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\title[Algorithm for constructing general  % The reduced tittle is indicated
solution of a hyperbolic system\dots]{Algorithm for constructing
general
solution of an $n$--component hyperbolic system of equations with Laplace zero invariants and boundary value problems}% Title of the article

\author{A.V. Ivanov, Yu.P. Petrov}%Indicate authors

\address{Aleksandr Vasil'evich Ivanov,
\newline\hphantom{iii} Institute of mathematics with Computer Centre RAS,% Institution
\newline\hphantom{iii} Chernyshevskii Str., 112, % Adress (street, house number, etc.)
\newline\hphantom{iii} 450008, г. Уфа, Россия}%  Адрес (postal code, city, Country)
\email{Ivanov@mail.ru}% your e-mail for correspondence

\address{Yurii Petrovich Petrov,
\newline\hphantom{iii} Bashkir State University,% Institution
\newline\hphantom{iii} Z.Validi Str., 32, % Address (street, building, etc.)
\newline\hphantom{iii} 450074, Ufa, Russia} %Address (postal code, city, country)
\email{Petrov@mail.ru}% your e-mail for correspondence


\thanks{\sc A.V. Ivanov, Yu.P. Petrov, %   authors' names in English
Algorithm of building the solution of hyperbolic system of equation.}% title of the article in English
\thanks{\copyright \ Ivanov A.B., Petrov Y.P. 2009}
\thanks{\rm The work is supported  by the Russia Foundation for Basic Research (grants 07-01-00111-а, 08-01-05550-а)}
\thanks{\it Submitted on 24 August 2010.}


\maketitle {\small
\begin{quote}
\noindent{\bf Abstract. } Explicit formulae for solving a
hyperbolic system are obtained in the paper ... The volume of
abstracts of at least 500 characters and no more than 1500
characters.
\medskip

\noindent{\bf Key words:}{ invariants and generalized Laplace
invariants, Goursat problem, Tody chains}
\medskip
\end{quote}
\begin{quote}
\noindent{\bf Abstract. } Explicit formulae for solving a
hyperbolic system are obtained in the paper ... The volume of
abstracts of at least 500 characters and no more than 1500
characters..\medskip

\noindent{\bf Keywords:} invariants and generalized Laplace
invariants, Goursat problem, Tody chains
\end{quote} }

\section{Introduction}%The main text of the article

One of the classical tool for constructing general solutions of
linear hyperbolic equations of the form
\begin{equation}
u_{xy}+a(x,y)u_x+b(x,y)u_y+c(x,y)u=0 \label{eqo}
\end{equation}
is the cascade Laplace method (see, e.g. \cite{ZHIBER1}). The
basis of this method is the sequence of the Laplace invariants
\begin{equation}
\ldots,h_{-3},h_{-2},h_{-1},h_0,h_1,h_2,h_3,\ldots \label{invv}
\end{equation}
and Laplace transformations related to it.

Equation \eqref{eqo} represents \dots


\bigskip
\begin{thebibliography}{1}

\bibitem{ZHIBER1} E. Goursat {\it Legon sur J'integretion des equations aux derivees partielles du second ordre a
deux variables independantes}  Hermann. Paris. 1896. 200 p.





\end{thebibliography}
\end{document}
