggl\{H^{-4}[ - dt \otimes dt+ H^2 g^1]
+  H_1^{-1} g^2 +
\\ \nn H_1^{-2}  g^3 + H_1^{-1} g^4  \biggr\},
\\  \label{4.4a}
F= Q_1 H_1^{- 1}dt  \wedge \tau_2 \wedge \tau_3 \pm Q_1 \tau_1 \wedge
\tau_2,  \ear
where $H$ is defined in  (\ref{2.9a})-(\ref{2.9c}) and
 \beq{4.4b}
H_1 = \left[ |Q_1|  \frac{\ch(M (t-t_1))}{M}\right]^{2}.
 \eeq

This solution is also a special case of $A_2$ Toda chain
solution (with $Q_1^2 = Q_2^2$) from subsection 3.4.


\section{Kasner-like asymptotical behaviour}

Here we consider asymptotical behaviour of  $S$-brane solutions
in the limit $t \to +\infty$, when
\beq{s.1}
H \sim {\rm const} \exp(Mt), \qquad  H_s \sim {\rm const} \exp(M_s t),
\eeq
$s \in S$.

It may be verified by a straightforward calculation for one-brane
case that  in the limit $t \to +\infty$
the metric and scalar fields have asymptotical
Kasner-like behaviour
\bear{s.2}
g_{as}=
- d\tau \otimes d\tau + \sum_{i =1}^{n} A_i \tau^{2 \alpha^i}g^i, \\
\varphi^\beta_{as}  = \alpha^{\beta} \ln \tau + const,
\ear
as $\tau \to +0$ ($\tau$ is synchronous time variable)
with  the set of Kasner parameters $\alpha =
(\alpha^{i}, \alpha^{\gamma})$, obeying
\beq{s.3}
\sum_{i=1}^{n} d_i \alpha^i =
\sum_{i=1}^{n} d_i (\alpha^i)^2 +
\alpha^{\beta} \alpha^{\gamma} h_{\beta \gamma}= 1.
\eeq
Here $\tau \sim {\rm const} \exp(- {\cal M}t)$ for $t \to +\infty$,
where ${\cal M} > 0$ is a linear combination of "masses" $M$, $M_s$.
Analogous (though more tedious) calculations for several branes with
orthogonal intersections give an analogous result.

Here we show that the  Kasner-like asymptotical behaviour may be proved
using the billiard representation for multidimensional cosmology with
branes \cite{IMb1} under  restrictions imposed
in Section 3 and the following additional
condition added: the  matrix $(h_{\alpha\beta})$ is supposed to be
positive definite.
According to results of refs. \cite{IMb1}
for the proof of the Kasner-like asymptotical
behaviour it is sufficient to verify
that there exists a Kasner set obeying the  inequalities
\beq{5.1}
U^s(\alpha) =   \sum_{i\in I_s}d_i\alpha^i
 -\chi_s\lambda_{a_s \gamma}\alpha^{\gamma} > 0,
\eeq
$s \in S$.
Indeed, let us consider the following set  of Kasner parameters
\bear{5.2}
 \alpha^1 =   (d_1 - \Delta)/[d_1(D-1)], \\
\alpha^2 =...= \alpha^n =  (d + \Delta)/[d(D-1)], \\
\alpha^{\gamma} = 0,
\ear
where $d = d_2 +...+ d_n$ and $\Delta = \sqrt{d_1 d (D-2)}$. Due to
condition $1 \notin I_s$ this Kasner set  does satisfy the relations
(\ref{5.1}).

When all $d_i > 1$ and
$\alpha^{\beta} \alpha^{\gamma} h_{\beta \gamma} = 0$,
then the Riemann tensor squared of the Kasner-type metric
(\ref{s.2}) diverges for $\tau \to +0$ \cite{IMsing} and, hence,
the Riemann tensor squared of the original metric $g$
diverges as $t \to + \infty$.
For  $\alpha^{\beta} \alpha^{\gamma} h_{\beta \gamma} \neq 0$
the scalar curvature of (\ref{s.2}) diverges as $\tau \to +0$
that implies $R[g] \to \infty$  for $t \to + \infty$.

Analogous consideration may be carried out for the solution
with $H$ from (\ref{2.9b}) in the limit  $t \to -\infty$.


{\bf Remark. Conical singularity.}
Let us consider  the asymptotical behaviour of the
solution with $H$ from
(\ref{2.9a}) when  $t \to +0$. In general case we get
a conical singularity in this limit. This singularity
is absent in special case when
$(M_1,g^1)$ is unit Lobachevsky space of dimension
$d_1$ (and $\xi_1 = -(d_1 -1)$).
This is a well-known Milne-type resolution of singularity.


{\bf Remark. $S$-brane cosmology with oscillating
behaviour near the singularity.}
In  the pioneering paper on billiard approach
for multidimensional cosmology
with $p$-branes \cite{IMb1} a large variety
of composite $S$-brane solutions with never-ending
oscillating behaviour near the singularity
was described (and certain examples were considered).
The open problem is to find examples of such type solutions
(with block-orthogonal metrics) in string cosmology.
We note that recent results  of Damour and Henneaux
on chaotic behaviour in  $D=10,11$ supergravities
\cite{DamH1,DamH2} (based on inequalities (\ref{5.1}))
deal with  $1$-dimensional manifolds $M_1 = \dots = M_n$
and sets of (composite) electric branes with "forbidden" intersections
(see subsection 2.4) that lead to additional constraints
\cite{IMC}. The recent mathematical analysis carried out
in the paper \cite{DHRW} gives
certain arguments in favour of asymptotically
block-orthogonal (but not obviously block-orthogonal) form of metric
in the vicinity of singularity
for pure cosmological solutions in string cosmology with
composite $p$-brane configurations from \cite{DamH1,DamH2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions and discussions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this paper  a family of Toda-like
composite $S$-brane solutions was presented.
These ($S$-brane) solutions are special
case of more general cosmological-type solutions from  \cite{IK}.
They generalize $S$-brane solutions from refs. \cite{S1,S2,S3,S4,S5}
to a composite  configurations with next to arbitrary
intersecting rules.

Here  several subclasses of $S$-brane solutions
with "orthogonal",  "block-orthogonal"
and $A_m$ intersections were considered and  certain examples  of
solutions in $D= 11$ supergravity (describing $SM$-branes with
$A_1$, $A_1 \oplus A_1$ and  $A_2$ intersections),  were singled out.
%(For non-Ricci-flat space $(M_1,g^1)$ general
%classical and quantum $S$-brane solutions with
%"orthogonal" intersection rules were considered earlier in \cite{IMJ}.)

The solutions under consideration have (in general position) an
asymptotical  Kasner-like behaviour near the singularity when $t \to
+ \infty$.  This fact was shown here using the billiard approach
for $p$-brane cosmology \cite{IMb1} under the assumption
$1 \notin I_s$ imposed (i.e.
when all branes do not contain $M_1$-submanifold).
The relaxing of this assumption  may lead
to $S$-brane configurations with oscillating
behaviour near the singularity (see  \cite{IMb1}).

We remind that in refs. \cite{IMJ,IMAJ}  the Wheeler-DeWitt (WDW) equation
for the quantum cosmology with composite electro-magnetic $p$-branes
(e.g. $S$-branes) defined on product of Einstein spaces
was obtained (for non-composite  electric case see also \cite{GrIM}).
In these papers the WDW equation
was integrated for intersecting $p$-branes with orthogonal $U$-vectors,
when $n-1$ internal spaces are Ricci-flat and one is
the Einstein space of a
non-zero curvature (for non-composite electric case see \cite{GrIM}).
A slightly different approach with classical field of forms (and
a special brane setup) was suggested in \cite{LMMP}.
An open problem is to generalize classical
and quantum $S$-brane solutions  from \cite{IMJ}
(corresponding to the Lie algebra $A_1 \oplus \dots \oplus A_1$)
to other semisimple Lie algebras.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\begin{center}
{\bf Acknowledgments}
\end{center}

This work was supported in part by the Russian Ministry of
Science and Technology, Russian Foundation for Basic Research
(RFFI-01-02-17312-a) and Project DFG (436 RUS 113/678/0-1(R)).
The author thanks Prof. H. Dehnen
and colleagues from Physical Department of the University of Konstanz for
hospitality during his visit in May-July 2002.

\vspace{10pt}

{\large \bf Appendix}

\renewcommand{\theequation}{A.\arabic{equation}}

\vspace{10pt}

{\bf Solutions for "block-orthogonal" intersections.}
Here we consider "block-orthogonal" case (\ref{2.13a}), (\ref{2.13b}).
In this  case the moduli functions (\ref{1.19})
read  \cite{IMJ2,IMJ1,IMtop}
\beq{1.4.3}
f_s = (\bar{f}_s)^{b_s},
\eeq
with powers $b_s$ defined in (\ref{1.4.4}) and
\bear{1.4.5}
\bar{f}_s(u)=
R_s \sh(\sqrt{C_s}(u-u_s)), \;
C_s>0, \; \eta_s\eps_s<0; \\ \label{1.4.7}
R_s \sin(\sqrt{|C_s|}(u-u_s)), \;
C_s<0, \; \eta_s\eps_s<0; \\ \label{1.4.8}
R_s \ch(\sqrt{C_s}(u-u_s)), \;
C_s>0, \; \eta_s\eps_s>0; \\ \label{1.4.9}
|Q^s||b_s h_s|^{-1/2}(u-u_s), \; C_s=0, \; \eta_s\eps_s<0,
\ear
where $R_s = |Q_s||b_s h_s C_s|^{-1/2}$,
$\eta_s = {\rm sign}(b_s h_s) = \pm 1$,
$C_s$, $u_s$ -- are constants, $s \in S$, coinciding
inside "blocks":
$u_s = u_{s'}$, $C_s = C_{s'}$,
$s,s' \in S_i$; $i = 1, \ldots, k$.

The charges should be proportional to each other inside blocks
\bear{1.4.10}
\frac{\eps_s Q_s^2}{b_s h_s} = \frac{\eps_{s'} Q_{s'}^2}{b_{s'} h_{s'}},
\ear
$s,s' \in S_i$, $i = 1, \ldots, k$.

The Toda part of the energy reads in this case
\beq{1.4.11}
E_{TL} = \frac{1}{2} \sum_{s \in S} C_s b_s h_s.
\eeq

{\bf "Orthogonal" case. }
In the "orthogonal" case when all blocks consist of one element,
i.e.  $|S_1| = \ldots = |S_k| = 1$ \cite{IMJ},
the quasy-Cartan matrix is diagonal   $A = {\rm
 diag}(2, \ldots,2)$, all $b_s = 1$
and
\beq{1.4.3a}
f_s = \bar{f}_s,
\eeq
$s \in S$.


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String Kaluza-Klein cosmology,
{\it Nucl. Phys.} {\bf B 430} 441 (1994) 441;
hep-th/9403179 (see also hep-th/9312167, hep-th/9704013).

\bibitem{LOW}
A. Lukas, B.A. Ovrut and D. Waldram, Cosmological solutions in type
II string theory, {\it Phys. Lett.} {\bf B 393}, 65 (1997); hep-th/9608195.

\bibitem{LMPX}
H. L\"u, S. Mukherji, C.N. Pope and K.-W. Xu,
Cosmological solutions in string theories,
{\it Phys. Rev.} {\bf D 55 }, 7926-7935 (1997);
hep-th/9610107.

\bibitem{LMP}
H. L\"u, S. Mukherji and C.N. Pope,
>From p-branes to cosmology,
{\it Int. J. Mod. Phys.} {\bf A 14} 4121-4142 (1999);
hep-th/9612224.

\bibitem{GrIM}
M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov,
Integrable multidimensional quantum cosmology  for intersecting p-branes,
{\it Grav. and Cosmol.\/} {\bf 3}, No 3 (11), 243-249 (1997),
gr-qc/9708031.

\bibitem{BGIM}
K.A. Bronnikov, M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov,
Integrable multidimensional cosmology for
intersecting $p$-branes,
{\it Grav.  Cosmol. } {\bf  3}, No 2(10), 105-112 (1997).

\bibitem{IMJ}
V.D. Ivashchuk and V.N. Melnikov,
Multidimensional classical and quantum cosmology
with intersecting $p$-branes, hep-th/9708157;
{\it J. Math. Phys.}, {\bf 39}, 2866-2889 (1998).

\bibitem{GM}
V.R. Gavrilov and V.N. Melnikov,
Toda chains  associated with   Lie algebras  $A_m$ in multidimensional
gravitation and cosmology with intersecting $p$-branes,
{\it Theor. Math. Phys.} {\bf 123}, No 3, 374-394 (2000) (in Russian).

\bibitem{IK}
V.D. Ivashchuk and S.-W. Kim,
Solutions with intersecting p-branes related to Toda chains,
{\it J. Math. Phys.} {\bf 41}, 444-460 (2000); hep-th/9907019.

\bibitem{IMtop}
V.D. Ivashchuk and V.N. Melnikov, Exact solutions in
multidimensional gravity with antisymmetric forms, topical review,
{\it Class. Quantum Grav.},  {\bf 18},  R87-R152 (2001);
hep-th/0110274.

\bibitem{LPX}
H. L\"u, C.N. Pope and K-W. Xu,
Liouville and Toda solutions of M-theory,
{\it Mod. Phys. Lett.} {\bf A 11}, 1785-1796 (1996); hep-th/9604058.

\bibitem{LP1}
H. L\"u and C.N. Pope,
SL(N+1,R) Toda solitons in supergravities,
{\it Int. J. Mod. Phys.} {\bf A 12}, 2061 -2074 (1997);
hep-th/9607027.

\bibitem{IMC}
V.D. Ivashchuk and V.N. Melnikov,
Sigma-model for the generalized  composite p-branes,
{\it Class. Quantum Grav.} {\bf 14}, 3001 (1997);
Corrigenda {\it ibid.} {\bf 15 } (12), 3941 (1998);
hep-th/9705036.

\bibitem{IMBl}
V.D. Ivashchuk and V.N. Melnikov,
Madjumdar-Papapetrou type solutions in sigma-model
and intersecting p-branes, {\it Class. Quantum Grav.} {\bf 16},
849 (1999); hep-th/9802121.

\bibitem{Br1}
K.A. Bronnikov, Block-orthogonal brane systems, black
holes and wormholes,
{\it Grav.  Cosmol.} {\bf 4}, No 1 (13),  49 (1998);
hep-th/9710207.

\bibitem{IMJ1}
V.D.Ivashchuk and V.N.Melnikov. Multidimensional cosmological and
spherically symmetric solutions with intersecting p-branes.
In Lecture Notes in Physics, Vol. 537,
"Mathematical and Quantum Aspects of Relativity and Cosmology
Proceedings of the Second Samos Meeting on Cosmology, Geometry and
Relativity held at Pythagoreon, Samos, Greece, 1998,
eds: S. Cotsakis, G.W. Gibbons., Berlin, Springer, 2000;
gr-qc/9901001.

\bibitem{IMJ2}
V.D.Ivashchuk and V.N.Melnikov, Cosmological and spherically
symmetric solutions with Intersecting p-branes, {\it J. Math. Phys.},
{\bf 40}, 6558-6576 (1999).

\bibitem{IMb1}
V.D. Ivashchuk and V.N. Melnikov, Billiard representation for
multidimensional cosmology with intersecting p-branes near the singularity,
{\it J. Math. Phys.},  {\bf 41}, No 8, 6341-6363 (2000);
hep-th/9904077.

\bibitem{IMsing}
V.D. Ivashchuk and V.N. Melnikov,
On singular solutions in multidimensional gravity,
{\it Grav. and Cosmol.} {\bf 1}, 204 (1996);
hep-th/9612089.

\bibitem{CJS}
E. Cremmer, B. Julia, J. Scherk,
{\it Phys. Lett. } {\bf B 76}, 409 (1978).

\bibitem{T}
M. Toda, {\it Progr. Theor. Phys.} {\bf 45}, 174 (1970).

\bibitem{And}
A. Anderson, {\it J. Math. Phys.} {\bf 37}, 1349 (1996);
hep-th/9507092.


\bibitem{FS}
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and
Representations,  A graduate course for physicists
(Cambridge University Press, Cambridge, 1997).

\bibitem{DamH1}
T. Damour and M. Henneaux,
Chaos in superstring cosmology, {\it Phys. Rev. Lett.}
{\bf 85}, 920 (2000); hep-th/0003139.

\bibitem{DamH2}
T. Damour and M. Henneaux,
Oscillatory behaviour in homogeneous string cosmology
models, {\it Phys. Lett.}
{\bf B 488}, 108 (2000); {\it Erratum ibid.} {\bf 491} 377;
hep-th/0006171.

\bibitem{DHRW}
T. Damour, M. Henneaux, A.D. Rendall and M. Weaver,
Kasner-like behaviour for subcritical Einstein-matter systems,
models, gr-qc/0202069.

\bibitem{IMAJ}
V.D. Ivashchuk and V.N. Melnikov,
Multidimensional quantum cosmology with
intersecting p-branes, {\it Hadronic J.} {\bf 21}, 319-335 (1998).

\bibitem{LMMP}
H. L\"u, J. Maharana, S. Mukherji  and C.N. Pope,
Cosmological solutions, p-branes and the Wheeler De Witt
equation, {\it Phys. Rev.} {\bf D 57 }, 2219-2229 (1997);
hep-th/9707182.


\end{thebibliography}

\end{document}


hep-th0208101.extracted_bib
\begin{thebibliography}{99}

%%%%%%%%%%%%%%%S-BRANES%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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A. Strominger,  The dS/CFT correspondence,
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J.M. Maldacena,  The large N limit of
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E. Witten,  Anti-de Sitter space and holography,
{\it Adv. Theor. Math. Phys.} {\bf 2}, 253 (1998); hep-th/9802150.

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K. Behrndt and S. Forste,
String Kaluza-Klein cosmology,
{\it Nucl. Phys.} {\bf B 430} 441 (1994) 441;
hep-th/9403179 (see also hep-th/9312167, hep-th/9704013).

\bibitem{LOW}
A. Lukas, B.A. Ovrut and D. Waldram, Cosmological solutions in type
II string theory, {\it Phys. Lett.} {\bf B 393}, 65 (1997); hep-th/9608195.

\bibitem{LMPX}
H. L\"u, S. Mukherji, C.N. Pope and K.-W. Xu,
Cosmological solutions in string theories,
{\it Phys. Rev.} {\bf D 55 }, 7926-7935 (1997);
hep-th/9610107.

\bibitem{LMP}
H. L\"u, S. Mukherji and C.N. Pope,
>From p-branes to cosmology,
{\it Int. J. Mod. Phys.} {\bf A 14} 4121-4142 (1999);
hep-th/9612224.

\bibitem{GrIM}
M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov,
Integrable multidimensional quantum cosmology  for intersecting p-branes,
{\it Grav. and Cosmol.\/} {\bf 3}, No 3 (11), 243-249 (1997),
gr-qc/9708031.

\bibitem{BGIM}
K.A. Bronnikov, M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov,
Integrable multidimensional cosmology for
intersecting $p$-branes,
{\it Grav.  Cosmol. } {\bf  3}, No 2(10), 105-112 (1997).

\bibitem{IMJ}
V.D. Ivashchuk and V.N. Melnikov,
Multidimensional classical and quantum cosmology
with intersecting $p$-branes, hep-th/9708157;
{\it J. Math. Phys.}, {\bf 39}, 2866-2889 (1998).

\bibitem{GM}
V.R. Gavrilov and V.N. Melnikov,
Toda chains  associated with   Lie algebras  $A_m$ in multidimensional
gravitation and cosmology with intersecting $p$-branes,
{\it Theor. Math. Phys.} {\bf 123}, No 3, 374-394 (2000) (in Russian).

\bibitem{IK}
V.D. Ivashchuk and S.-W. Kim,
Solutions with intersecting p-branes related to Toda chains,
{\it J. Math. Phys.} {\bf 41}, 444-460 (2000); hep-th/9907019.

\bibitem{IMtop}
V.D. Ivashchuk and V.N. Melnikov, Exact solutions in
multidimensional gravity with antisymmetric forms, topical review,
{\it Class. Quantum Grav.},  {\bf 18},  R87-R152 (2001);
hep-th/0110274.

\bibitem{LPX}
H. L\"u, C.N. Pope and K-W. Xu,
Liouville and Toda solutions of M-theory,
{\it Mod. Phys. Lett.} {\bf A 11}, 1785-1796 (1996); hep-th/9604058.

\bibitem{LP1}
H. L\"u and C.N. Pope,
SL(N+1,R) Toda solitons in supergravities,
{\it Int. J. Mod. Phys.} {\bf A 12}, 2061 -2074 (1997);
hep-th/9607027.

\bibitem{IMC}
V.D. Ivashchuk and V.N. Melnikov,
Sigma-model for the generalized  composite p-branes,
{\it Class. Quantum Grav.} {\bf 14}, 3001 (1997);
Corrigenda {\it ibid.} {\bf 15 } (12), 3941 (1998);
hep-th/9705036.

\bibitem{IMBl}
V.D. Ivashchuk and V.N. Melnikov,
Madjumdar-Papapetrou type solutions in sigma-model
and intersecting p-branes, {\it Class. Quantum Grav.} {\bf 16},
849 (1999); hep-th/9802121.

\bibitem{Br1}
K.A. Bronnikov, Block-orthogonal brane systems, black
holes and wormholes,
{\it Grav.  Cosmol.} {\bf 4}, No 1 (13),  49 (1998);
hep-th/9710207.

\bibitem{IMJ1}
V.D.Ivashchuk and V.N.Melnikov. Multidimensional cosmological and
spherically symmetric solutions with intersecting p-branes.
In Lecture Notes in Physics, Vol. 537,
"Mathematical and Quantum Aspects of Relativity and Cosmology
Proceedings of the Second Samos Meeting on Cosmology, Geometry and
Relativity held at Pythagoreon, Samos, Greece, 1998,
eds: S. Cotsakis, G.W. Gibbons., Berlin, Springer, 2000;
gr-qc/9901001.

\bibitem{IMJ2}
V.D.Ivashchuk and V.N.Melnikov, Cosmological and spherically
symmetric solutions with Intersecting p-branes, {\it J. Math. Phys.},
{\bf 40}, 6558-6576 (1999).

\bibitem{IMb1}
V.D. Ivashchuk and V.N. Melnikov, Billiard representation for
multidimensional cosmology with intersecting p-branes near the singularity,
{\it J. Math. Phys.},  {\bf 41}, No 8, 6341-6363 (2000);
hep-th/9904077.

\bibitem{IMsing}
V.D. Ivashchuk and V.N. Melnikov,
On singular solutions in multidimensional gravity,
{\it Grav. and Cosmol.} {\bf 1}, 204 (1996);
hep-th/9612089.

\bibitem{CJS}
E. Cremmer, B. Julia, J. Scherk,
{\it Phys. Lett. } {\bf B 76}, 409 (1978).

\bibitem{T}
M. Toda, {\it Progr. Theor. Phys.} {\bf 45}, 174 (1970).

\bibitem{And}
A. Anderson, {\it J. Math. Phys.} {\bf 37}, 1349 (1996);
hep-th/9507092.


\bibitem{FS}
J. Fuchs and C. Schweigert, Symmetries, Lie algebras and
Representations,  A graduate course for physicists
(Cambridge University Press, Cambridge, 1997).

\bibitem{DamH1}
T. Damour and M. Henneaux,
Chaos in superstring cosmology, {\it Phys. Rev. Lett.}
{\bf 85}, 920 (2000); hep-th/0003139.

\bibitem{DamH2}
T. Damour and M. Henneaux,
Oscillatory behaviour in homogeneous string cosmology
models, {\it Phys. Lett.}
{\bf B 488}, 108 (2000); {\it Erratum ibid.} {\bf 491} 377;
hep-th/0006171.

\bibitem{DHRW}
T. Damour, M. Henneaux, A.D. Rendall and M. Weaver,
Kasner-like behaviour for subcritical Einstein-matter systems,
models, gr-qc/0202069.

\bibitem{IMAJ}
V.D. Ivashchuk and V.N. Melnikov,
Multidimensional quantum cosmology with
intersecting p-branes, {\it Hadronic J.} {\bf 21}, 319-335 (1998).

\bibitem{LMMP}
H. L\"u, J. Maharana, S. Mukherji  and C.N. Pope,
Cosmological solutions, p-branes and the Wheeler De Witt
equation, {\it Phys. Rev.} {\bf D 57 }, 2219-2229 (1997);
hep-th/9707182.


\end{thebibliography}