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:

$$\frac{\partial^2\varphi_1^{\vphantom{+}}(x,\tilde t)}{\partia... ...=1}^{\infty} R_{nj}({\bf a},{\bf b})\sin(nx)\sin(j\tilde t)=0.$$

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 $\varphi_0^{\vphantom{+}}(x,\tilde t)$:

$$\forall j\in\hbox{\normalsize\cal I\hskip-2pt\bf N}\;{ }:{ }\; R_{jj}({\bf a})=0, \eqno(5)$$

where

$$\begin{array}{rcl} \displaystyle R_{jj}&\displaystyle \!\!\eq... ...tom{+}}a_{p}^{\vphantom{+}}a_{j-s-p}^{\vphantom{+}} \end{array}$$

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 Fourier coefficients of the function $\varphi_0^{\vphantom{+}}(x,\tilde t)$ 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 $R_{jj}({\bf a})$ are proportional to a₁³:
$R_{jj}({\bf a})=a_1^3R_{jj}({\bf c})$ and, therefore, the coefficient $a_1^{\vphantom{+}}$ can be selected arbitrarily.