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):

$$\varphi_1^{\vphantom{+}}(x,\tilde t)=\sum_{n=1}^\infty \:\su... ...um_{n=1}^{\infty}b_{nn}^{\vphantom{+}}\sin(nx)\sin(n\tilde t).$$

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 $\varphi_1^{\vphantom{+}}(x,\tilde t)$ are proportional to A³.