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\begin{document}

%\begin{center}
%{ \Huge   \bf
\title{Approximate Standing Wave Solutions in Massless $\varphi^4$ Theory}
%
%}
%\vspace{5mm}


\tocauthor{O.~A.~Khrustalev and S.~Yu.~Vernov}
\author{O.~A.~Khrustalev\footnote{E-mail address:
    khrust@sunny.bog.msu.su }\\
\small\it  Institute for Theoretical
Problems of Microphysics, Moscow\  State\  University;\\
\small\it Department of
Physics, Moscow\  State\ University,\ \ Moscow,\ \ 119899,\ \ Russia\\
and
S.~Yu.~Vernov\footnote {E-mail address:
svernov@theory.npi.msu.su}\\
{\small\it  Institute\ of\ Nuclear\ Physics,
Moscow\  State\  University,\  Moscow,\  119899,\  Russia}}

\date{}
\maketitle
%\end{center}

%\vspace{5mm}

%\renewcommand{\abstractname}{\normalsize \bf Abstract}

\begin{abstract}
Doubly periodic solutions for  the Lagrange--Euler
equation of the $(1+1)$-dimensional scalar $\varphi^4$
theory are studied. Provided that nonlinear term is small, the
Poincare--Lindstedt asympto\-tic method is used to find asymptotic
solutions in the standing wave form. It is proved that using the Jacobi
elliptic function {\bf cn }  as a zero approximation one can solve the
problem of the main resonance, appearing in the case of zero mass, and
construct a uniform expansion both to the first and to the second order.
\end{abstract}

 \section {Two kinds of doubly periodic solutions}


 Our investigation is dedicated to the construction
 of doubly periodic classical fields in the $(1+1)$-dimen\-sional
$\varphi^4$ theory.  We study the model of an isolated real scalar field
$\varphi(x,t)$, described by the Lagrangian density:
  $$
 {\cal L}(\varphi)
  = \frac{1}{2}\left(\varphi_{,t}^2(x,t)-\varphi_{,x}^2(x,t)-
  M^2\varphi^2(x,t) - \frac{\varepsilon}{2}\varphi^4(x,t)\right).
  $$


Let us consider the Lagrange--Euler equation:
$$
 \frac{\partial^2\varphi(x,t)}{\partial
 x^2}-\frac{\partial^2\varphi(x,t)} {\partial t^2}-M^2\varphi(x,t)
-\varepsilon\varphi^3(x,t)=0.      \eqno(1)
$$

 There are two classes of doubly periodic (periodic both in time
and in space) solutions for this equation.  If we  seek fields in the
traveling wave form: $  \varphi(x,t)=\varphi(x-vt), $ where $v$ is the
velocity of the wave motion, then equation~$(1)$ reduces to the Duffing's
   equation~\cite{5}.  As is known~\cite{2}, periodic solutions for this
  equation are Jacobi elliptic functions.  Such solutions are well known
  for both $(1+1)$-dimensional~\cite{15} and $(3+1)$-dimensional~\cite{46}
  models of $\varphi^4$  theory.


The second class of doubly periodic solutions consists of functions in  the
standing wave form:
$$
  \varphi(x,t)\equiv\sum_{n=1}^{\infty}\sum_{j=1}^{\infty}
{\cal C}_{nj}^{\vphantom{+}}\sin(n(x-x_0^{\vphantom{+}}))
\sin(j\!\cdot\!\omega(t-t_0^{\vphantom{+}})),\eqno(2)
$$
where $x_0^{\vphantom{+}}$ and $t_0^{\vphantom{+}}$ are
constants determined by boundary and initial conditions.
Equation~$(1)$
is a translation-invariant one, so, we can restrict our consideration to
the case of zero $x_0^{\vphantom{+}}$ and $t_0^{\vphantom{+}}$  without
loss of generality. We suppose that the function $\varphi(x,t)$ is
$2\pi$-periodic in space and seek its period in time.

 Exact standing wave solutions aren't known. Approximate
solutions for equation~$(1)$ with $\varepsilon=1$ and $M=1$ have been
found under the assumption that all highest harmonics are zeros:
$\forall n,j > N $ : ${\cal C}_{nj}=0$, where $N$ is a large
number. Numerical calculations have been made~\cite{8} for several
values of $N$. The obtained solutions are very close to each other, but
convergence of the sequence of these solutions as $N$
tends to infinity has yet to be proved.


 Provided that $\varepsilon\ll 1$, C.~Eugene~Wayne~\cite{1} has considered
the problem of the construction of periodic and quasi-periodic in time
solutions for the equation~$(1)$ with the mass term, depending on
space-coordinate $x$.  He has defined non-resonance conditions for
eigenvalues of the operator $L_M^{\vphantom{+}}\equiv
\frac{\partial^2}{\partial x^2}-M^2(x)$ and proved that if these
conditions are satisfied, then one can construct periodic and
quasi-periodic solutions for equation~$(1)$, using a variant of the
Kolmogorov, Arnold, Moser (KAM) scheme. C.~Eugene~Wayne has remarked that
although his theorem gives one many periodic solutions, one has no
information whether or not they occur in smooth families.

\section { Asymptotic periodic solutions}

Let us consider periodic solutions for equation~$(1)$, provided
that $\varepsilon$ is small or, equivalently, that these solutions have
small amplitudes.

An asymptotic expansion, containing only bounded functions,
is called {\it a uniform expansion}.
The possibility to obtain a uniform expansion, using standard
asymptotic methods, for example, Poincare--Lindstedt~\cite{10}  or
Krylov--Bogoliu\-bov~\cite{11} ones, depends on values of the frequencies
in zero approximation.  If $\varepsilon=0$, then equation~$(1)$ is a
linear one and has periodic solutions in the form $(2)$ with frequencies
in time $\Omega_j=\sqrt{j^2+M^2}$, where $j \in \natur$. There are two
fundamentally different cases.

If

{\LARGE  \vspace{-1.1cm}
$$
  {\scriptstyle \forall  i,j
\scriptscriptstyle \in\;\natu}
\mbox{ \ \  \  } {\not\:\scriptstyle\exists \; k,n \scriptscriptstyle\in
\;\natu} \mbox{ \ \normalsize  such that \ } {
\scriptstyle\frac{\Omega_i}{\vphantom{\Omega^+}\Omega_j}=\frac{k}{n}}
  \mbox{ \normalsize , }
$$      \vspace{-2mm}
}

then it is a non-resonance case.

The periodic asymptotic solutions in non-resonance case are well known.
To obtain these solutions with any degree of
exactness standard asymptotic methods can be used.

The resonance case, when there exist two frequencies $ \Omega_j$, whose
relation is rational, is more difficult. The Krylov--Bogoliubov method and
the standard variant of the Poincare--Lindstedt method allow to find
periodic solutions only in a few leading orders in $\varepsilon$.

The important example of resonance case is the massless $\varphi^4$ theory
  when $\Omega_j=j$ and all relations of
  frequencies are rational.  It is the main resonance case.  Using the
standard asymptotic methods one can't construct a periodic solution even
to the first order in $\varepsilon$.

 In this case it is possible to transform
differential equations in a system of nonlinear algebraic equations in
Fourier coefficients and frequencies using the Poincare--Dulac's normal
form method.  The algorithm of this procedure was constructed~\cite{9} and
realized in the software for symbolic and algebraic computation {\bf
REDUCE}~\cite{14} (the system {\bf NORT}~\cite{64,65}).  But algorithm to
solve the obtained algebraic system has yet to be realized.



  \section{ Construction of standing wave solutions  }

 \subsection { The Poincare--Lindstedt method }

 The purpose of this article is the construction of standing wave
 solutions of equation~$(1)$ with $M=0$, using asymptotic methods. They
  can only be applied, provided $\varepsilon\ll 1$.

  We use the Poincare--Lindstedt method: introduce the new time $\tilde t\equiv
 \omega t$ and seek a  doubly  periodic solution of equation~$(1)$ \
$\varphi(x,\tilde t)$ and the frequency (in time) $\omega$ in the form of
power series in $\varepsilon$:
$$
 \begin{array}{r@{\;\equiv\;}l}
  \displaystyle \varphi(x,\tilde t,\varepsilon) &\displaystyle
\sum_{n=0}^\infty\varphi_n^{\vphantom{+}} (x,\tilde t)\varepsilon^n,
\\[0.4cm] \displaystyle \omega(\varepsilon)& \displaystyle
1+\sum_{n=1}^\infty\omega_n^{\vphantom{+}} \varepsilon^n.  \end{array} $$

 After we expand the Lagrange--Euler equation in a power series in
$\varepsilon$ and obtain a sequence of equations. Write out
two leading equations:

\noindent 1) To zero order in $\varepsilon$ the equation in
  $\varphi_0^{\vphantom{+}}$ is:
$$
  \frac{\partial^2\varphi_0^{\vphantom{+}}(x,\tilde t)}{\partial x^2}-
\frac{\partial^2\varphi_0^{\vphantom{+}}(x,\tilde t)}
{\partial \tilde t^2}=0,   \eqno(3)
$$

\noindent 2) To first order in $\varepsilon$ the equation in
  $\varphi_0^{\vphantom{+}}$,
$\varphi_1^{\vphantom{+}}$ and  $\omega_1^{\vphantom{+}}$ is:
 $$
   \frac{\partial^2\varphi_1^{\vphantom{+}}(x,\tilde t)}{\partial x^2}-
\frac{\partial^2\varphi_1^{\vphantom{+}}(x,\tilde
t)}{\partial \tilde t^2} =2\omega_1^{\vphantom{+}}
\frac{\partial^2\varphi_0(x,\tilde t)}{\partial
\tilde t^2} +\varphi_0^3(x,\tilde t).  \eqno(4)
$$

  Equation~$(3)$ has many periodic solutions.
If we select as a solution for this equation  the function
$\varphi_0^{\vphantom{+}}(x,\tilde t)=\sin(x)\sin(\tilde
t)$,  then the second equation has no periodic solutions,
because the frequency  of the external force $\sin(3x)\sin(3\tilde t)$
is equal to the frequency of its own oscillations and it is impossible to
put this resonance harmonic to zero selecting only
 $\omega_1^{\vphantom{+}}$.

In massless case correct selection of not only the frequency
$\omega(\varepsilon)$, but also the function
$\varphi_0^{\vphantom{+}}(x,\tilde t)$ allows to find a uniform expansion.


\subsection{ The condition of existence of periodic solutions }

   The general solution in the standing wave form $(2)$ for equation~$(3)$
 is the function
 $$
 \varphi_0^{\vphantom{+}}(x,\tilde t)=
 \sum_{n=1}^{\infty}a_n^{\vphantom{+}}\sin(nx)\sin(n\tilde t)
$$
 with arbitrary $a_n^{\vphantom{+}}$.  We have to find such coefficients
$a_n^{\vphantom{+}}$ that the function
$\varphi_1^{\vphantom{+}}(x,\tilde t)$ is  a periodic solution for  equation
$(4)$.  If we select $\varphi_1^{\vphantom{+}}(x,\tilde t)$ as a double sum:
$$ \varphi_1^{\vphantom{+}}(x,\tilde
t)\equiv\sum_{n=1}^{\infty}\sum_{j=1}^{\infty}
b_{nj}^{\vphantom{+}}\sin(nx)\sin(j\tilde t)
$$
with arbitrary $b_{nj}^{\vphantom{+}}$, then equation~$(4)$ can be
presented in the form of Fourier series:
$$
\begin{array}{rcl} \displaystyle
R(x,\tilde t)&\displaystyle\equiv&
\displaystyle\frac{\partial^2\varphi_1^{\vphantom{+}}(x,\tilde
  t)}{\partial x^2}  -\frac{\partial^2\varphi_1^{\vphantom{+}}(x,\tilde
 t)}{\partial \tilde t^2} -2\omega_1^{\vphantom{+}}
\frac{\partial^2\varphi_0^{\vphantom{+}}(x,\tilde
t)}{\partial \tilde t^2}
-\varphi_0^3(x,\tilde t)\;\equiv \\[5mm]
&\displaystyle\equiv& \displaystyle\sum_{n=1}^{\infty}\sum_{j=1}^{\infty}
R_{nj}({\bf a},{\bf b})\sin(nx)\sin(j\tilde t)\;=\;0
\end{array}
$$
 and is equivalent to the following infinite system of the algebraic
equations in Fourier coefficients of the functions
$\varphi_0^{\vphantom{+}}(x,\tilde t)$ and
 $\varphi_1^{\vphantom{+}}(x,\tilde t)$:
 $$
 \forall n,j \;{}:{}\;  R_{nj}({\bf a},{\bf b})=0.\eqno(4')
 $$

 This system has a subsystem of the equations
 in Fourier coefficients of the function
$\varphi_0^{\vphantom{+}}(x,\tilde t)$:
$$
\forall j\in\natu\;{ }:{ }\;
  R_{jj}({\bf a})=0, \eqno(5) $$ where $$ \begin{array}{rcl} \displaystyle
R_{jj}&\displaystyle \!\!\equiv\!\! & \displaystyle
9a_j^3+3a_j^2a_{3j}^{\vphantom{+}} +a_j^{\vphantom{+}}\!\left(6\!
 \sum_{\vbox{\hbox{$\scriptstyle s=1$}\vspace{1mm}\hbox{$\scriptstyle
s\neq j$}}}^\infty\;(2a_s^2+ a_s^{\vphantom{+}}a_{2j+s}^{\vphantom{+}})+
 3\!\sum_{\vbox{\hbox{$\scriptstyle s=1$}
\vspace{1mm}\hbox{$\scriptstyle s\neq j$}}}^{2j-1}\!
a_s^{\vphantom{+}}a_{2j-s}^{\vphantom{+}}-
 32j^2\omega_1^{\vphantom{+}}\right)+\\[1cm]
&\displaystyle \!\!+\!\!&\displaystyle
3\sum_{\vbox{\hbox{$\scriptstyle s=1$}
\vspace{1mm}\hbox{$\scriptstyle s\neq j$}}}^{\infty}
\;\sum_{\vbox{\hbox{$\scriptstyle p=1$}\vspace{0.5mm}\hbox{$\scriptstyle
p\neq j$}}} ^\infty a_s^{\vphantom{+}} a_p^{\vphantom{+}}
a_{j+s+p}^{\vphantom{+}} +3 \sum_{\vbox{\hbox{$\scriptstyle s=1$}
\vspace{1mm}\hbox{$\scriptstyle s\neq j$}}}^{\infty}\!\!\!\!\!\!
\sum_{\vbox{\hbox{\ \ \ \ $\scriptstyle p=1$}\vspace{0.5mm}\hbox{\ \ \ \
$\scriptstyle p\neq j$}\vspace{1mm}\hbox{\ \ \ \ $\scriptstyle p\neq
2j-s$}}}^{\infty}\!\!\!\!\!\!\!a_s^{\vphantom{+}}
a_p^{\vphantom{+}}a_{s+p-j}^{\vphantom{+}}+
\sum_{s=1}^{j-2}\;\sum_{p=1}^{j-2}
a_{s}^{\vphantom{+}}a_{p}^{\vphantom{+}}a_{j-s-p}^{\vphantom{+}}
\end{array}
$$

We have obtained the necessary and sufficient condition of the existence of
periodic solutions for equation~$(4)$: there exists a periodic function
$\varphi_1^{\vphantom{+}}(x,\tilde t)$, satisfying equation~$(4)$, if and
only if the coefficients of $\varphi_0^{\vphantom{+}}(x,\tilde t)$ Fourier
series satisfy system~$(5)$.

Coefficient $a_1^{\vphantom{+}}$ is a parameter, determining the
oscillation amplitude. In fact, let
$a_j^{\vphantom{+}}=c_j^{\vphantom{+}}a_1^{\vphantom{+}}$ and
$\omega_1^{\vphantom{+}}=c_\omega^{\vphantom{+}}a_1^2$, then all
polynomials $R_{jj}$ are proportional to $a_1^3$:\
$R_{jj}({\bf a})=a_1^3R_{jj}({\bf c})$ and, therefore, the coefficient
$a_1^{\vphantom{+}}$ can be selected arbitrarily.

This system of equations
is very difficult to solve. On the one hand, all
$R_{jj}$ are infinite series and the number of equations is infinite too.
On the other hand, each equation of this system is a nonlinear one. We have
restricted ourselves to finding a particular solution. Our goal is to find a
real solution and we seek $c_j^{\vphantom{+}}\in\real$.



  \subsection{ The zero approximation}

The aim of this section is to find the analytical form of the function
$\varphi_0^{\vphantom{+}}(x,\tilde t)$, which  Fourier coefficients obey
system~$(5)$. It is easy to see that if  the function
$\varphi_0^{\vphantom{+}}(x,\tilde t)$ is a sum of finite number of
 harmonics, then the resonance problem can't be solved, so, only an
infinite sequence can be a solution of system~$(5)$.  On the other hand,
we can consider sequences, including only odd harmonics.




 For arbitrary $q\in(0,1)$ let us define the following
sequence:
$$
  {\bf f}\stackrel{\rm def}{=}\{\;\forall n\in
  \natu { }:{ } f_{2n-1}^{\vphantom{+}}=\frac{q^{n-1/2}}{1+q^{2n-1}},{ }
  {    }{ } f_{2n}^{\vphantom{+}}=0\;\}.
$$
 The terms of the sequence ${\bf f}$ are proportional to Fourier
coefficients of the Jacobi elliptic function $\bf cn$~\cite{2}:
$$
\cn(z,k)=\frac{\gamma}{k}\sum_{n=1}^\infty
f_{2n-1}^{\vphantom{+}}\cos\left((2n-1)\frac{\gamma z}{4} \right)
\mbox{, \ \ where \ } \gamma\equiv\frac{2\pi}{K} \mbox{ , \  \  \  }
z\in \real.
\eqno (6)
$$

Let us clarify introduced designations and point out some properties of
elliptic cosine:

\noindent 1) Basic periods of the doubly periodic function $\cn(z,k)$
are  $4K(k)$ and $2K(k)+2\dot\imath K'(k)$, where $K(k)$ is a full elliptic
integral, $K'(k)\equiv K(k')$ and $k'=\sqrt{1-k^{2^{\vphantom{+}}}}$.

\noindent 2) The parameter $q$ in the Fourier expansion
can be expressed in terms of elliptic integrals:
$q\equiv e^{-{\displaystyle\pi}\frac{K'}{K^{\vphantom{+}}}}$.\vspace{1mm}

\noindent 3) The Fourier-series expansion of the function $\cn(z,k)$
does not include even harmonics.  This expansion is valid in the following
domain of the complex plane: $-K'<\Im m\: z<K'$,   in particular, for
$z\in \real$.  \vspace{1mm}


\noindent 4) If $z \in \real$ and $k\in(0,1)$, then  $\cn(z,k)\in \real$.
\vspace{1mm}

\noindent 5) The function $\cn(z,k)$ is a solution of the following
 differential equation:
 $$
 \frac{d^2\cn(z,k)}{dz^2}=(2k^2-1)\cn(z,k)-2k^2\cn^3(z,k).
 \eqno (7)
 $$

The latest property means that infinite sequence of $\cn(z,k)$ Fourier
coefficients is a solution of some infinite system of nonlinear algebraic
equations.

 Let us find this system.
On the one hand, it is clear from $(6)$ that the
Fourier-series expansion for the function $\cn^3(z,k)$ is:
$$
\cn^3(z,k)=\frac{\gamma^3}{4k^3}\sum_{n=1}^\infty F_j^{(3)}({\bf f})
\cos\left(j\frac{\gamma z}{4}\right)\mbox{, \ \ where } j=1,3,5,\dots,+\infty
\mbox{\ ; \ \ \ }
$$
$$
\begin{array}{rcl}
\displaystyle
F_{j}^{(3)}({\bf f})&\displaystyle\! \equiv \! & \displaystyle
3f_j^3+3f_j^2f_{3j}^{\vphantom{+}} +f_j^{\vphantom{+}}\left(6\!\sum_{s\neq
 j}^{\infty}\;(f_s^2+ f_s^{\vphantom{+}}f_{2j+s}^{\vphantom{+}})+
 3\!\sum_{s\neq j}^{2j-1}f_s^{\vphantom{+}}f_{2j-s}^{\vphantom{+}}
 \right)+\\[0.5cm]
\!\!\!\!\!\!\!\!\!\!\!\!&
\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle +\!&
%\!\!\!\!\!\!\!\!\!\!\!\! 
\!\!\!\!\!\!\!\!\!\!\!\!\displaystyle\!\!\! 3\sum_{s\neq j}^{\infty}\;\sum_{p\neq
j}^{\infty}\;f_s^{\vphantom{+}} f_p^{\vphantom{+}}
f_{j+s+p}^{\vphantom{+}} +3 \sum_{s\neq
j}^{\infty}\!\!\!\!\!\!\sum_{\vbox{\hbox{\ \ \ \ $\scriptstyle p\neq
j$}\vspace{1mm}\hbox{\ \ \ \  $\scriptstyle p\neq 2j-s$
}}}^{\infty}\!\!\!\!\!\!\!f_s^{\vphantom{+}}
f_p^{\vphantom{+}}f_{s+p-j}^{\vphantom{+}}+
\sum_{s=1}^{j-2}\;\sum_{p=1}^{j-2}\;
f_{s}^{\vphantom{+}}f_{p}^{\vphantom{+}}f_{j-s-p}^{\vphantom{+}}
\end{array}
$$
 (in all sums we summarize over only odd numbers).

 On the other hand, from differential equation~$(7)$ it follows that
$F_j^{(3)}({\bf f})$ is proportional to $f_j^{\vphantom{+}}${\it ,}  with
coefficients of proportionality depending on $j$:
$$
\forall j \mbox{ \ \
:  \ \ } F_j^{(3)}({\bf
f})=\left(\frac{2(2k^2-1)}{\gamma^2}+\frac{j^2}{8}\right)
f_j^{\vphantom{+}}.    \eqno(8)
$$

 Thus, the sequence ${\bf f}$ is a nonzero
solution of system~$(8)$ at all $q\in(0,1)$.  The following
lemma proves the existence of a preferred value of $q$.

\vspace{5mm}
\noindent $\!\!\!\!$
\begin{tabular}{rp{0.8\textwidth}}
\large\bf Lemma. & \large\it  There exists such value of parameter
$q\in(0,1)$ that   the sequence ${\bf f}$ is a  real solution of
system~$(5)$,   in addition a value of \ $\omega_1^{\vphantom{+}}$  also
is real. \\[2mm]
 \bf Proof. &  Inserting the
sequence ${\bf f}$ into system  $(5)$ :
$a_j^{\vphantom{+}}=f_j^{\vphantom{+}}$ and using system  $(8)$, we
 obtain:
\end{tabular}
  $$ \begin{array}{r@{\displaystyle \;=\;}l} \displaystyle \mbox{
\ \  } \forall j \mbox{ \ \   :  \ \ } R_{jj}({\bf f})& \displaystyle
 \left\{ F_j^{(3)}({\bf f})+ f_j^{\vphantom{+}}\left(6\sum_{n=1}^\infty
 f_n^2 -32j^2\omega_1^{\vphantom{+}} \right)\right\}=\\[0.7cm]
 &\displaystyle f_j^{\vphantom{+}}
 \left\{6\sum_{n=1}^\infty f_n^2+
 \frac{2(2k^2-1)}{\gamma^2}+j^2\left(\frac{1}{8}-32
 \omega_1^{\vphantom{+}}\right)\right\}=0.
\end{array}
 $$

 System  $(5)$ has a nonzero solution if and only if
 $$
 \left\{ \begin{array}{r@{\; =\;}l}
 \displaystyle\omega_1^{\vphantom{+}}&\displaystyle
 \frac{1}{256},\\[0.3cm]
 \displaystyle\sum\limits_{n=1}^\infty
    f_n^2 & \displaystyle\frac{(1-2k^2)}{3\gamma^2}. \\
\end{array}
\right.
$$

 We have obtained the value of $\omega_1^{\vphantom{+}}$. The second
equation of this system is equivalent to the following equation in
parameter $q$:
$$
3\sum_{n=1}^\infty\left(\vphantom{\sum_{n=1}^\infty}
\frac{q^{n-1/2}}{1+q^{2n-1}}\right)^2
-\left(\frac{1}{4}+ \sum_{n=1}^\infty
\frac{q^n}{1+q^{2n}}\right)^2+2\left(\sum_{n=1}^\infty
\frac{q^{n-1/2}}{1+q^{2n-1}}\right)^2=0.     \eqno (9)
$$

This equation has the following solution on interval (0,1):
$$ q=1.42142623201\times10^{-2}\pm1\times10^{-13}.  $$

\noindent {\bf Thus the lemma is proved.}


 Now it is easy to construct the required zero approximation of the
function $\varphi(x,\tilde t)$:
$$
\varphi_0^{\vphantom{+}}(x,\tilde t)= A\Bigl\{\cn(\alpha (x-\tilde
t),k)-\cn(\alpha (x+\tilde t),k)\Bigr\}.
$$

For arbitrary $k\in(0,1)$ this function
  is a real solution of equation  $(4)$. If
$\alpha=\frac{2K}{\displaystyle\pi}$, then periods
  $\varphi_0^{\vphantom{+}}(x,\tilde t)$ in $x$ and in $\tilde t$ are
  equal to $2\pi$. Using the Fourier-series expansion for the function
$\cn(z,k)$ (formula $(6)$), we obtain the following  expansion for
 the function $\varphi_0^{\vphantom{+}}(x,\tilde t)$:
$$
\varphi_0^{\vphantom{+}}(x,\tilde
t)=2A\frac{\gamma}{k}\sum_{n=1}^{\infty}f_{2n-1}^{\vphantom{+}}
\sin((2n-1)x)\sin((2n-1)\tilde t).
$$

 If $q=1.42142623201\times10^{-2}\pm1\times10^{-13}$, then $q$ is a
solution of equation~$(9)$ and the sequence ${\bf f}$ is a real solution
of system~$(5)$.  The middle value of $q$ corresponds to
$k=0.451075598811$, $\gamma=3.78191440007$ and $\alpha=1.0576653982$.  All
  equation in system~$(5)$ are homogeneous ones, hence, for  these values
of parameters, the sequence of $\varphi_0^{\vphantom{+}}(x,\tilde t)$
Fourier coefficients also is a solution of system~$(5)$, with $$
  \omega_1^{\vphantom{+}}=\frac{\gamma^2}{64k^2}A^2=1.0983600974A^2.
$$

 Thus we have proved that the function
$$
 \varphi_0^{\vphantom{+}}(x,\tilde t)=
 A\Bigl\{\cn(\alpha (x-\tilde t),k)-\cn(\alpha (x+\tilde t),k)\Bigr\},
$$
 with $k=0.451075598811$ and $\alpha=1.0576653982$
is such a standing wave solution of equation~$(3)$ that equation~$(4)$ has
a periodic solution.


\subsection { The first  approximation}

 Now it is easy to find this periodic
 solution $\varphi_1^{\vphantom{+}}(x,\tilde t)$.  Let us designate  the
 Fourier coefficients of the function $\varphi_0^3(x,\tilde t)$ as
$D_{nj}$:  $$ \varphi_0^3(x,\tilde t)\equiv\sum_{n=1}^{\infty}
 \:\sum_{j=1}^{\infty}D_{nj}\sin(nx) \sin(j\tilde
 t).
 $$

 Equation~$(4)$ gives the following result ($b_{nn}^{\vphantom{+}}$
are arbitrary numbers):
$$
\varphi_1^{\vphantom{+}}(x,\tilde
 t)=\sum_{n=1}^\infty \:\sum_{\vbox{\hbox{$\scriptstyle j=1$}\vspace{-1mm}
 \hbox{$\scriptstyle j\neq n$}}}^\infty\frac{D_{nj}}{j^2-n^2}
\sin(nx)\sin(j\tilde t)+
\sum_{n=1}^{\infty}b_{nn}^{\vphantom{+}}\sin(nx)\sin(n\tilde t).
$$

 It should be noted that the function  $\varphi_1^{\vphantom{+}}(x,\tilde
  t)$ with arbitrary diagonal coefficients $b_{nn}^{\vphantom{+}}$ is a
solution of equation~$(4)$ and that all off-diagonal coefficients of
the function $\varphi_1^{\vphantom{+}}(x,\tilde t)$ are
proportional to $A^3$.

 \subsection { The second approximation}

 Let us consider the equation to second order in $\varepsilon$:
$$
 \frac{\partial^2\varphi_2^{\vphantom{+}}}{\partial x^2}
-\frac{\partial^2\varphi_2^{\vphantom{+}}}{\partial \tilde t^2}
=2\omega_1^{\vphantom{+}}
\frac{\partial^2\varphi_1^{\vphantom{+}}}{\partial\tilde
t^2}+(2\omega_2^{\vphantom{+}}+\omega_1^2)
\frac{\partial^2\varphi_0^{\vphantom{+}}}{\partial \tilde t^2}
+3\varphi_1^{\vphantom{+}}
\varphi_0^2.    \eqno(10)
$$


If  all diagonal
 coefficients of $\varphi_1^{\vphantom{+}}(x,\tilde t)$ are zeros:
 $\forall n$ :\  $b_{nn}^{\vphantom{+}}=0$,
then \linebreak $\forall j,n$ :\
$b_{jn}^{\vphantom{+}}=-b_{nj}^{\vphantom{+}}$, and the function
$\varphi_1^{\vphantom{+}}(x,\tilde t)\varphi_0^2(x,\tilde t)$  hasn't
 diagonal harmonics. So, selecting
$\omega_2^{\vphantom{+}}=-\frac{1}{2}\omega_1^2$
we obtain a periodic solution of equation~$(10)$:
$$
\varphi_2^{\vphantom{+}}(x,\tilde t)\equiv\sum_{n=1}^\infty
 \:\sum_{\vbox{\hbox{$\scriptstyle j=1$}\vspace{-1mm} \hbox{$\scriptstyle
j\neq n$}}}^\infty\frac{H_{nj}}{j^2-n^2}\sin(nx)\sin(j\tilde t)+
\sum_{n=1}^\infty h_{nn}^{\vphantom{+}}\sin(nx)\sin(n\tilde t)
\mbox{, \  \   \  where }
 $$
 $$
 H(x,\tilde t)\equiv
2\omega_1^{\vphantom{+}}\frac{\partial^2 \varphi_1^{\vphantom{+}}(x,\tilde t)}
{\partial \tilde t^2} -3\varphi_1^{\vphantom{+}}(x,\tilde t)\varphi_0^2(x,\tilde t)
\equiv\sum_{n=1}^\infty\:\sum_{j=1}^\infty H_{nj}
  \sin(nx)\sin(j\tilde t).
 $$
It should be noted that all diagonal coefficients $h_{nn}^{\vphantom{+}}$
are arbitrary numbers.

\section { Conclusions}

 Using massless $\varphi^4$ theory as an example, we show that a uniform
 expansion of solutions for quasilinear Klein--Gordon equations can be
 constructed even in the main resonance case.
 To construct the uniform expansion we have used the  Poincare--Lindstedt
method and the nontrivial zero approximation: the function
 $$
 \varphi_0^{\vphantom{+}}(x,t)=
 A\Bigl\{\cn(\alpha (x-\omega t),k)-\cn(\alpha (x+\omega t),k)\Bigr\},
 $$
 with $k=0.451075598811$ and $\alpha=1.0576653982$.

Thus, using the Jacobi elliptic function {\bf cn} instead of
the trigonometric function {\bf cos}, we have put the main resonance to
zero and constructed with accuracy ${\cal O}(\varepsilon^3)$ the doubly
periodic solution in the standing wave form:  $$ \varphi(x,\omega
t)=\varphi_0^{\vphantom{+}}(x,\omega t)+
\varepsilon\varphi_1^{\vphantom{+}}(x,\omega t)+
\varepsilon^2\varphi_2^{\vphantom{+}}(x,\omega t)+
{\cal O}(\varepsilon^3)\mbox{, \  with the frequency }
$$
$$
 \omega=1+1.0983600974A^2\varepsilon
 -0.6031974518A^4\varepsilon^2 +{\cal O}(\varepsilon^3).
$$
\vspace{1mm}
{\large \bf Acknowledgement. } \ \  The authors are grateful to\ \
 M.~V.~Chichikina, V.~F.~Edneral\, and\, P.~K.~Silaev\,  for valuable
discussions.


\begin{thebibliography}{99}
\bibitem{5}
G.~Duffing, {\it Erzwungene Schwingungen bei ver\"anderlicher
Eigenfrequenz und ihre technische Bedeu\-tung}, Vieweg, Braunschweig, 1918
\vspace{-2.5mm}
\bibitem{2}
H.~Bateman, A.~Erdelyi, ed., {\it  Higher transcendental functions.
Volume~3}, MC Graw-Hill Book Company, Inc., New York, Toronto, London,~1955
\vspace{-2.5mm}
\bibitem{15}
S.~Aubry, J. of Chem.
Phys., {\bf  64} (1976) 3392--3402 \vspace{-2.5mm} \bibitem{46}
D.~F.~Kurdgelaidze, JETF (Rus. J. of Exper. $\&$ Theor. Phys.),
 {\bf  36} (1959) 842--849 \{in~Russian\}
\vspace{-2.5mm}
\bibitem{8}
O.~A.~Khrustalev,
P.~K.~Silaev, TMF (Rus. J. of Theor. $\&$ Math.  Phys.), {\bf 117} (1998)
{\the\textfont2 N}\lower-0.4ex\hbox{\small\underline{$\circ$}}~2,
in~print, \{in~Russian\} \vspace{-2.5mm} \bibitem{1} C.~E.~Wayne, Commun.
Math. Phys., {\bf  127} (1990) 479--528 \vspace{-2.5mm} \bibitem{10}
 H.~Poincare.
{\it New Methods of Celestial Mechanics}, Volume 1--3.  NASA TTF--450,
1967
\vspace{-2.5mm}
\bibitem{11}
N.~N.~Bogoliubov, Yu.~A.~Mitropolskiy,
{\it Asymptotic methods in theory of nonlinear oscillations}, Moscow,
Nayka, 1974 \{in~Russian\}
\vspace{-2.5mm}
\bibitem{9}
A.~D.~Bruno, {\it
Local method in Nonlinear Differential Equations}, Springer Series in
Soviet Mathema\-tics, ISBN-3-540-18926-2, 1989
\vspace{-2.5mm}
\bibitem{14}
A.~C.~Hearn, {\bf REDUCE. \it User's Manual.
Version 3.6}, RAND Publication CP78, 1995
\vspace{-2.5mm}
\bibitem{64}
V.~F.~Edneral, O.~A.~Khrustalev, Proc.
of the Int. Conf. on Computer Algebra and its
Applications in Theoretical Physics. (USSR, Dubna, 1985), Dubna: JINR
publ., (1985) 219--224 \{in~Russian\}
\vspace{-2.5mm}
\bibitem{65}
V.~F.~Edneral, O.~A.~Khrustalev, Programirovanie (Rus. J. of Programming)
\# {\bf 5}  (1992) 73--80 \{in~Russian\}
\end{thebibliography}
\end{document}


