stuff-to-do

Limits :

1. When the period is much bigger than the half-life $latex \Omega\ll\gamma$ then the field A simplies to :

$latex \displaystyle{ \large A(x,t)=\frac{\alpha}{2\sqrt{D\gamma}}\exp(-\sqrt{\gamma/D}|x|) R \exp(i\Omega t+i\phi) }$

For several oscillators :

$latex \displaystyle{ \large A(x,t)=\frac{\alpha}{2\sqrt{D\gamma}}\sum_{j}\exp(-\sqrt{\gamma/D}|x-x_{j}|) R \exp(i\Omega t+i\beta j) }$

• How does the dispersion relation looks like in this case ?
• Does the inifinite sum $latex \large \displaystyle{ \sum_{j} \; \; \exp(-\sqrt{\gamma/D} \; |x-x_{j}|)  }$ (for $latex \beta=0$) converge ?

2. When the period is much smaller than the half-life $latex \Omega\gg\gamma$ then the field A simplies to :

$latex \displaystyle{ \large \frac{1}{2\sqrt{D\Omega}}R\alpha\exp(i\Omega t+i\phi)\exp(-i\left[\frac{1}{\sqrt{2}}\sqrt{\frac{\Omega}{D}}|x|+\frac{\pi}{4}\right]-\frac{1}{\sqrt{2}}\sqrt{\frac{\Omega}{D}}|x|) }$

• Same questions.

3. Stability of dispersion relation : done

4. plot Z as a vector in 2D and plot A to measure the angle

5. add perturbations at the beginning to see how they evolve