The exact solution of system (5)

The aim of this section is to find the analytical form of the function, which Fourier coefficients satisfy system (5). For arbitrary $q\in(0,1)$ let us define the following sequence:

$${\bf f}\stackrel{\rm def}{=}\{\;\forall n\in \hbox{\normalsi... ...^{n-1/2}}{1+q^{2n-1}},{ } { }{ } f_{2n}^{\vphantom{+}}=0\;\}.$$

The terms of the sequence ${\bf f}$ are proportional to Fourier coefficients of the Jacobi elliptic function $\bf cn$ [2]:

$$\mathop{\rm cn}\nolimits(z,k)=\frac{\gamma}{k}\sum_{n=1}^\inf... ...} \mbox{ , \ \ \ } z\in \hbox{\rm I\hskip-2pt\bf R}. \eqno (6)$$

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

1) Basic periods of the doubly periodic function $\mathop{\rm cn}\nolimits(z,k)$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 $k'=\sqrt{1-k^{2^{\vphantom{+}}}}$.

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

3) The Fourier-series expansion of the function $\mathop{\rm cn}\nolimits(z,k)$ 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 $z\in \hbox{\rm I\hskip-2pt\bf R}$.

4) If $z\in \hbox{\rm I\hskip-2pt\bf R}$ and $k\in(0,1)$, then $\mathop{\rm cn}\nolimits(z,k)\in \hbox{\rm I\hskip-2pt\bf R}$.

5) The function $\mathop{\rm cn}\nolimits(z,k)$ is a solution of the following differential equation:

$$\frac{d^2\mathop{\rm cn}\nolimits(z,k)}{dz^2}=(2k^2-1)\mathop... ...nolimits(z,k)-2k^2\mathop{\rm cn}\nolimits^3(z,k). \eqno (7)$$

The latest property means that the infinite sequence of the Fourier coefficients for $\mathop{\rm cn}\nolimits(z,k)$ 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 $\mathop{\rm cn}\nolimits^3(z,k)$ is:

$$\mathop{\rm cn}\nolimits^3(z,k)=\frac{\gamma^3}{4k^3}\sum_{j=... ...ht)\mbox{, \ \ where } j=1,3,5,\dots,+\infty \mbox{\ ; \ \ \ }$$

$$\begin{array}{rl} \displaystyle F_{j}^{(3)}({\bf f})\!&\displ... ...tom{+}}f_{p}^{\vphantom{+}}f_{j-s-p}^{\vphantom{+}} \end{array}$$

(in all sums we summarize over only odd numbers).

On the other hand, from the differential equation (7) it follows that $F_j^{(3)}({\bf f})$ is proportional to $f_j^{\vphantom{+}}$, with coefficients of proportionality depending on j:

$$\forall j \mbox{ \ \ : \ \ } F_j^{(3)}({\bf f})=\left(\frac{2... ...)}{\gamma^2}+\frac{j^2}{8}\right) f_j^{\vphantom{+}}. \eqno(8)$$

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.

$\!\!\!\!\!$

Lemma.	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.
Proof.	Inserting the sequence ${\bf f}$ into system (5) : $a_j^{\vphantom{+}}=f_j^{\vphantom{+}}$ and using system (8), we obtain:

$$\begin{array}{r@{\displaystyle \;=\;}l} \displaystyle \mbox{ ... ...}{8}-32 \omega_1^{\vphantom{+}}\right)\right\}=0. \end{array}$$

System (5) has a nonzero solution if and only if

$$\left\{ \begin{array}{r@{\; =\;}l} \displaystyle\omega_1^{\v... ...displaystyle\frac{(1-2k^2)}{3\gamma^2}. \\ \end{array}\right.$$

We have obtained the value of $\omega_1^{\vphantom{+}}$. The second equation of this system is equivalent to the following equation in parameter q:

$$3\sum_{n=1}^\infty\left(\vphantom{\sum_{n=1}^\infty} \frac{q^... ...n=1}^\infty \frac{q^{n-1/2}}{1+q^{2n-1}}\right)^2=0. \eqno (9)$$

This equation has the following solution on interval (0,1):

$$q=1.42142623201\times10^{-2}\pm1\times10^{-13}.$$

Thus the lemma is proved.

Now it is easy to construct the required zero approximation for the function $\varphi(x,\tilde t)$:

$$\varphi_0^{\vphantom{+}}(x,\tilde t)= A\Bigl\{\mathop{\rm cn}... ... t),k)-\mathop{\rm cn}\nolimits(\alpha (x+\tilde t),k)\Bigr\}.$$

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

$$\varphi_0^{\vphantom{+}}(x,\tilde t)=2A\frac{\gamma}{k}\sum_{... ...fty}f_{2n-1}^{\vphantom{+}} \sin((2n-1)x)\sin((2n-1)\tilde t).$$

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 the Fourier coefficients of the function $\varphi_0^{\vphantom{+}}(x,\tilde t)$ 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

$$\varphi_0^{\vphantom{+}}(x,\tilde t)= A\Bigl\{\mathop{\rm cn... ... t),k)-\mathop{\rm cn}\nolimits(\alpha (x+\tilde t),k)\Bigr\},$$

with k=0.451075598811 and $\alpha=1.0576653982$is such a standing wave solution of equation (3) that equation (4) has a periodic solution.