Asymptotic solutions in the standing wave form

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 a uniform expansion. The possibility to obtain a uniform expansion, using standard asymptotic methods [7], for example, Poincare-Lindstedt [8] or Krylov-Bogoliubov [9] ones, depends on values of the frequencies in zero approximation. If $\varepsilon=0$, then equation (1) is a linear one and has periodic solutions of the form (2) with frequencies in time $\Omega_j=\sqrt{j^2+M^2}$, where $j \in \hbox{\cal I\hskip-2pt\bf N}$. There are two fundamentally different cases.

If

$${\scriptstyle \forall i,j \scriptscriptstyle \in\;\hbox{\norm... ...hantom{\Omega^+}\Omega_j} =\frac{k}{n}} \mbox{ \normalsize , }$$

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 cannot construct a periodic solution even to the first order in $\varepsilon$. 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 [10,11] and realized in the software for symbolic and algebraic computation REDUCE [12] (the system NORT [13,14]). But algorithm to solve the obtained algebraic system has yet to be realized.