From 6c9967a13e5f8d080b9b2ebcfaa61af31e12e47c Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 11 Nov 2021 15:39:11 +0100
Subject: Some writing.

---
 stokes.tex | 39 ++++++++++++++++++++++++++++++---------
 1 file changed, 30 insertions(+), 9 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index 91f7f18..e574e92 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -442,15 +442,6 @@ imaginary energy join.
 \section{Numerics}
 
 To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically.
-\begin{equation}
-  \mathcal L(z(t), z'(t))
-  = 1-\frac{\operatorname{Re}(\dot z^*_iz'_i)}{|\dot z||z'|}
-\end{equation}
-\begin{equation}
-  \mathcal C[\mathcal L]=\int dt\,\mathcal L(z(t), z'(t))
-\end{equation}
-$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ whose tangent is everywhere parallel to the direction of the dynamics. Therefore, finding functions which reach this minimum will find time-reparameterized Stokes lines. For the sake of numeric stability, we look for functions whose tangent has constant norm.
-
 
 
 \begin{acknowledgments}
@@ -517,4 +508,34 @@ Putting these pieces together, we find
   \frac{\partial\phi}{\partial\bar z^{\bar\jmath}}\frac\partial{\partial z^i}
 \end{equation}
 
+\section{Numerics}
+
+To study Stokes lines numerically, we approximated them by parametric curves.
+If $z_0$ and $z_1$ are two stationary points of the action with
+$\operatorname{Re}\mathcal S(z_0)>\operatorname{Re}\mathcal S(z_1)$, then we
+take the curve
+\begin{equation}
+  z(t)
+  =(1-t)z_0+tz_1+(1-t)t\sum_{i=0}^mg_it^i
+\end{equation}
+where the $g$s are undetermined complex vectors. These are fixed by minimizing
+a cost function, which has a global minimum only for Stokes lines. Defining
+\begin{equation}
+  \mathcal L(t)
+  = 1-\frac{\operatorname{Re}[\dot z^*(z(t))\cdot z'(t)]}{|\dot z(z(t))||z'(t)|}
+\end{equation}
+this cost is given by
+\begin{equation}
+  \mathcal C=\int_0^1 dt\,\mathcal L(t)
+\end{equation}
+$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$
+whose tangent is everywhere parallel to the direction $\dot z$ of the dynamics.
+Therefore, functions that minimize $\mathcal C$ are time-reparameterized Stokes
+lines.
+
+We explicitly computed the gradient and Hessian of $\mathcal C$ with respect to
+the parameter vectors $g$. Stokes lines are found or not between points by
+using the Levenberg--Marquardt algorithm starting from $g^{(i)}=0$ for all $i$,
+and approximating the cost integral by a finite sum.
+
 \end{document}
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