# Statistical mechanics of elastica for the shape of supercoiled DNA:
hyperelliptic elastica of genus three

Shigeki Matsutani

###### Abstract.

This article studies the statistical mechanics of elastica as a model of the
shapes of the supercoiled DNA, and shows that its excited states can be
characterized by the focusing modified KdV (MKdV) equation due to thermal
fluctuation. Following the previous paper (Matsutani and Previato, Physica D
430 (2022) 133073), the hyperelliptic solutions of the focusing modified KdV
(MKdV) equation of genus three are considered. There appears a pattern as a
repetition of the modulation of figure-eight and the inverse ‘S’ as a thermal
fluctuation of elastica, called the S-eight mode. Our model states that the
excited states of elastica due to the thermal effect have the S-eight mode,
which reproduces the shapes of the AFM image of the supercoiled DNAs observed
by Japaridze et al. (Nano Lett. 17 3, (2017) 1938).

##  1\. Introduction

In the previous paper [34], the author with Emma Previato investigated an
algebro-geometric model for the shape of supercoiled DNA. As mentioned there,
the mathematical description of the shape of the supercoiled DNA is a
challenging problem in which no one can find the shape mathematically. Since
the shape of the supercoiled DNA plays crucial roles in life [6, 10, 24, 38,
43], there are many studies on the shape [4, 19, 23, 24, 41, 42]. The electron
microscope images of DNA on a plane show that, except in a few cases, the
shapes of the loop are much more complicated than Euler’s elastica, a circle
and eight-figure. Furthermore, it is neither squeezed nor dense, but is
characterized by voids between intersections that are weakly governed by
elastic forces. These properties mean that it cannot be realized as a minimal
state of its Euler-Bernoulli energy functional even by considering its three-
dimensional effect; the voids cannot appear mathematically as long as we
consider the minimal state of a given energy functional. The minimal state
cannot have any other parameters and is therefore expressed by the elliptic
functions, which have only double periods and no ability to express
complicated shapes.

In other words, we should move on to consider the excited state of supercoiled
DNA due to thermal effect, i.e., the statistical mechanics of geometric
objects. The author sometimes called this model quantized elastica due to the
analogy between the Planck constant $\hbar$ and the inverse of the temperature
$\beta$, since the statistical mechanics of geometric objects corresponds to
the quantization of geometric objects after Wick rotation; (Two-dimensional
quantum gravity is related to the statistical mechanics of random
triangulation of some Riemann surfaces [20, 44]. The statistical mechanics of
the liquid crystal observed in the laboratory is related to the $O(3)$ gauge
field theory [17].)

Since the elastica model has the constraint that it does not stretch, it
cannot be regarded as an extension of the random walk model [11], but it has
an interesting internal symmetry related to the hyperelliptic curves. The
author proposed a model of the statistical mechanics of elastica to express
the shapes of the supercoiled DNA in 1998 [26]. The shapes can only be
realized if thermal effects are taken into account, and must be the excited
states of the elastica rather than minimal states. The excited states of the
elastica on the plane are well described by curves whose tangential angles
$\phi$ obey the modified KdV (MKdV) equation [1],

$(\partial_{t}+\alpha\partial_{s})\phi+\frac{1}{8}\left(\partial_{s}\phi\right)^{3}+\frac{1}{4}\partial_{s}^{3}\phi=0,$
(1.1)

where $t$ and $s$ are the real axes and $\alpha$ is a real parameter. Here $t$
does not mean the physical time axis, but one of the inner-space directions of
the excited states due to thermal fluctuations. (The auxiliary time $t$
corresponds to the Schwinger time of the quantum field theory [39, 40] in this
system, which shows the propagation from an excited state to the others. ) It
is shown that the inner symmetry of the statistical mechanics of elastica is
governed by the hyperelliptic curves and their moduli whereas the symmetry of
the classical elastica at zero-temperature is determined by the elliptic
curves and their moduli [26, 28, 32, 33]. Thus we have also referred to this
model as generalized elasticae [33].

The statistical mechanics of elastica is a kind of solvable model of
statistical mechanics, which can be expressed by solutions of the integrable
system, the MKdV equation. Finding the solutions, however, requires advanced
knowledge of modern algebraic geometry and the computational techniques of
hyperelliptic functions. Therefore, no one has attacked this problem.

The author and Emma Previato decided to solve the problem of elastica in 2004
based on the papers [28, 37]. The author has developed the tools together with
co-authors [28, 33, 34, 13, 15] following Baker’s approach and a similar
movement [3, 7, 9, 8]. To solve the problem, they considered that a novel
approach was needed that directly connects the algebraic curves, rather than
the theta-function approach [36]. They have spent two decades refining and
reconstructing Abelian function theory, including hyperelliptic function
theory, as problems in algebraic geometry [29, 34, 31]. (The computational
techniques of hyperelliptic functions have the ability to describe some
complicated systems that cannot be described by elliptic functions. We note
that the development of computational techniques provided solutions to the
problem of the motion of a particle around the black hole [14]).

Since supercoiled DNA exists in three-dimensional space, its statistical
mechanical model of excited states of elastica is governed by the nonlinear
Schrödinger (NLS) equation and the complex MKdV (CMKdV) equation rather than
the MKdV equation [27]. However, since the observed shapes of supercoiled DNA
are on a plane and the solutions of the NLS and CMKdV equations are much more
complicated than the MKdV equation, we focus on the two-dimensional model.
Though the hyperelliptic solutions have not been concretely obtained, the
elliptic function solutions of (two-dimensional) elastica, figure-eight due to
Euler [12], have been studied well which correspond to the ground state of the
model, and are often observed in AFM [38].

Thus the author attacked this two-dimensional problem in the previous paper
with Previato [34] to find the real analytic solution of the MKdV equation in
terms of the data of hyperelliptic curves of genus two [28]. Then the attempt
failed in [34]. It was clarified that any hyperelliptic non-degenerate curves
of genus two could not provide the excited states of elastica because of the
reality conditions [34]. As a conclusion of [34], hyperelliptic curves of
higher genus ($g\geq 3$) should be investigated to find the solution of (1.1).
Therefore, the associated paper [30] is devoted to finding the solutions of
(1.1) in terms of the meromorphic functions of hyperelliptic curves of genus
three. There we find some approximate solutions of (1.1) of the gauged MKdV
equation.

This paper, based on the results in the associated paper [30], shows that some
hyperelliptic curves of genus three supply the forms of the excited states of
elastica which have never been obtained. They exhibit typical shapes,
modulation of a repetition of figure-eight and inverse of ‘S’. We call it the
S-eight mode. In other words, our model states that the excited states of the
statistical mechanics of elastica have the S-eight mode. It is quite
surprising that we find a similar shape in some of the AFM images of a
supercoiled DNA in [21, Figure 4]. The observed shapes of supercoiled DNA have
the S-eight mode.

The figure-eight given by Euler in 1744 which appears similar shapes of the
short closed supercoiled DNAs, e.g. in [38] but no one has ever mathematically
reproduce any more complicated shape of supercoiled DNA with voids. We
emphasize that this demonstration shows that an important first step has been
made toward the complete mathematical expression of the supercoiled DNAs.

The content is following: Section 2 reviews the statistical mechanics of
elastica as in [26]. previous results, which is the same as [34]. Section 3 is
devoted to the geometry of the hyperelliptic curves and their relation to the
MKdV equation over ${\mathbb{C}}$ and the gauge MKdV equation. Section 4
provides hyperelliptic solutions of the gauged MKdV equation of genus three.
Section 5 shows the numerical evaluation of the solutions and provides the
discussion of the solutions. We consider the computational results and their
relations to the supercoiled DNA. Section 6 gives the conclusion of this
paper.

##  2\. Statistical Mechanics of elastica

We consider that the supercoiled DNA is caused by thermal fluctuations. As a
model, we could assume that the stretching must be suppressed and obey the
elastic force known as the Euler-Bernoulli energy function. When supercoiled
DNAs are observed, e.g. by atomic force microscopy, they lie on a plane.
Therefore, we consider their shape of the flat plane. In other words, we
consider the statistical mechanics of elastica on a plane.

Let us consider an analytic and isometric immersion $Z$ of a loop $S^{1}$ into
the complex plane ${\mathbb{C}}$ [33],

${\mathcal{M}}:=\\{Z:S^{1}\hookrightarrow{\mathbb{C}}\ |\ \mbox{analytic,
isometric}\\}.$

Let $ds^{2}=\overline{dZ}dZ$ be the differential of arclength $s$ such that
$\displaystyle{\int ds=2\pi}$. Then its tangential vector is given by
$\partial_{s}Z$ satisfying $|\partial_{s}Z|=1$ and its curvature is
$\displaystyle{k:=\frac{1}{{\sqrt{-1}}}\partial_{s}\log\partial_{s}Z}$ for
$\partial_{s}:=\partial/\partial s$. Let
$\partial_{s}Z={\mathrm{e}}^{{\sqrt{-1}}\psi}$ where $\psi$ is a real value
and $k=\partial_{s}\psi$. The elastica is given as the minimal point of the
Euler-Bernoulli energy

${\mathcal{E}}[Z]:=\frac{1}{2}\oint_{S^{1}}\,k^{2}(s)ds=\oint_{S^{1}}\\{Z,s\\}_{\mathrm{SD}}ds,$
(2.1)

where $\displaystyle{\\{Z,s\\}_{\mathrm{SD}}={\small{\frac{\partial}{\partial
s}\left(\frac{\partial^{2}Z/\partial s^{2}}{\partial Z/\partial
s}\right)-\frac{1}{2}\left(\frac{\partial^{2}Z/\partial s^{2}}{\partial
Z/\partial s}\right)^{2}}}}$ is the Schwarzian derivative.

Let us consider the partition function ${\mathcal{Z}}[\beta]$ of the
statistical mechanics of elastica, which is given by

$\displaystyle{\mathcal{Z}}[\beta]=\int_{\mathcal{M}}DZ\exp(-\beta{\mathcal{E}}[Z])$
(2.2)

$DZ$ is the Feynman measure of this system, and $1/\beta$ is the temperature.
Due to ${\mathcal{M}}$ we only consider the isometric deformation. We remark
that this isometric condition is strong and this model is completely different
from the well-known random chain model [11].

To evaluate this partition function ${\mathcal{Z}}[\beta]$, we consider the
infinitesimal deformation $Z\to Z+\delta Z$ parameterized by the infinite
dimensional virtual time parameter ${\mathfrak{t}}=(t_{1},t_{2},\ldots)$. We
apply the effective action method to this evaluation (The time
${\mathfrak{t}}$ corresponds to the Schwinger time of the quantum field theory
[39, 40] in this system, which shows the propagation from one excited state to
another). In other words, $\delta Z=\partial_{t}Z\delta
t=(u_{1}(s)+{\sqrt{-1}}u_{2}(s))\partial_{s}Z\delta t$ must preserve the
isomorphic deformation, i.e. $[\partial_{t},\partial_{s}]=0$. By the
Goldstein-Petrich condition $[\partial_{t},\partial_{s}]=0$ [18], we have
$\partial_{t}\partial_{s}Z={\sqrt{-1}}\partial_{t}\psi\partial_{s}Z$, while
$\partial_{s}\partial_{t}Z=[(\partial_{s}u_{1}+ku_{2})+{\sqrt{-1}}(\partial_{s}u_{2}-ku_{1})]\partial_{s}Z$.
The isometric deformation is given by

$\partial_{t}k=\Omega u_{2},\quad
u_{1}=\partial_{s}^{-1}ku_{2},\quad\Omega:=\partial_{s}^{2}+\partial_{s}k\partial_{s}^{-1}k.$

Following the theory of the effective action for the functional integral
method [39], we consider the deformation such that the Euler-Bernoulli energy
changes as $E[Z+\delta Z]=E[Z]+\displaystyle{\delta t\int
k(\partial_{t}k)ds}+O(\delta t^{2})$. We require the condition that $\delta
E:=\displaystyle{\delta t\int k(\partial_{t}k)ds}=0$.

The condition $|\partial_{s}Z|=1$, namely $\partial_{s}Z\in{\mathrm{U}}(1)$,
allows us to define an action $g_{s_{0}}Z(s)=Z(s-s_{0})$ of ${\mathrm{U}}(1)$.
It can be expressed as $(\partial_{s}-\partial_{t})\partial_{s}Z=0$ and
$(\partial_{s}-\partial_{t})\partial_{s}k=0$. This deformation can be regarded
as the trivial isometric deformation and the isoenergy deformation, i.e. the
trivial gauge transformation. Thus we introduce the infinite deformations with
infinite proper time ${\mathfrak{t}}=(t_{1},t_{2},\cdots)$ by starting the
trivial transformation as the initial deformation,

$\partial_{t_{n+1}}k=\Omega\partial_{t_{n}}k,\quad\partial_{t_{1}}k=\partial_{s}k.$

Then $\partial_{t_{n}}k=\Omega^{n}\partial_{s}k$ is consistent with the $n$-th
focusing modified KdV (MKdV) equation [33]. For the deformation, the Euler-
Bernoulli energy is preserved, i.e., $E[Z+\delta Z]=E[Z]+\delta
t_{n}\displaystyle{\int k(\partial_{t_{n}}k)ds}=E[Z]$. The requirement that
the vanishing of the first-order deformation $\delta E=\delta
t\displaystyle{\int k(\partial_{t}k)ds}=0$ guarantees complete energy
preservation due to the integrability of the MKdV equation, i.e.,
$\partial_{t_{n}}E[Z]=0$. In other words, the statistical mechanics of
elastica has infinite gauge symmetry, in general. Since the MKdV equation
preserves the Euler-Bernoulli energy of the deformation of curves.
${\mathcal{M}}$ is decomposed to

${\mathcal{M}}=\bigsqcup_{E\in{\mathbb{R}}}{\mathcal{M}}_{E},\quad{\mathcal{M}}_{E}:=\\{Z\in{\mathcal{M}}\
|\ {\mathcal{E}}[Z]=E\\},$

which naturally includes the Euler’s elastica as its minimal point (the state
of the zero temperature.)

It has the orbital decomposition in terms of the loop soliton whose curvature
obeys the modified KdV equation;

Then ${\mathcal{Z}}[\beta]$ can be evaluated like

${\mathcal{Z}}[\beta]=\int
dE\frac{\partial\mathrm{Vol}({\mathcal{M}}_{E})}{\partial
E}{\mathrm{e}}^{-\beta E},$

where $\mathrm{Vol}({\mathcal{M}}_{E})$ is the volume of ${\mathcal{M}}_{E}$.

Thus to evaluate the statistical mechanics of the elastica is reduced to
finding the solutions of the modified KdV equations.

The evaluation of the statistical mechanics of elastica is given as follows.

  1. (0)

to regard the elastica model as the minimal point or the state at the zero
temperature.

  2. (1)

to find the solutions of the (focusing) MKdV equation as the excited state of
the Euler-Bernoulli energy associated with the hyperelliptic curves $X$ of
genus $g$,

  3. (2)

to find the energy-preserving orbit consisting of the iso-energy state
${\mathcal{M}}_{E}$, which is a real subspace ${\mathcal{M}}_{X,E}$ of the
Jacobi varieties $J_{X}$ of $X$ (given by real period matrices),

  4. (3)

to evaluate the moduli dependence of the volume of ${\mathcal{M}}_{X,E}$ for
the moduli parameters of $X$ as subspace of the moduli space of the
hyperelliptic curves.

  5. (4)

to assign the topology in ${\mathcal{M}}$ and to define a suitable measure
with respect to the Euler-Bernoulli energy, and

  6. (5)

to analyze ${\mathcal{M}}$ for a given $\beta$.

The author with Previato investigated the geometrical structure of
${\mathcal{M}}$ as a generalized elastica problem [29, 32, 33, 37]. However
finding the solutions of the focusing MKdV equation is quite difficult. The
reality conditions CI, CII, CIII as mentioned after Theorem 3.1 restrict the
space ${\mathcal{M}}$. The infinite symmetry can regarded as the loop group
action [2] but reality condition restricts the group action, which quite
differs from the loop group over ${\mathbb{C}}$.

There is a mystery in the research of the shapes of supercoiled DNA, that
though we have found the circle types, the eight-figure types, and much
complicated shapes with several holes of supercoiled DNA in the AFM
observations, we cannot find twisted loops on a plane with two or three holes.
As in [34], the real analytic solution of the focusing MKdV equation can be
found for the cases corresponding to the circle and the eight-figure of the
elliptic curve $g=1$ but cannot found for the hyperelliptic curves of genus
two due to the reality condition. The real hyperelliptic solution of genus two
might be related to twisted loops with two or three holes [34] if exists. We
require higher genus curves to obtain the real analytic solution of the
focusing MKdV equation. Since the higher twisted loops observed in the
laboratory are much more complicated than the figure eight, we conjecture that
this phenomenon must be related to the difficulty of the existence of the real
analytic solution of the focusing MKdV equations.

The shape of the supercoiled DNA might be determined by profound reality of
the algebraic geometry over the complex field. We believe that there is a deep
relationship between nature (life sciences) and mathematics, and that symmetry
is crucial.

If we replace $\beta$ with ${\sqrt{-1}}/\hslash$, the problem is reduced to a
functional integration representation of a quantum system related to a
geometrical object. In theoretical physics, the method to sum a weight with
respect to an invariant functional over possible states is known as the
Feynman path integral (functional integral) method for statistical mechanics
and quantum mechanics. The statistical mechanics of elastica [26] can be seen
as a problem in which we quantize geometric objects with a non-trivial
(classical or ground state) equation, i.e. the static MKdV equation, and a
non-trivial constraint, i.e. isometry. Thus the author called it quantized
elastica [28]. A similar idea appeared in [5], but it was too difficult to
solve in the 1990s [26]. However, the patient efforts of the last three
decades to develop the study of Abelian function theory [28, 33, 34, 13, 15,
7, 9, 8, 31] have allowed a concrete attack on the solution. For example, by
Buchstaber Leykin [9], we can study the moduli space of hyperelliptic curves
by using the heat equation on the moduli parameter $b_{i}$ in (3.1). In other
words, due to the recent development, this model could be explicitly evaluated
using these mathematical tools and could be compared with the experimental
results in the laboratory [34]. Thus, this model itself is quite crucial and
interesting for the life science and theoretical physics.

Hence the solution of the MKdV equation can be regarded as an excited state of
elastica due to thermal effect. We investigate the explicit solutions using
the hyperelliptic function, while the ground state of the elastica (at zero
temperature) is expressed by the elliptic function. Though in the above
formulation we consider a loop (closed curve), in this paper we consider
curves whose curvature obeys the MKdV equation.

##  3\. Hyperelliptic solutions of the focusing modified KdV equation over
${\mathbb{C}}$

We review the solution of the generalized elastica problem of excited states
of elastica [28] for a hyperelliptic curve $X_{g}$ of genus $g$ over
${\mathbb{C}}$,

$X_{g}:=\left\\{(x,y)\in{\mathbb{C}}^{2}\ |\
y^{2}=(x-b_{1})(x-b_{2})\cdots(x-b_{2g+1})\right\\}\cup\\{\infty\\},$ (3.1)

where $b_{i}$’s are mutually distinct complex numbers. Let
$\lambda_{2g}=\displaystyle{-\sum_{i=1}^{2g+1}b_{i}}$ and $S^{k}X_{g}$ be the
$k$-th symmetric product of the curve $X_{g}$. The Abelian integral
$v:S^{k}X_{g}\to{\mathbb{C}}^{g}$, $(k=1,\ldots,g)$ is defined by its $i$-th
component $v_{i}$ $(i=1,\ldots,g)$,

$v_{i}((x_{1},y_{1}),\ldots,(x_{k},y_{k}))=\sum_{j=1}^{k}v_{i}(x_{j},y_{j}),\quad
v_{i}(x,y)=\int^{(x,y)}_{\infty}{\nu^{\mathrm{I}}_{i}},\quad{\nu^{\mathrm{I}}_{i}}=\frac{x^{i-1}dx}{2y}.$
(3.2)

[28] shows the hyperelliptic solutions of the MKdV equation over
${\mathbb{C}}$,

######  Theorem 3.1.

[28] For $((x_{1},y_{1}),\cdots,(x_{g},y_{g}))\in S^{g}X_{g}$, a fixed branch
point $b_{a}$ $(a=1,2,\ldots,2g+1)$, and $u:=v((x_{1},y_{1}),$
$\cdots,(x_{g},y_{g}))$,

$\displaystyle{\psi(u):=-{\sqrt{-1}}\log(b_{a}-x_{1})(b_{a}-x_{2})\cdots(b_{a}-x_{g})}$

satisfies the MKdV equation over ${\mathbb{C}}$,

$(\partial_{u_{g-1}}-\frac{1}{2}(\lambda_{2g}+3b_{a})\partial_{u_{g}})\psi-\frac{1}{8}\left(\partial_{u_{g}}\psi\right)^{3}-\frac{1}{4}\partial_{u_{g}}^{3}\psi=0,$
(3.3)

where $\partial_{u_{i}}:=\partial/\partial u_{i}$ as an differential identity
in $S^{g}X_{g}$ and ${\mathbb{C}}^{g}$.

We, here, emphasize the difference between the MKdV equations (1.1) over
${\mathbb{R}}$ and (3.3) over ${\mathbb{C}}$. The difference is crucial since
we want to obtain solutions of (1.1), not (3.3). However, the latter is
expressed well in terms of the hyperelliptic function theory. In the
associated paper [30], we have investigated the solutions of (1.1) based on
the solutions of (3.3).

As mentioned in [34, (11)], we describe the difference. By introducing real
and imaginary parts,
$u_{b}=u_{b\,{\mathrm{r}}}+{\sqrt{-1}}u_{b\,{\mathrm{i}}}$ and
$\psi=\psi_{{\mathrm{r}}}+{\sqrt{-1}}\psi_{{\mathrm{i}}}$, the real part of
(3.3) is reduced to the gauged MKdV equation with gauge field
$A(u)=(\lambda_{2g}+3b_{a}-\frac{3}{4}(\partial_{u_{g}\,{\mathrm{r}}}\psi_{\mathrm{i}})^{2})/2$,

$-(\partial_{u_{g-1}\,{\mathrm{r}}}-A(u)\partial_{u_{g}\,{\mathrm{r}}})\psi_{\mathrm{r}}+\frac{1}{8}\left(\partial_{u_{g}\,{\mathrm{r}}}\psi_{\mathrm{r}}\right)^{3}+\frac{1}{4}\partial_{u_{g}\,{\mathrm{r}}}^{3}\psi_{\mathrm{r}}=0$
(3.4)

by the Cauchy-Riemann relations as mentioned in [34, (11)]. The conserved
energy $\displaystyle{\frac{1}{2}\int(\partial_{s}\psi)^{2}ds=}$
$\displaystyle{\frac{1}{2}\int[(\partial_{s}\psi_{\mathrm{r}})^{2}-(\partial_{s}\psi_{\mathrm{i}})^{2}]ds}$
$+{\sqrt{-1}}\displaystyle{\int[(\partial_{s}\psi_{\mathrm{r}})(\partial_{s}\psi_{\mathrm{i}})]ds}$.

In order to obtain a solution of (1.1) or excited states of elastica in terms
of the data in Theorem 3.1, the following conditions must be satisfied [34]:

  1. CI

$\prod_{i=1}^{g}|x_{i}-b_{a}|=$ a constant $(>0)$ for all $i$ in Theorem 3.1,

  2. CII

$du_{g\,{\mathrm{i}}}=du_{g-1\,{\mathrm{i}}}=0$ in Theorem 3.1, and

  3. CIII

$A(u)$ is a real constant: if $A(u)=$ constant, (3.4) is reduced to (1.1).

##  4\. Hyperelliptic Curves of Genus Three

For the reality conditions we consider the hyperelliptic curves $X$ of genus
three. According to the associated paper [30], we show the results of the real
solution of the gauged MKdV equation and the MKdV equation in [30] without
proofs.

Let $\gamma{\mathrm{e}}^{2{\sqrt{-1}}\varphi}:=(x-b_{0})$ in (3.1) of $g=3$.
We rewrite (3.1) for the angle expression,

$\displaystyle X:=\\{(x,y)\ |\
y^{2}=-64\frac{\gamma^{4}{\mathrm{e}}^{8{\sqrt{-1}}\varphi}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}}(1-k_{1}^{2}\sin^{2}\varphi)(1-k_{2}^{2}\sin^{2}\varphi)(1-k_{3}^{2}\sin^{2}\varphi)\\},$
(4.1)

where
$\displaystyle{k_{a}=\frac{2{\sqrt{-1}}\sqrt[4]{e_{2a-1}e_{2a}}}{\sqrt{e_{2a-1}}-\sqrt{e_{2a}}}=\frac{\sqrt{\gamma}}{\beta_{a}}}$,
$(a=1,2,3)$.

We note that (4.1) means the double covering
${\widehat{\varpi}}:{\widehat{X}}\to X$ and there exists the double covering
Jacobi variety ${\widehat{\varpi}}_{J}:{\widehat{J}}_{X}\to J_{X}$ [30, 31].
We have the holomorphic one-form in the covering Jacobi variety
${\widehat{J}}_{X}$ for $S^{g}{\widehat{X}}$.

######  Lemma 4.1.

For $(\gamma{\mathrm{e}}^{{\sqrt{-1}}\varphi_{i}},K_{j})_{j=1,2,3}\in
S^{3}{\widehat{X}}$, where
$K_{j}:={\widetilde{\gamma}}_{j}{\widetilde{K}}(\varphi_{j})$, ($j=1,2,3$),
and

$\displaystyle{{\widetilde{K}}(\varphi):=\frac{\sqrt{\gamma(1-k_{1}^{2}\sin^{2}\varphi)(1-k_{2}^{2}\sin^{2}\varphi)(1-k_{3}^{2}\sin^{2}\varphi)}}{k_{1}k_{2}k_{3}}},\quad{\widetilde{\gamma}}=\pm
1,$

the following holds:

$\displaystyle{\begin{pmatrix}du_{1}\\\ du_{2}\\\
du_{3}\end{pmatrix}=-\begin{pmatrix}\frac{{\mathrm{e}}^{-2{\sqrt{-1}}\varphi_{1}}}{8\gamma^{2}K_{1}}&\frac{{\mathrm{e}}^{-2{\sqrt{-1}}\varphi_{2}}}{8\gamma^{2}K_{2}}&\frac{{\mathrm{e}}^{-2{\sqrt{-1}}\varphi_{3}}}{8\gamma^{2}K_{3}}\\\
\frac{{\sqrt{-1}}{\mathrm{e}}^{-{\sqrt{-1}}\varphi_{1}}\sin(\varphi_{1})}{4\gamma
K_{1}}&\frac{{\sqrt{-1}}{\mathrm{e}}^{-{\sqrt{-1}}\varphi_{2}}\sin(\varphi_{2})}{4\gamma
K_{2}}&\frac{{\sqrt{-1}}{\mathrm{e}}^{-{\sqrt{-1}}\varphi_{3}}\sin(\varphi_{3})}{4\gamma
K_{3}}\\\
\frac{-\sin^{2}(\varphi_{1})}{2K_{1}}&\frac{-\sin^{2}(\varphi_{2})}{2K_{2}}&\frac{-\sin^{2}(\varphi_{3})}{2K_{3}}\\\
\end{pmatrix}\begin{pmatrix}d\varphi_{1}\\\ d\varphi_{2}\\\
d\varphi_{3}\end{pmatrix}}.$

Let the matrix be denoted by ${\mathcal{L}}$. Then
$\det({\mathcal{L}})=\displaystyle{\frac{\sin(2\phi_{2}-2\phi_{1})+\sin(2\phi_{3}-2\phi_{2})+\sin(2\phi_{1}-2\phi_{3})}{4^{4}K_{1}K_{2}K_{3}}}$.

As we are concerned with the situation that
$y/{\mathrm{e}}^{4{\sqrt{-1}}\varphi}$ is real or pure imaginary, we assume
that the branch points surround the circle whose center is $(b_{0},0)$ and
radius is $\gamma=1$. Noting that we handle the double covering
${\widehat{X}}$ of $X$ with twelve branch points, we define
$\varphi_{{\mathfrak{b}}a}^{+\pm}:=\pm\sin^{-1}(1/k_{a})$ and
$\varphi_{{\mathfrak{b}}a}^{-\pm}=\pi-\varphi_{{\mathfrak{b}}a}^{+\pm}$ as in
Figure 1.

Further, for the case Figure 1 (a) $k_{1}>k_{2}>k_{3}>1.0$, we assume
$\varphi_{\mathfrak{b}}^{\pm}:=\varphi_{{\mathfrak{b}}1}^{+\pm}$ whereas for
the case Figure 1 (b) we let
$\varphi_{\mathfrak{b}}^{\pm}:=\varphi_{{\mathfrak{b}}1}^{\pm+}$.

(a) (b)

Figure 1.  The orbits of each $\varphi_{i}$ in the quadrature: (a):
$k_{1}>k_{2}>k_{3}>1.0$. (b): $k_{3}>k_{2}>k_{1}>1.0$.

We also have the inverse of Lemma 4.1:

######  Lemma 4.2.

For $\varphi_{j}\in[\varphi_{\mathfrak{b}}^{-},\varphi_{\mathfrak{b}}^{+}]$,
$(j=1,2,3)$ such that $\varphi_{\neq}\varphi_{j}$ $(i\neq j)$, we have

$\displaystyle{\begin{pmatrix}d\varphi_{1}\\\ d\varphi_{2}\\\
d\varphi_{3}\end{pmatrix}={\mathcal{K}}{\mathcal{M}}\begin{pmatrix}du_{1}\\\
du_{2}\\\
du_{3}\end{pmatrix}},\qquad{\mathcal{L}}^{-1}={\mathcal{K}}{\mathcal{M}},$
(4.2)

where
$\displaystyle{{\mathcal{K}}:=-\begin{pmatrix}\frac{K_{1}}{\sin(\varphi_{2}-\varphi_{1})\sin(\varphi_{3}-\varphi_{1})}&0&0\\\
0&\frac{K_{2}}{\sin(\varphi_{3}-\varphi_{2})\sin(\varphi_{1}-\varphi_{2})}&0\\\
0&0&\frac{K_{3}}{\sin(\varphi_{1}-\varphi_{3})\sin(\varphi_{2}-\varphi_{3})}\end{pmatrix}}$
and,

$\displaystyle{{\mathcal{M}}:=\begin{pmatrix}8\gamma^{2}\sin\varphi_{2}\sin\varphi_{3}&-4{\sqrt{-1}}\gamma(2{\sqrt{-1}}\sin\varphi_{2}\sin\varphi_{3}-\sin(\varphi_{2}+\varphi_{3}))&-2{\mathrm{e}}^{-{\sqrt{-1}}(\varphi_{2}+\varphi_{3})}\\\
8\gamma^{2}\sin\varphi_{1}\sin\varphi_{3}&-4{\sqrt{-1}}\gamma(2{\sqrt{-1}}\sin\varphi_{1}\sin\varphi_{3}-\sin(\varphi_{3}+\varphi_{1}))&-2{\mathrm{e}}^{-{\sqrt{-1}}(\varphi_{1}+\varphi_{3})}\\\
8\gamma^{2}\sin\varphi_{1}\sin\varphi_{2}&-4{\sqrt{-1}}\gamma(2{\sqrt{-1}}\sin\varphi_{1}\sin\varphi_{2}-\sin(\varphi_{1}+\varphi_{2}))&-2{\mathrm{e}}^{-{\sqrt{-1}}(\varphi_{1}+\varphi_{2})}\\\
\end{pmatrix}.}$

###### Proof.

The straightforward computations show it.

Let us focus on the differential equation,

$\begin{pmatrix}d\varphi_{1,{\mathrm{r}}}\\\ d\varphi_{2,{\mathrm{r}}}\\\
d\varphi_{3,{\mathrm{r}}}\end{pmatrix}=\begin{pmatrix}\frac{2K_{1}{\widetilde{\gamma}}_{1}\cos(\varphi_{2}-\varphi_{3})}{\sin(\varphi_{2}-\varphi_{1})\sin(\varphi_{3}-\varphi_{1})}\\\
\frac{2K_{2}{\widetilde{\gamma}}_{2}\cos(\varphi_{3}-\varphi_{1})}{\sin(\varphi_{3}-\varphi_{2})\sin(\varphi_{1}-\varphi_{2})}\\\
\frac{2K_{3}{\widetilde{\gamma}}_{3}\cos(\varphi_{1}-\varphi_{2})}{\sin(\varphi_{1}-\varphi_{3})\sin(\varphi_{2}-\varphi_{3})}\end{pmatrix}ds$
(4.3)

where $ds$ is the one-form on the real axis, or $s\in{\mathbb{R}}$. Then we
have $\begin{pmatrix}0\\\ -ds/2\\\
ds\end{pmatrix}={\mathcal{L}}\begin{pmatrix}d\varphi_{1,{\mathrm{r}}}\\\
d\varphi_{2,{\mathrm{r}}}\\\ d\varphi_{3,{\mathrm{r}}}\end{pmatrix}$ Here $s$
corresponds to the arclength of the excited states of elastica and
$u_{g,{\mathrm{r}}}$ in (3.4) of $g=3$.

Theorem 4.4 in [30] reads as follows.

######  Theorem 4.3.

For a solution of the differential equation (4.3),
$(\varphi_{1}(s),\varphi_{2}(s),\varphi_{3}(s))\in{\mathbb{R}}^{3}$ if exists,
we let
$\psi_{{\mathrm{r}}}(s)=2(\varphi_{1}(s)+\varphi_{2}(s)+\varphi_{3}(s))$, and
then $\psi_{\mathrm{r}}(s)$ is a cross-section of
$\psi_{\mathrm{r}}(u_{2},u_{3})|_{u_{3}=s,u_{2}=-s/2}$ of
$\psi_{\mathrm{r}}(u_{2},u_{3})$ which satisfies the gauged MKdV equation,

$(\partial_{u_{1}}+{\widetilde{A}}(u_{2},u_{3})\partial_{u_{2}})\psi_{\mathrm{r}}(u_{2},u_{3})+\frac{1}{8}\left(\partial_{u_{3}}\psi_{\mathrm{r}}(u_{2},u_{3})\right)^{3}+\frac{1}{4}\partial_{u_{3}}^{3}\psi_{\mathrm{r}}(u_{2},u_{3})=0,$
(4.4)

where the gauge field is
${\widetilde{A}}(u_{1},u_{2})=(\lambda_{6}-3-\frac{3}{4}(\partial_{u_{3}}\psi_{\mathrm{i}}(u_{2},u_{3}))^{2})/2$
given by
$\partial_{u_{3}}\psi_{\mathrm{i}}(u_{2},u_{3})=\partial_{u_{3}}\varphi_{1,{\mathrm{i}}}+\partial_{u_{3}}\varphi_{2,{\mathrm{i}}}+\partial_{u_{3}}\varphi_{3,{\mathrm{i}}}$,
and

$\begin{pmatrix}\partial_{u_{3}}\varphi_{1,{\mathrm{i}}}\\\
\partial_{u_{3}}\varphi_{2,{\mathrm{i}}}\\\
\partial_{u_{3}}\varphi_{3,{\mathrm{i}}}\end{pmatrix}=\begin{pmatrix}\frac{2K_{1}{\widetilde{\gamma}}_{1}\sin(\varphi_{2}+\varphi_{3})}{\sin(\varphi_{2}-\varphi_{1})\sin(\varphi_{3}-\varphi_{1})}\\\
\frac{2K_{2}{\widetilde{\gamma}}_{2}\sin(\varphi_{3}+\varphi_{1})}{\sin(\varphi_{3}-\varphi_{2})\sin(\varphi_{1}-\varphi_{2})}\\\
\frac{2K_{3}{\widetilde{\gamma}}_{3}\sin(\varphi_{1}+\varphi_{2})}{\sin(\varphi_{1}-\varphi_{3})\sin(\varphi_{2}-\varphi_{3})}\end{pmatrix}.$
(4.5)

We note that (4.4) is a differential identity of $\psi_{{\mathrm{r}}}$ for
every hyperelliptic curve as in [28].

It means the following theorem:

######  Theorem 4.4.

$\psi_{{\mathrm{r}}}:=2(\varphi_{1}+\varphi_{2}+\varphi_{3})$ of the
quadrature $d\varphi_{i,{\mathrm{r}}}$ $(i=1,2,3)$ of (4.3) shows a local
solution of the MKdV equation (1.1) for the region that
$\partial_{u_{3}}\psi_{\mathrm{i}}$ is constant.

Further on the region that $\partial_{u_{3}}\psi_{\mathrm{i}}$ is
approximately constant, the $\psi_{{\mathrm{r}}}$ shows the approximate
solutions of the MKdV equation.

##  5\. Numerical results and discussion

We demonstrate the shapes of excited states of elastica of genus three, a
closed solution and three open solutions.

We show the numerical integration (4.3) as follows.

  1. (1)

We set $(k_{1},k_{2},k_{3})$ to determine a hyperelliptic curve. We assume
Figure 1 (a) or (b). $k_{1}>k_{2}>k_{3}>1.0$ and $k_{3}>k_{2}>k_{1}>1.0$. We
explain mainly the case (a). Let
$\varphi_{{\mathfrak{b}}}:=\varphi_{{\mathfrak{b}}1}^{++}$.

  2. (2)

We set the initial condition $(\varphi_{1},\varphi_{2},\varphi_{3})|_{s=0}$
that the imaginary part of
$2\partial(\varphi_{1,{\mathrm{i}}}+\varphi_{2,{\mathrm{i}}}+\varphi_{3,{\mathrm{i}}})$
in (4.5) vanishes.

  3. (3)

We employ the Euler method of the numerical quadrature method for a
sufficiently small real value $\delta s$ to solve (4.3) numerically so that
the orbit of $\varphi_{a}$ moves back and forth between the branch points
$[-\varphi_{\mathfrak{b}},\varphi_{\mathfrak{b}}]$ in (4.1) as in Figure 1
(a).

We note that at the branch point, the orbit of $\varphi_{a}$ turns the
direction by changing the sign of ${\widetilde{\gamma}}_{a}$ so that
${\widetilde{\gamma}}_{a}\sin^{2}(\varphi_{a})\delta\varphi_{a}/2K_{a}$ is
positive as in Figure 1; it moves the different leaf of the Riemann surface
with respect to the projection $X_{3}\to{\mathbb{P}}$ $((x,y)\mapsto x)$ after
passing the branch points.

Following the Euler method of the numerical quadrature method for the
$n$-step, we obtain the $\varphi_{a}$ development,

$\varphi_{a,n+1}:=\varphi_{a,n}+\delta\varphi,\quad(a=1,2,3).$

We let $\psi_{n}=2(\varphi_{1,n}+\varphi_{2,n}+\varphi_{3,n})+\psi_{c}$ so
that $\psi_{0}$ is a certain value (in the following results, $\psi_{0}=0$),
and numerically integrate

$X_{n+1}=X_{n}+\cos(\psi_{n})\delta s,\quad Y_{n+1}=Y_{n}+\sin(\psi_{n})\delta
s,$

to obtain the excited states of elastica $(X(s),Y(s))$ of a certain
$(u_{1},u_{2},u_{3})$ point in $J_{X}$.

  4. (4)

We monitor the imaginary part of
$\psi_{\mathrm{i}}:=\displaystyle{\int^{s}\partial_{u_{3}}\psi_{{\mathrm{i}}}ds}$
for $\partial_{u_{3}}\psi_{\mathrm{i}}$
$=2\partial(\varphi_{1,{\mathrm{i}}}+\varphi_{2,{\mathrm{i}}}+\varphi_{3,{\mathrm{i}}})$
in (4.5). Though the above theorem can hold if
$\partial_{u_{3}}\psi_{\mathrm{i}}=$constant number $c\in{\mathbb{R}}$, we
focus on the case where $c\sim 0$. Thus we monitor the flatness of
$\psi_{{\mathrm{i}}}$.

The first result is displayed in Figure 2. For the hyperelliptic curve given
by $(k_{1},k_{2},k_{3})=(1.04,1.0392,1.010)$, we set the initial condition
$(\varphi_{1},\varphi_{2},\varphi_{3})=(\varphi_{\mathfrak{b}},-0.90,-0.90)$.
Figure 2 (a) shows the profile of $\psi_{\mathrm{r}}$ and $\psi_{\mathrm{i}}$
and (b) shows the shape of the excited states of elastica. The maximum of
$\partial_{u_{3}}\psi_{\mathrm{i}}$ is $1.00952\times 10^{-02}$. In other
words, the orbit is considered to be one of the MKdV equation (1.1) rather
than the gauged MKdV equation (3.4).

(a) (b)

Figure 2.  An open excited state of elastica:
$(k_{1},k_{2},k_{3})=(1.04,1.0392,1.010)$, and the initial condition is
$(\varphi_{1},\varphi_{2},\varphi_{3})=(\varphi_{\mathfrak{b}},-0.90,-0.90)$.
(a): $\psi_{\mathrm{r}}$ and $\psi_{\mathrm{i}}$, and (b): a shape of the
excited state of elastica.

The second result is presented in Figure 3. We used the hyperelliptic curve
given by $(k_{1},k_{2},k_{3})=(1.0260,1.0259,1.0008)$. The initial condition
is given by
$(\varphi_{1},\varphi_{2},\varphi_{3})=(\varphi_{\mathfrak{b}},0.0,0.0)$.
Figure 3 (a) shows the profile of $\psi_{\mathrm{r}}$ and $\psi_{\mathrm{i}}$
and (b) shows the shape of the excited state of elastica. The orbit could be
considered as an approximation of the MKdV equation (1.1) rather than the
measured MKdV equation (3.4). Figures 3 (e) and (f) show the respective parts
of $\partial_{u_{3}}\varphi_{i,{\mathrm{r}}}$ and
$\partial_{u_{3}}\varphi_{i,{\mathrm{i}}}$.

(a) (b)

Figure 3.  An open excited state of elastica:
$(k_{1},k_{2},k_{3})=(1.0260,1.0259,1.0008)$
$(\varphi_{1},\varphi_{2},\varphi_{3})=(\varphi_{\mathfrak{b}},0.0,0.0)$ (a)
is the profile of $\psi_{\mathrm{r}}$ and $\psi_{\mathrm{i}}$ and (b) is the
shape of the excited state of elastica.

(a) (b)

(c)

Figure 4.  An open curve of excited state of elastica and shape of supercoiled
DNA: $(k_{1},k_{2},k_{3})=(6.00,7.00,8.00)$, and the initial condition is
$(\varphi_{1},\varphi_{2},\varphi_{3})=(\pi-\varphi_{\mathfrak{b}},1.4,1.4)$.
(a) is the profile of $\psi_{\mathrm{r}}$ and $\psi_{\mathrm{i}}$, (b) is a
shape of the excited state of elastica, and (c) is the shape of a supercoiled
DNA, which is a part of the AFM images in [21, Figure 4].

Similarly, we can use ${\sqrt{-1}}ds$ instead of $ds$ and consider the case in
Figure 1 (b). Since replacing $ds$ with ${\sqrt{-1}}ds$ does not seriously
affect the MKdV equation (1.1) by replacing $(dt,\alpha)$ with
$(-{\sqrt{-1}}dt,-\alpha)$, we can use the same algorithm to compute the shape
of the excited state of elastica by considering ${\sqrt{-1}}K_{a}$ instead of
$K_{a}$ for Figure 1 (b). The result is Figure 4. Figure 4 (a) and (b) show
profile of $\psi_{\mathrm{r}}$ and $\psi_{\mathrm{i}}$, and the shape of the
excited state of elastica with parameters
$(k_{1},k_{2},k_{3})=(6.00,7.00,8.00)$ and initial condition
$(\varphi_{1},\varphi_{2},\varphi_{3})=(\pi-\varphi_{\mathfrak{b}},1.4,1.4)$.
Due to $\psi_{{\mathrm{r}}}$ and $\psi_{{\mathrm{i}}}$ in Figure 4 (a), when
$\partial_{u_{3}}\psi_{{\mathrm{i}}}$ is constant, $\psi_{{\mathrm{r}}}$ is a
solution of the MKdV equation (1.1). The third example also consists of
figure-eight and the inverse of the figure ‘S’, a repetition of the modulation
of the figure-eight and the inverse ‘S’, as in Figure 4 (b). We call it the
S-eight mode.

In other words, we conclude that our model of the statistical mechanics of
elastica shows that in the excited states of elastica due to thermal effect,
we find the S-eight mode. This fact has never been reported anywhere.

On the other hand, in the shape of the supercoiled DNAs in Figure 4(c) [21,
Figure 4], it is surprising that we can find the similar shape, a repetition
of the modulation of the figure-eight and the inverse ‘S’, or the S-eight
mode. In their study of a new type of highly ordered DNA organization,
Japaridze et al.ĩnvestigated the conformation of DNA both experimentally and
theoretically. They found the new type of order called ”hyperplectonemes”. The
shape in the figure is shown as the case that has fewer hyperplectonemes. In
fact, in the case there are similar shapes whose parts have figure-eight and
inverse S shapes, or the S-eight mode. After the study, some of the authors
found a parameter which governs the order and controls the degree of order in
[35]. Then we also find the shapes similar to Figure 4 (c) in [35, Figure 7
(c)], whose part is expressed by the modulation of figure-eight and inverse
‘S’, as in Figure 4 (a).

We emphasize that except for the figure-eight given by Euler in 1744, which is
found as short closed supercoiled DNAs, e.g. in [38], no one has ever
mathematically reproduced any shape of supercoiled DNA with voids. Thus, this
demonstration provides the first step towards the mathematical representation
of the conformations of supercoiled DNA. In other words, the excited states of
elastica of genus three of the statistical mechanics of elastica reproduces
the geometric property of supercoiled DNA. The shapes of supercoiled DNA are
not rigid, but generally obey weak elastic forces.

Since the MKdV equation preserves the Euler-Bernoulli energy
$\displaystyle{\int(\partial_{s}\psi)^{2}ds}$, it can be considered an excited
state of elasticity rather than a ground state or minimum energy point. This
means that we are beginning to go beyond the form of Euler’s elastica,
including thermal effects. [29].

Thus, we conclude our model of the statistical mechanics of elastica to
express a certain class of supercoiled DNA shapes observed in the laboratory.
The S-eight mode exits in the class of the shapes of the supercoiled DNAs. It
shows that there is a deep and beautiful relationship between nature (life
sciences) and algebraic geometry in mathematics.

##  6\. Conclusion

In this paper, we employed a novel algebro-geometric method in [30] to obtain
the solutions of the excited states of elastica of the gauged MKdV equation
(3.4) as in Proposition 4.3. Theorem 4.4 shows that they can also be regarded
as approximate solutions of the MKdV equation (1.1) if
$\partial_{u_{3}}\psi_{{\mathrm{i}}}$ is approximately constant. Based on
them, we provided a concrete algorithm to obtain the numerical solutions of
the excited states of elastica in Section 5. We demonstrate typical
conformations of the excited states of elastica by numerical computations,
which no one ever draw such a complicated shape.

We found that there is a mode, a repetition of the modulation of the figure-
eight and the inverse ‘S’. We call it the S-eight mode. It means that our
model of the statistical mechanics of elastica shows that the excited states
of elastica due to thermal effect has the S-eight mode. Then we found that
there exist the shapes with the S-eight mode in the AFM image of the
supercoiled DNAs in [21, Figure 4]. We conclude that our model expresses a
certain class of the shape of supercoiled DNA. Only the shape of the
supercoiled DNA related to figure-eight and circle of Euler’s has been
expressed by theoretically and numerically. Though the more complicated ones
with void has never obtained, this paper shows such a shape by the algebro-
geometric method.

There are so many mathematical models of algebraic geometry related to
elementary particle physics and string theory. This paper shows that there is
a deep and beautiful relationship between the biophysics of DNAs and algebraic
geometry as mentioned in [29].

However, while the results of this paper are certainly novel and intriguing, a
number of issues need to be resolved in the future. Though we have computed
it, the behavior of the imaginary part $\partial_{u_{3}}\psi_{{\mathrm{i}}}$
is unclear. The behavior should be considered more precisely in the future to
find the solutions of the MKdV equation. Based on the knowledge, we should
find the shapes of the excited states of elastica with the higher genus $g>3$.

Furthermore, we should find the hyperelliptic solutions of the NLS equation
beyond [36] to obtain the excited states of elastica in ${\mathbb{R}}^{3}$ as
in [27].

Acknowledgment: This project was started with Emma Previato 2004 in Montreal
and had been collaborated until she passed away June 29, 2022. Though the
author started to step to genus three curves without her, he appreciate her
contributions and suggestions which she gave him to this project during her
lifetime. Further, it is acknowledged that John McKay who passed way April
2022 invited the author and her to his private seminar in Montreal 2004 since
he considered that this project [26] must have been related to his Monster
group problem [25, 33]. This problem might be related to Witten conjecture of
loop space via the sigma function [33]. Thus this study is devoted to Emma
Previato and John McKay. The author thanks to Junkichi Satsuma, Takashi
Tsuboi, and Tetsuji Tokihiro for inviting him to the Musashino Center of
Mathematical Engineering Seminar and for valuable discussions and to Yuta
Ogata, Yutaro Kabata and Kaname Matsue for helpful discussions and
suggestions. He is also grateful to Aleksandre Japaridze, Giovanni Longo, and
Giovanni Dietler, the authors of [21] for helpful comments on Figure 4 in [21]
and sending him its follow-up interesting article [35]. He also acknowledges
support from the Grant-in-Aid for Scientific Research (C) of Japan Society for
the Promotion of Science, Grant No.21K03289.

## References

  * [1] M. J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, SIAM 1981.
  * [2] M. R. Adams, J. Harnad, and E. Previato, Isospectal Hamiltonian Flows in Finite and Infinite Dimensions, Comm. Math. Phys., (1988) 117 451-500.
  * [3] H.F. Baker, Abelian functions, Cambridge Univ. Press, Cambridge, 1995, Reprint of the 1897 original.
  * [4] C. Bouchiat and M. Mézard, Elastic rod model of a supercoiled DNA molecule, Eur. Phys. J. E 2 (2000) 377-402.
  * [5] J. -L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics 107, Birkhäuser, Boston, 1993\.
  * [6] T. Brouns et al.,  Free Energy Landscape and Dynamics of Supercoiled DNA by High-Speed Atomic Force Microscopy, ACS Nano. 2018 Dec 26;12(12):11907-11916.
  * [7] V.M. Buchstaber, V.Z. Enolskiĭ and D.V. Leĭkin, Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math. Math. Phys., 10 (1997) 1–103.
  * [8] V. M. Buchstaber, V. Z. Enolski, and D. V. Leykin, $\sigma$ functions: old and new results, in Integrable systems and algebraic geometry vol.2, edited by R. Donagi and T. Shaska, London Math Soc. Lect. Note Series 459 (2020) 175–214.
  * [9] V. M. Buchstaber and D. V. Leykin, Heat equations in a nonholonomic frame, Funct. Anal. Appl., 38 (2004) 88–101.
  * [10] C. R. Calladine, H. Drew, B. Luisi and A. Travers, Understanding DNA, 3rd ed., Academic Press, 2004.
  * [11] M. Doi S. F. Edwards, The Theory of Polymer Dynamics (International Series of Monographs on Physics), Clarendon Press, 1988.
  * [12] L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, 1744.
  * [13] J. C. Eilbeck, V. Z. Enol’skii, S. Matsutani, Y. Ônishi, and E. Previato, Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties, J. Reine Angew Math. , 619 (2008) 37–48.
  * [14] V. Enolski, B. Hartmann, V. Kagramanova, J. Kunz, C. Lämmerzahl, P. Sirimachan,  Inversion of a general hyperelliptic integral and particle motion in Ho?ava?Lifshitz black hole space-times, J. Math. Phys., 53 (2012) 012504.
  * [15] V. Z. Enolskii, S. Matsutani and Y. Ônishi, The addition law attached to a stratification for a hyperelliptic Jacobian variety, Tokyo J. Math., 31 (2008) 27–38.
  * [16] H. M. Farkas and I. Kra, Riemann Surfaces (GTM 71), Springer-Verlag, New York, 1991.
  * [17] D. Frenkel,  Statistical Mechanics of Liquid Crystals, in Liquids, Freezing and the Glass Transition, Volume Part II (Les Houches) , ed. by J. -P. Hansen, D. Levesque and J. Zinn-Justin, North-Holland, 1991. pp. 691-762
  * [18] R. E. Goldstein and D. M. Petrich, The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett., 67 (1991) 3203–3206.
  * [19] S. Goyal et al., Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables, J. Comp. Phys., 209 (2005) 371.
  * [20] C. Itzykson, J.-M. Drouffe, Statistical Field Theory: Vol. 2, Cambridge Univ., 1989.
  * [21] A. Japaridze, et al., Hyperplectonemes: A Higher Order Compact and Dynamic DNA Self-Organization, Nano Lett., 17 3, (2017) 1938–1948
  * [22] J. Komeda, S. Matsutani, E. Previato, , Algebraic construction of the sigma function for general Weierstrass curves, Mathematics (MDPI) , 10 (16) (2022) 10, 32.
  * [23] E. A. Kümmerle, and E. Pomplun, A computer-generated supercoiled model of the pUC19 plasmid, Eur Biophys J., 34 (2005) 13.
  * [24] Y. L. Lyubchenko and L. S. Shlyakhtenko, Visualization of supercoiled DNA with atomic force microscopy in situ, Proc. Natl. Acad. Sci. USA, 94 (1997) 496.
  * [25] J. McKay and Y-H. He, Kashiwa lectures on new approaches to the Monster, Notices, Int. Cons. Chinese Math., 10 (2022) 71-88.
  * [26] S. Matsutani, Statistical mechanics of elastica on a plane, J. Phys. A: Math. & Gen., 31 (1998) 2705.
  * [27] S. Matsutani, Statistical mechanics of non-stretching elastica in three dimensional space, J. Geom. Phys., 29 (1999) 243.
  * [28] S. Matsutani, Hyperelliptic loop solitons with genus $g$: investigation of a quantized elastica, , J. Geom. Phys., 43 (2002) 146.
  * [29] S. Matsutani, Euler’s Elastica and Beyond, J. Geom. Symm. Phys, 17 (2010) 45–86.
  * [30] S. Matsutani, On real hyperelliptic solutions of focusing modified KdV equation, arXiv:2309.04904,
  * [31] S. Matsutani, The Weierstrass sigma function in higher genus and applications to integrable equations, to appear as ‘Monographs in Mathematics’ Springer 2024, (p.484).
  * [32] S. Matsutani and Y. Ônishi, On the moduli of a quantized elastica in ${\mathbb{P}}$ and KdV flows: study of hyperelliptic curves as an extension of Euler’s perspective of elastica I, Rev. Math. Phys., 15 (2003) 559–628.
  * [33] S. Matsutani and E. Previato, From Euler’s elastica to the mKdV hierarchy, through the Faber polynomials, J. Math. Phys., 57 (2016) 081519.
  * [34] S. Matsutani and E. Previato, An algebro-geometric model for the shape of supercoiled DNA, Physica D, 430 (2022) 133073.
  * [35] N T. Odermatta, et.al., Structural and DNA binding properties of mycobacterial integration host factor mIHF, J. Struct. Biology, 209 (2020) 107434.
  * [36] E. Previato,  Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation, Duke Math. J., 52 (1985) 329-377.
  * [37] E. Previato,  Geometry of the modified KdV equation, in  LNP 424: Geometric and quantum aspects of integrable systems Ed. by G. F. Helminck, Springer 1993, 43-65.
  * [38] A. L. B. Pyne et al.  Base-pair resolution analysis of the effect of supercoiling on DNA flexibility and major groove recognition by triplex-forming oligonucleotides, Nature Comm. 12 (2021) 1053.
  * [39] P. Ramond, Field Theory: A Modern Primer, 2nd ed., Addison-Wesley, 1989.
  * [40] J. Schwinger, On Gauge Invariance and Vacuuwn Polarization, Phys. Rev., 82 (1951) 664-678
  * [41] D. Swigon, et.al., The Elastic Rod Model for DNA and Its Application to the Tertiary Structure of DNA Minicircles in Mononucleosomes Biophysical J. 74 (1998) 2515.
  * [42] H. Tsuru and M. Wadati, Elastic Model of Highly Supercoiled DNA, Biopolymers, 25 (1986) 2083.
  * [43] D Voet and J. G. Voet Fundamentals of Biochemistry: Life at the Molecular Level, 4th ed. Wiley 2015.
  * [44] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surv. Diff. Geom., 1 (1991) 243–310


2^{-(s_{0}-1-\sigma)m}\|{b-\overline{b}}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}}\Big{(}\sum_{k}\Big{(}2^{s_{0}k}\|{P_{k}b}\|_{\ell^{1}_{\mathcal{I}}X_{k}}\sum_{j<k-m}2^{-(s_{0}-\frac{7}{2})j}\Big{)}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim
2^{-(s_{0}-1-\sigma)m}M\|{b-\overline{b}}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}}.$

For the second term, we use (6.20) to estimate

$\displaystyle\|{\nabla\times\sum_{k}P_{[k-m,k-10)}(b-\overline{b})\times(\nabla\times
P_{k}b)}\|_{\ell^{1}_{\mathcal{I}}Y^{\sigma}}$ $\displaystyle\lesssim
T\Big{(}\sum_{k}\Big{(}2^{(\sigma+\frac{7}{2})k}\sum_{j\in[k-m,k-10)}\|{P_{j}(b-\overline{b})}\|_{\ell^{1}_{\mathcal{I}}X_{j}}\|{P_{k}b}\|_{\ell^{1}_{\mathcal{I}}X_{k}}\Big{)}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim T2^{\sigma
m}\|{b-\overline{b}}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}}\Big{(}\sum_{k}\Big{(}2^{-(s_{0}-\frac{7}{2})k}2^{s_{0}k}\|{P_{k}b}\|_{\ell^{1}_{\mathcal{I}}X_{k}}\Big{)}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim T2^{\sigma
m}M_{0}\|{b-\overline{b}}\|_{\ell^{1}_{\mathcal{I}}X^{\sigma}}.$

Choosing $m$ so that $T=2^{-(s_{0}-1)m}$, the desired estimate follows for
$\nabla\times(T_{b-\overline{b}}\times(\nabla\times b))$.

The remaining term $\nabla\times(T_{\nabla\times(b-\overline{b})}\times b)$ is
handled similarly (in fact, the numerology is more favorable compared to the
previous case since $\nabla\times$ now falls on $b-\overline{b}$) and will be
omitted. ∎

###### Proof of (5.11).

The desired bounds for the paraproduct terms
$\nabla\times(T_{b}\times(\nabla\times b))$ and $\nabla\times(T_{\nabla\times
b}\times b)$ follow by writing out these expressions, then estimating the
first (i.e., low frequency) factor of $b$ in $L^{\infty}$ (to which
$\ell^{1}_{\mathcal{I}}H^{s_{1}}$ embeds) and the second (i.e., high
frequency) factor of $b$ in $\ell^{1}_{\mathcal{I}}H^{s_{1}}$. For $G(b)$, we
use (7.66) and bound (using Littlewood–Paley theory, Corollary 6.2 and
Hölder),

$\displaystyle\|{\nabla\times\sum_{k}P_{[k-10,k+10]}b\times(\nabla\times
P_{k}b)}\|_{\ell^{1}_{\mathcal{I}}H^{s_{1}-2}}$
$\displaystyle\lesssim\sum_{\ell}\sum_{k:k>\ell-20}2^{(s_{1}-1)\ell}\|{P_{\ell}(P_{[k-10,k+10]}b\times(\nabla\times
P_{k}b))}\|_{\ell^{1}_{\mathcal{I}}L^{2}}$
$\displaystyle\lesssim\sum_{\ell}\sum_{k:k>\ell-20}2^{(s_{1}-1)\ell}2^{\frac{3}{2}\ell}2^{k}\|{P_{[k-10,k+10]}b}\|_{\ell^{1}_{\mathcal{I}}L^{2}}\|{P_{k}b}\|_{\ell^{1}_{\mathcal{I}}L^{2}}$
$\displaystyle\lesssim\|{b}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}}^{2}\sum_{\ell}\sum_{k:k>\ell-20}2^{(s_{1}+\frac{1}{2})\ell}2^{-(2s_{1}-1)k}\lesssim\|{b}\|_{\ell^{1}_{\mathcal{I}}X^{s_{1}}}^{2},$

where we used $s_{1}>\frac{1}{2}$ in the last inequality. ∎

## References

  * [1] Marion Acheritogaray, Pierre Degond, Amic Frouvelle, and Jian-Guo Liu, _Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system_ , Kinet. Relat. Models 4 (2011), no. 4, 901–918. MR 2861579
  * [2] Timur Akhunov, _A sharp condition for the well-posedness of the linear KdV-type equation_ , Proc. Amer. Math. Soc. 142 (2014), no. 12, 4207–4220. MR 3266990
  * [3] David M. Ambrose, Gideon Simpson, J. Douglas Wright, and Dennis G. Yang, _Ill-posedness of degenerate dispersive equations_ , Nonlinearity 25 (2012), no. 9, 2655–2680. MR 2967120
  * [4] David M. Ambrose and J. Douglas Wright, _Dispersion vs. anti-diffusion: well-posedness in variable coefficient and quasilinear equations of KdV type_ , Indiana Univ. Math. J. 62 (2013), no. 4, 1237–1281. MR 3179690
  * [5] Hantaek Bae and Kyungkeun Kang, _On the existence and temporal asymptotics of solutions for the two and half dimensional Hall MHD_ , J. Math. Fluid Mech. 25 (2023), no. 2, Paper No. 24, 30. MR 4547410
  * [6] D. Biskamp, E. Schwarz, and J. F. Drake, _Two-fluid theory of collisionless magnetic reconnection_ , Physics of Plasmas 4 (1997), no. 4, 1002–1009.
  * [7] Dongho Chae, Pierre Degond, and Jian-Guo Liu, _Well-posedness for Hall-magnetohydrodynamics_ , Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, 555–565. MR 3208454
  * [8] Dongho Chae, In-Jee Jeong, and Sung-Jin Oh, _Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities_ , 2023.
  * [9] Dongho Chae and Jihoon Lee, _On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics_ , J. Differential Equations 256 (2014), no. 11, 3835–3858. MR 3186849
  * [10] Dongho Chae and Maria Schonbek, _On the temporal decay for the Hall-magnetohydrodynamic equations_ , J. Differential Equations 255 (2013), no. 11, 3971–3982. MR 3097244
  * [11] Dongho Chae and Shangkun Weng, _Singularity formation for the incompressible Hall-MHD equations without resistivity_ , Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 4, 1009–1022. MR 3519529
  * [12] Dongho Chae and Jörg Wolf, _On partial regularity for the 3D nonstationary Hall magnetohydrodynamics equations on the plane_ , SIAM J. Math. Anal. 48 (2016), no. 1, 443–469. MR 3455137
  * [13] by same author, _Regularity of the $3D$ stationary hall magnetohydrodynamic equations on the plane_, Comm. Math. Phys. 354 (2017), no. 1, 213–230. MR 3656516
  * [14] Hiroyuki Chihara, _Local existence for semilinear Schrödinger equations_ , Math. Japon. 42 (1995), no. 1, 35–51. MR 1344627
  * [15] Jungyeon Cho, Jungyeon Cho, and Alex Lazarian, _Simulations of electron magnetohydrodynamic turbulence_ , The Astrophysical Journal 701 (2009), 236 – 252.
  * [16] Walter Craig, Thomas Kappeler, and Walter Strauss, _Microlocal dispersive smoothing for the Schrödinger equation_ , Comm. Pure Appl. Math. 48 (1995), no. 8, 769–860. MR 1361016
  * [17] Mimi Dai, _Local well-posedness of the Hall-MHD system in $H^{s}(\mathbb{R}^{n})$ with $s>\frac{n}{2}$_, Math. Nachr. 293 (2020), no. 1, 67–78. MR 4060362
  * [18] by same author, _Local well-posedness for the Hall-MHD system in optimal Sobolev spaces_ , J. Differential Equations 289 (2021), 159–181. MR 4248458
  * [19] by same author, _Nonunique weak solutions in Leray-Hopf class for the three-dimensional Hall-MHD system_ , SIAM J. Math. Anal. 53 (2021), no. 5, 5979–6016. MR 4328510
  * [20] by same author, _Blowup of Dyadic MHD Models with Forward Energy Cascade_ , International Mathematics Research Notices 2023 (2022), no. 24, 21805–21837.
  * [21] by same author, _Global existence of 2D electron MHD near a steady state_ , preprint (2023).
  * [22] Mimi Dai and Susan Friedlander, _Uniqueness and Non-Uniqueness Results for Forced Dyadic MHD Models_ , J Nonlinear Sci 33 (2023), no. 10.
  * [23] Mimi Dai, Jacob Krol, and Han Liu, _On uniqueness and helicity conservation of weak solutions to the electron-MHD system_ , J. Math. Fluid Mech. 24 (2022), no. 3, Paper No. 69, 17. MR 4434207
  * [24] Mimi Dai and Han Liu, _Long time behavior of solutions to the 3d hall-magneto-hydrodynamics system with one diffusion_ , preprint (2017).
  * [25] Shin-ichi Doi, _Remarks on the Cauchy problem for Schrödinger-type equations_ , Comm. Partial Differential Equations 21 (1996), no. 1-2, 163–178. MR 1373768
  * [26] J. C. Dorelli, Alex Glocer, Glyn Collinson, and Gábor Tóth, _The role of the hall effect in the global structure and dynamics of planetary magnetospheres: Ganymede as a case study_ , Journal of Geophysical Research: Space Physics 120 (2015), no. 7, 5377–5392.
  * [27] James Drake, Robert G. Kleva, and M. E. Mandt, _Structure of thin current layers: Implications for magnetic reconnection._ , Physical review letters 73 9 (1994), 1251–1254.
  * [28] J. Dreher, V. Runban, and R. Grauer, _Axisymmetric flows in Hall-MHD: a tendency towards finite-time singularity formation_ , Phys. Scr. 72 (2005).
  * [29] Baoying Du, _Global regularity for the $2\frac{1}{2}$d incompressible Hall-MHD system with partial dissipation_, J. Math. Anal. Appl. 484 (2020), no. 1, 123701, 27. MR 4042622
  * [30] Jishan Fan, Shuxiang Huang, and Gen Nakamura, _Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations_ , Appl. Math. Lett. 26 (2013), no. 9, 963–967. MR 3069946
  * [31] S. Peter Gary and Charles W. Smith, _Short‐wavelength turbulence in the solar wind: Linear theory of whistler and kinetic alfvén fluctuations_ , Journal of Geophysical Research 114 (2009).
  * [32] Pierre Germain, Benjamin Harrop-Griffiths, and Jeremy L. Marzuola, _Existence and uniqueness of solutions for a quasilinear KdV equation with degenerate dispersion_ , Comm. Pure Appl. Math. 72 (2019), no. 11, 2449–2484. MR 4011864
  * [33] Peter Goldreich and Andreas Reisenegger, _Magnetic Field Decay in Isolated Neutron Stars_ , Astrophysical Journal 395 (1992), 250.
  * [34] K. N. Gourgouliatos and A. Cumming, _Hall effect in neutron star crusts: evolution, endpoint and dependence on initial conditions_ , Monthly Notices of the Royal Astronomical Society 438 (2013), no. 2, 1618–1629.
  * [35] E. H. Hall, _On a New Action of the Magnet on Electric Currents_ , Amer. J. Math. 2 (1879), no. 3, 287–292. MR 1505227
  * [36] Benjamin Harrop-Griffiths and Jeremy L. Marzuola, _Local well-posedness for a quasilinear Schrödinger equation with degenerate dispersion_ , Indiana Univ. Math. J. 71 (2022), no. 4, 1585–1626. MR 4481094
  * [37] Nakao Hayashi and Tohru Ozawa, _Remarks on nonlinear Schrödinger equations in one space dimension_ , Differential Integral Equations 7 (1994), no. 2, 453–461. MR 1255899
  * [38] Holger Homann and Rainer Grauer, _Bifurcation analysis of magnetic reconnection in Hall-MHD-systems_ , Phys. D 208 (2005), no. 1-2, 59–72. MR 2167907
  * [39] Mihaela Ifrim and Daniel Tataru, _Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension_ , Nonlinearity 28 (2015), no. 8, 2661–2675. MR 3382579
  * [40] Juhi Jang and Nader Masmoudi, _Derivation of Ohm’s law from the kinetic equations_ , SIAM J. Math. Anal. 44 (2012), no. 5, 3649–3669. MR 3023426
  * [41] Eunji Jeong, Junha Kim, and Jihoon Lee, _Local well-posedness and blow-up for the solutions to the axisymmetric inviscid Hall-MHD equations_ , Adv. Math. Phys. (2018), Art. ID 5343824, 16. MR 3875724
  * [42] In-Jee Jeong and Sung-Jin Oh, _On illposedness of the hall and electron magnetohydrodynamic equations without resistivity on the whole space_.
  * [43] by same author, _On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity II: geometric conditions ensuring wellposedness_.
  * [44] by same author, _On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: illposedness near degenerate stationary solutions_ , Annals of PDE 8 (2019).
  * [45] In-Jee Jeong and Sung-Jin Oh, _Illposedness for dispersive equations: Degenerate dispersion and Takeuchi–Mizohata condition_ , 2023.
  * [46] P. B. Jones, _Neutron star magnetic field decay - Hall drift and Ohmic diffusion_ , Monthly Notices of the Royal Astronomical Society 233 (1988), 875–885.
  * [47] C. E. Kenig, G. Ponce, C. Rolvung, and L. Vega, _Variable coefficient Schrödinger flows for ultrahyperbolic operators_ , Adv. Math. 196 (2005), no. 2, 373–486. MR 2166312
  * [48] Carlos E. Kenig, Gustavo Ponce, Christian Rolvung, and Luis Vega, _The general quasilinear ultrahyperbolic Schrödinger equation_ , Adv. Math. 206 (2006), no. 2, 402–433. MR 2263709
  * [49] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, _Small solutions to nonlinear Schrödinger equations_ , Ann. Inst. H. Poincaré C Anal. Non Linéaire 10 (1993), no. 3, 255–288. MR 1230709
  * [50] by same author, _Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations_ , Invent. Math. 134 (1998), no. 3, 489–545. MR 1660933
  * [51] by same author, _The Cauchy problem for quasi-linear Schrödinger equations_ , Invent. Math. 158 (2004), no. 2, 343–388. MR 2096797
  * [52] M. J. Lighthill, _Studies on magneto-hydrodynamic waves and other anisotropic wave motions_ , Philos. Trans. Roy. Soc. London Ser. A 252 (1960), 397–430. MR 0148337
  * [53] Wee Keong Lim and Gustavo Ponce, _On the initial value problem for the one dimensional quasi-linear Schrödinger equations_ , SIAM J. Math. Anal. 34 (2002), no. 2, 435–459. MR 1951782
  * [54] Jeremy L. Marzuola, Jason Metcalfe, and Daniel Tataru, _Quasilinear Schrödinger equations I: Small data and quadratic interactions_ , Adv. Math. 231 (2012), no. 2, 1151–1172. MR 2955206
  * [55] by same author, _Quasilinear Schrödinger equations, II: Small data and cubic nonlinearities_ , Kyoto J. Math. 54 (2014), no. 3, 529–546. MR 3263550
  * [56] by same author, _Quasilinear Schrödinger equations III: Large data and short time_ , Arch. Ration. Mech. Anal. 242 (2021), no. 2, 1119–1175. MR 4331023
  * [57] Laura F. Morales, Sergio Dasso, Daniel O. Gómez, and Pablo Mininni, _Hall effect on magnetic reconnection at the earth’s magnetopause_ , Journal of Atmospheric and Solar-Terrestrial Physics 67 (2005), no. 17, 1821–1826, Space Geophysics.
  * [58] Hans Pécseli, _Waves and oscillations in plasmas_ , Series in Plasma Physics (2012).
  * [59] Yi Peng, Huaqiao Wang, and Qiuju Xu, _Derivation of the Hall-MHD equations from the Navier-Stokes-Maxwell equations_ , J. Nonlinear Sci. 32 (2022), no. 6, Paper No. 90, 27. MR 4491095
  * [60] Ben Pineau and Mitchell A. Taylor, _Low regularity solutions for the general Quasilinear ultrahyperbolic Schrödinger equation_ , preprint (2023).
  * [61] Mohammad Mahabubur Rahman and Kazuo Yamazaki, _Remarks on the global regularity issue of the two-and-a-half-dimensional Hall-magnetohydrodynamics system_ , Z. Angew. Math. Phys. 73 (2022), no. 5, Paper No. 217, 29. MR 4490634
  * [62] Y. Raitses and N. J. Fisch, _Parametric investigations of a nonconventional Hall thruster_ , Physics of Plasmas 8 (2001), no. 5, 2579–2586.
  * [63] Philip Rosenau, _What is $\dots$a compacton?_, Notices Amer. Math. Soc. 52 (2005), no. 7, 738–739. MR 2159688
  * [64] Philip Rosenau and James M. Hyman, _Compactons: Solitons with finite wavelength_ , Physical Review Letters 70 (1993), no. 5, 564–567.
  * [65] Philip Rosenau and Steven Schochet, _Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit_ , Chaos 15 (2005), no. 1, 015111, 18. MR 2133462
  * [66] Philip Rosenau and Alon Zilburg, _On a strictly compact discrete breathers in a Klein-Gordon model_ , Phys. Lett. A 379 (2015), no. 43-44, 2811–2816. MR 3402749
  * [67] M. A. Shay, J. F. Drake, B. N. Rogers, and R. E. Denton, _Alfvénic collisionless magnetic reconnection and the hall term_ , Journal of Geophysical Research: Space Physics 106 (2001), no. A3, 3759–3772.
  * [68] Artem Smirnov, Yegeny Raitses, and Nathaniel J. Fisch, _Experimental and theoretical studies of cylindrical Hall thrustersa)_ , Physics of Plasmas 14 (2007), no. 5, 057106.
  * [69] Elias M. Stein, _Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals_ , Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 1232192
  * [70] Michael E. Taylor, _Pseudodifferential operators and nonlinear pde_ , Progress in Mathematics, Birkhäuser Boston, MA, 1991.
  * [71] Renhui Wan and Yong Zhou, _On global existence, energy decay and blow-up criteria for the Hall-MHD system_ , J. Differential Equations 259 (2015), no. 11, 5982–6008. MR 3397315
  * [72] by same author, _Global well-posedness, BKM blow-up criteria and zero $h$ limit for the 3D incompressible Hall-MHD equations_, J. Differential Equations 267 (2019), no. 6, 3724–3747. MR 3955610
  * [73] C J Wareing and Rainer Hollerbach, _Forward and inverse cascades in decaying two-dimensional electron magnetohydrodynamic turbulence_ , Physics of Plasmas 16 (2009), 042307.
  * [74] Toby S. Wood, Rainer Hollerbach, and Maxim Lyutikov, _Density-shear instability in electron magneto-hydrodynamics_ , Physics of Plasmas 21 (2014), 052110.
  * [75] Kazuo Yamazaki, _Stochastic Hall-magneto-hydrodynamics system in three and two and a half dimensions_ , J. Stat. Phys. 166 (2017), no. 2, 368–397. MR 3596853
  * [76] Rtimi Youness, Maxime Bonnet, Frédéric Messine, and Carole Hénaux, _Topology shape and parametric design optimization of Hall effect thrusters_ , Scientific computing in electrical engineering, Math. Ind., vol. 32, Springer, Cham, [2020] ©2020, pp. 255–264. MR 4214205
  * [77] Yong Zeng, _Steady states of Hall-MHD system_ , J. Math. Anal. Appl. 451 (2017), no. 2, 757–793. MR 3624766