Steady Progressing Wave Solutions

We consider here only the case where $s\in E^1$ and $\theta\in E^m$ where $m=1 \mbox{ or } 2$to illustrate several interesting features of such systems. The general case can be treated in similar ways.

The observables of this system are usually not $\theta$, but some periodic wave form or function of $\theta$ such as $\cos\theta$ or $\cos_+\theta$. This suggests that we seek a steady progressing wave solution in the form

$$\theta=\alpha s + c t$$

where $\alpha$ is called the wave number and c is a vector of wave speeds.

Direct substitution of the steady progressing wave solution into the equation gives

$$c_{\alpha}=\omega+ \int_{-L}^L { {K}}(s-s') f(\alpha(s-s')+\psi(s-s'))ds'$$

There is a unique constant solution of this equation for cprovided the components of $\alpha$ have the form $k\pi/L$ for some integer $k=0,\pm 1,\pm 2,\cdots$.

Let us fix $\alpha = N\pi/L$for some integer N. According to the calculation above, there is a steady progressing wave solution of the system having the form

$$\theta=N\pi s/L + c_N t$$

where c_N is given by the formula

$$c_{N}=\omega+ \int_{-L}^L K(s-s') f(N\pi(s-s')/L+\psi(s-s'))ds'$$

To test the stability of this steady progressing wave we define

$$x_n(s,t)\delta =\theta(s,t)-N\pi s/L-c_Nt$$

The result is

$$\delta \dot x(s,t)=\int_{-L}^LK(s-s') f(\delta(x(s,t)-x(s',t))+N\pi(s-s')/L+\psi(s-s'))ds'$$

Expanding the right hand side in powers of $\delta$ and ignoring higher order terms gives

$$\dot x(s,t)=\int_{-L}^LK(s-s') f'(N\pi(s-s')/L+\psi(s-s'))(x(s,t)-x(s',t))ds'$$

Expanding x in its Fourier series

$$x(s,t)=\sum_{n=-\infty}^{\infty} x_n(t)e^{in\pi s/L}$$

and substituting this into the equation gives

$$\dot x_n(t)=(A_0-A_n)x_n$$

where

$$K(s)f'(N\pi s/L+\psi(s))= \sum_{-\infty}^{\infty} A_ne^{in\pi s/L}\equiv A(s)$$

Thus, depending on the properties of the coefficients in A, we get the following stability results for the deviation x: If $K(s)=f(s)=\cos s$ and $\psi(s)\equiv 0$, then the leading components of x are harmonic and the remaining modes are damped. If $\psi\equiv\pi/2$, then all modes (except for n=0) are damped.