The approximate solution of system (5)

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. To simplify calculations we assume that the function $\varphi_0^{\vphantom{+}}(x,\tilde t)$ contains only odd harmonics. Our goal is to find a real solution and we seek $c_j^{\vphantom{+}}\in\hbox{\rm I\hskip-2pt\bf R}$.

To find an approximate solution we apply the Galerkin method: cut off higher diagonal harmonics $\forall j>N$ : $c_{2j-1}^{\vphantom{+}}=0$ and seek an approximation for $\varphi_0^{\vphantom{+}}(x,\tilde t)$ in the following form

$$\varphi_0^{\vphantom{+}}(x,\tilde t)=a_{1}^{\vphantom{+}} \l... ...2j-1}^{\vphantom{+}}\sin((2j-1)x)\sin((2j-1)\tilde t)\right\}.$$

We have obtained the finite system of nonlinear equations, with the number of equations three times as mush as that of the variables. We solve the first N equations and substitute the obtained values in the other equations. The system of the N leading nonlinear equations can be solved, using standard procedures of the computer algebra system REDUCE: the SOLVE operator and the Groebner basis package [15,16]. The SOLVE operator (in "on rounded" mode) round off numbers with accuracy $\delta=10^{-11}$ (the system (5) is too difficult to be solved in "off rounded" mode). We admit that the system of the first N equations is solved if for all $j\leq N$ the inequalities $\vert R_{jj}({\bf c})\vert< \delta$ are true. We also admit that the value of N is sufficient to solve system (5) if for all $j \in \hbox{\cal I\hskip-2pt\bf N}$ the inequalities $\vert R_{jj}({\bf c})\vert< \delta$ are true.

Using the SOLVE operator and computer with 128M bytes operating memory (RAM) we have solved the system (5) only for N<5, but due to a short program (see Appendix) we have found a particular solution for any N<50 and obtained that the minimal sufficient value of N for $\delta=10^{-11}$ is N=8 and that the value of the frequency is

$$\omega_1^{\vphantom{+}}=0.282680034541a_1^2.$$

We also have found numerical values of the fifteen leading Fourier coefficients of $\varphi_0^{\vphantom{+}}(x,\tilde t)$. The obtained values of $c_j^{\vphantom{+}}$ are very close to the values of the corresponding terms of the following finite sequence:

$${\bf d}=\{ d_{2j-1}^{\vphantom{+}}=\frac{f_{2j-1}^{\vphantom... ...}}; {} \;\;{ } d_{2j}^{\vphantom{+}}=0 ; { } \ \;{ } j<23 \},$$

with q=0.0142142623201. It is easy to verify that substitution of this finite sequence gives

$$\forall j\in\hbox{\cal I\hskip-2pt\bf N}\mbox{ \ \ : \ \ }\vert R_{jj}({\bf d})\vert<10^{-12}.$$

In other words, the finite sequence ${\bf d}$is an approximate solution of system (5). These numerical calculations help to find the analytical form of $\varphi_0^{\vphantom{+}}(x,\tilde t)$. The following table illustrates this interesting result:

$j^{\vphantom{+}}$	$c_{j_{\vphantom{j}}}^{\vphantom{+}}$	$R_{jj}({\bf c})$	$d_{j_{\vphantom{j}}}^{\vphantom{+}}$	$R_{jj}({\bf d})$
1	1	0	$1^{\vphantom{+}}$	0
3	$\;1.44162661711\times 10^{-2^{\vphantom{+}}}\;$	$3.5\times 10^{-12}$	$\;1.44162661711\times 10^{-2^{\vphantom{+}}}\;$	$- 8.0\times 10^{-14}$
5	$2.04917177408\times 10^{-4^{\vphantom{+}}}$	$2.1\times 10^{-12}$	$2.04917177419\times 10^{-4}$	$- 3.4\times 10^{-15}$
7	$2.91274649724\times 10^{-6^{\vphantom{+}}}$	$7.8\times 10^{-12}$	$2.91274651543\times 10^{-6}$	$- 9.7\times 10^{-17}$
9	$4.14025418115\times 10^{-8^{\vphantom{+}}}$	$8.3\times 10^{-13}$	$4.14025430425\times 10^{-8}$	$- 2.3\times 10^{-18}$
11	$5.88506592014\times 10^{-10^{\vphantom{+}}}$	$1.0\times 10^{-14}$	$5.88506607528\times 10^{-10}$	$- 4.9\times 10^{-20}$
13	$8.36488192079\times 10^{-12^{\vphantom{+}}}$	$4.6\times 10^{-13}$	$8.36518729655\times 10^{-12}$	$- 9.7\times 10^{-22}$
15	$1.18901919266\times 10^{-13^{\vphantom{+}}}$	$-7.8\times 10^{-22}$	$1.1890496659\times 10^{-13}$	$- 1.8\times 10^{-23}$
17	0	$4.4\times 10^{-12^{\vphantom{+}}}$	$1.69014638629\times 10^{-15}$	$- 3.4\times 10^{-25}$
19	0	$7.3\times 10^{-14^{\vphantom{+}}}$	$2.40241840942\times 10^{-17}$	$- 6.0\times 10^{-27}$
21	0	$1.1\times 10^{-15^{\vphantom{+}}}$	$3.41486054743\times 10^{-19}$	$- 1.0\times 10^{-28}$
23	0	$1.7\times 10^{-17^{\vphantom{+}}}$	$4.85397236079\times 10^{-21}$	$- 1.8\times 10^{-30}$
25	0	$2.4\times 10^{-19^{\vphantom{+}}}$	$6.89956364312\times 10^{-23}$	$- 3.0\times 10^{-32}$
27	0	$3.3\times 10^{-21^{\vphantom{+}}}$	$9.8072207518\times 10^{-25}$	$- 4.9\times 10^{-34}$
29	0	$4.2\times 10^{-23^{\vphantom{+}}}$	$1.39402408398\times 10^{-26}$	$- 8.1\times 10^{-36}$
31	0	$5.2\times 10^{-25^{\vphantom{+}}}$	$1.98150240103\times 10^{-28}$	$- 1.3\times 10^{-37}$
33	0	$5.7\times 10^{-27^{\vphantom{+}}}$	$2.81655949163\times 10^{-30}$	$- 2.1\times 10^{-39}$
35	0	$6.1\times 10^{-29^{\vphantom{+}}}$	$4.00353154544\times 10^{-32}$	$- 3.4\times 10^{-41}$
37	0	$6.2\times 10^{-31^{\vphantom{+}}}$	$5.69072475939\times 10^{-34}$	$- 5.4\times 10^{-43^{\vphantom{+}}}$
39	0	$5.9\times 10^{-33}$	$8.08894545219\times 10^{-36}$	$- 8.5\times 10^{-45^{\vphantom{+}}}$
41	0	$5.0\times 10^{-35}$	$1.14978392551\times 10^{-37}$	$- 1.3\times 10^{-46^{\vphantom{+}}}$
43	0	$3.6\times 10^{-37}$	$1.63433303287\times 10^{-39}$	$- 2.1\times 10^{-48^{\vphantom{+}}}$
45	0	$1.7\times 10^{-39^{\vphantom{+}}}$	$2.32308384477\times 10^{-41}$	$- 1.6\times 10^{-49^{\vphantom{+}}}$
47	0	0	0	$1.4\times 10^{-50^{\vphantom{+}}}$