Lecture 05 - Various Applications

#math

Let us consider the Heisenberg Spin chain equation:

S_{u} = S \times S_{x x}

If we identify with its tangent $S \to T$ then we obtain:

T_{u} = T \times T_{s s}

If we work on the Binormal frame, recall:

{(\begin{matrix} T \\ N \\ B \end{matrix})}_{s} = (\begin{matrix} 0 & κ & 0 \\ - κ & 0 & τ \\ 0 & τ & 0 \end{matrix}) (\begin{matrix} T \\ N \\ B \end{matrix})

Then we obtain an equation that is both a binormal and normal flow:

\begin{aligned} T_{u} & = T \times (κ N) \\ = κ_{s} B + κ τ N \end{aligned}

If we use the following compatibility condition:

\begin{aligned} T_{u s} & = T_{s u} \\ N_{u s} & = N_{s u} \\ B_{u s} & = B_{s u} \end{aligned}

We obtain the following result (Da Rios System):

\begin{aligned} κ_{u} & = - 2 κ_{s} τ - κ τ_{s} \\ τ_{u} & = [\frac{κ_{s s}}{κ} - τ^{2}]_{s} - κ κ_{s} \end{aligned}

Thus if we use our (favourite) Hasimoto map we obtain our NLSE again:

i ψ_{u} = ψ_{s s} + \frac{1}{2} | ψ |^{2} ψ

'Recipe'

Physical Equation $\leftrightarrow_{F r e n e t}^{S e r r e t}$ Moving Curve $\leftrightarrow_{m a p}^{H a s i m o t o}$ NLSE(+family)

$H = \sum f_{i} (s) S_{i} \cdot S_{i + 1}$ , where $f_{i}$ is a function of $s$ .
Continuum case:

\begin{aligned} S_{t} & = (f S \times S_{x})_{x} \\ \leftrightarrow T_{u} & = (f T \times T_{s})_{s} \\ = (f κ)_{s} B - f κ τ N \end{aligned}

('Homework') We will obtain after some computation:

i ψ_{u} + (f ψ)_{s s} + 2 ψ (f | ψ |^{2} + \int_{- \infty}^{\infty} f (s^{'}) | ψ |^{2} d s^{'}) = 0

This is a generalization of the NLSE.

Suppose $v_{i j}$ is a velocity vector for a small neighborhood around a point $γ_{i}$ , with $k$ is the vortex strength:

γ_{i j} = γ_{i} - γ_{j}

v_{i j} = \frac{- k}{4 π} \int | γ_{i j} |^{- 3} \frac{\partial γ_{i j}}{\partial s_{i}} \times γ_{i j} d s_{j}

LIA concept in the limiting case of a vortex filament of infinitesimal core size (long distance effects are considered)

\begin{aligned} γ_{i j} & = γ_{i j} (ξ, t) = γ_{i} (s_{i, t)} - γ_{j} (s_{i} + ξ, t) \\ γ_{i j} & = a_{1} ξ + a_{2} ξ^{2} + . . . \end{aligned}

where $a_{1} = \frac{\partial γ_{i j}}{\partial ξ}$ and $a_{2} = \frac{1}{2} \frac{\partial^{2} γ_{i j}}{\partial ξ}$ .

\begin{aligned} \frac{\partial γ_{i j}}{\partial s_{i}} & = \frac{\partial γ_{i j}}{\partial ξ} \\ \frac{\partial γ_{i j}}{\partial s_{i}} \times γ_{i j} & = a_{1} \times a_{2} ξ^{2} + O (ξ^{3}) \\ | γ_{i j} | & = \sqrt{γ_{i j} \cdot γ_{i j}} \\ = \sqrt{(a_{1} ξ + a_{2} x i^{2} + . . .) \cdot (a_{1} ξ + a_{2} x i^{2} + . . .)} \\ = | a_{1} | | ξ | \sqrt{1 + 2 \frac{a_{1} \cdot a_{2}}{| a_{1} |} ξ + . . .} \end{aligned}

Thus we obtain:

\begin{aligned} v_{i j} & = \frac{k}{4 π} \int [\frac{a_{1} \times a_{2}}{| a_{1} |^{3}} \frac{1}{ξ} + O (1)] d s \\ = \frac{k}{4 π} \frac{a_{1} \times a_{2}}{| a_{1} |^{3}} l o g (\frac{1}{| ξ |}) + O (1) \end{aligned}

Now we let $Γ = \frac{k}{4 Π | a_{1} |^{3}} l o g (1 / ξ)$ then the structure of our velocity is simply $v_{i j} = Γ γ_{ξ} \times γ_{ξ ξ}$ .