# 1.2.7. Dispersion Relation and Instabilities Using Green’s Function¶

This is a review

## 1.2.7.1. A Simple System¶

For a nonlinear system which has a linearized relation

$\mathscr D S = 0,$

where $$\mathscr D$$ is a derivative operator that contains terms like $$\partial_t$$ and $$\boldsymbol\nabla$$.

Assuming collective modes $$S = Q e^{i(\mathbf k \cdot \mathbf x - \omega t)}$$, the equation becomes

$D(\omega, \mathbf k) = 0,$

which is in fact the dispersion relation.

The picture behind these math is that we are treating the field as waves. By “solving” the dispersion relation, we find $$\omega(\mathbf k)$$ or $$\mathbf k(\omega)$$, which is substituted into the wave form,

$S = Q e^{i(\mathbf k(\omega) \cdot \mathbf x - \omega t)},$

or

$S = Q e^{i(\mathbf k \cdot \mathbf x - \omega(\mathbf k) t)}.$

However, we need to integrate over all Fourier modes in general. So we have the solution of the field

$S = \int d\omega Q_\omega e^{i(\mathbf k(\omega) \cdot \mathbf x - \omega t)},$

or

$S = \int d\mathbf k Q_{\mathbf k} e^{i(\mathbf k \cdot \mathbf x - \omega(\mathbf k) t)}.$

Instability of the system is related the convergence of the integrals. This was done by P. A. Sturrock. [Sturrock1958]

For the general equation of coherence

$\mathscr D S = f(\mathbf x,t),$

the authors consider the Green’s function of the equations, i.e.,

$\mathscr D G = \delta(\mathbf x)\delta(t).$

Fourier transform of $$G$$ leads to

$D(\omega,\mathbf k) \tilde G = \mathrm{Const},$

where the constant is chosen to be 1. The formal solution is

$\tilde G = \frac{1}{D(\omega,\mathbf k)}.$

Inverse Fourier transform of the solution

$G(t,\mathbf x) = \frac{1}{(2\pi)^2} \iint d\mathbf k d \omega \tilde G e^{i(\mathbf k \cdot \mathbf x - \omega t)} = \frac{1}{(2\pi)^2} \iint d\mathbf k d \omega \frac{1}{D(\omega,\mathbf k)} e^{i(\mathbf k \cdot \mathbf x - \omega t)}.$

Choice of Coefficient

The Fourier transform is chosen to be

$\tilde g = \frac{1}{(2\pi)^2} \iint d\omega d \mathbf k g e^{i(\mathbf k \cdot \mathbf x - \omega t)},$

which has coefficient $$1/(2\pi)^2$$. The inverse transform would have no coefficient.

Solution to S

The solution for coherence is

$S = \iint G f d\mathbf x dt,$

which indicates that the Fourier mode

$\tilde S = \tilde G \tilde f.$
 [Sturrock1958] Sturrock, P. A. (1958). Kinematics of Growing Waves. Physical Review, 112(5), 1488–1503.

One Pole Case

I have no idea how is Eqn. 27 derived.

Also have questions about Eqn. 34 for convective instability of 0 group velocity.

## 1.2.7.2. Neutrinos¶

The off diagonal elements of the density matrix $$S$$ obays the equation of the form

$i(\partial_t + \mathbf v \cdot \nabla) S = f(\mathbf x, t)$

in linear regime.

## 1.2.7.3. Gap¶

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