eurm10 msam10

# Density jump as a function of magnetic field for switch-on collisionless
shocks in pair plasmas

Antoine Bret1,2 Ramesh Narayan3,4 Email address for correspondence:
<EMAIL_ADDRESS>1ETSI Industriales, Universidad de Castilla-La
Mancha, 13071 Ciudad Real, Spain
2Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus
Universitario de Ciudad Real, 13071 Ciudad Real, Spain
3Harvard-Smithsonian Center for Astrophysics, Harvard University, 60 Garden
St., Cambridge, MA 02138 USA
4Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA
02138, USA

(?; revised ?; accepted ?. - To be entered by editorial office)

###### Abstract

The properties of collisionless shocks, like the density jump, are usually
derived from magnetohydrodynamics (MHD), where isotropic pressures are
assumed. Yet, in a collisionless plasma, an external magnetic field can
sustain a stable anisotropy. In Bret & Narayan (2018), we devised a model for
the kinetic history of the plasma through the shock front, allowing to self-
consistently compute the downstream anisotropy, hence the density jump, in
terms of the upstream parameters. This model dealt with the case of a parallel
shock, where the magnetic field is normal to the front both in the upstream
and the downstream. Yet, MHD also allows for shock solutions, the so-called
switch-on solutions, where the field is normal to the front only in the
upstream. This article consists in applying our model to these switch-on
shocks. While MHD offers only 1 switch-on solution within a limited range of
Alfvén Mach numbers, our model offers 2 kinds of solutions within a slightly
different range of Alfvén Mach numbers. These 2 solutions are most likely the
outcome of the intermediate and fast MHD shocks under our model. While the
intermediate and fast shocks merge in MHD for the parallel case, they do not
within our model. For simplicity, the formalism is restricted to non-
relativistic shocks in pair plasmas where the upstream is cold.

##  1 Introduction

Shock waves are fundamental processes in plasmas which are usually studied
within the context of magnetohydrodynamics (MHD). As an extension of fluid
dynamics to plasmas, MHD entails the same assumption of small mean-free-path
(see for example Gurnett & Bhattacharjee (2005) §5.4.4, Goedbloed et al.
(2010) chapters 2 & 3, or Thorne & Blandford (2017) §13.2). When fulfilled,
collisions ensure that the pressure is isotropic both in the upstream and
downstream, which simplifies the conservation equations.

In collisionless shock, where the mean-free-path is larger than the size of
the system, the isotropy assumption may not be fulfilled, possibly resulting
in a departure from the MHD predicted behavior. Such is especially the case in
the presence of an external magnetic field which can stabilize a temperature
anisotropy, as has been observed in the solar wind (Bale et al., 2009; Maruca
et al., 2011; Schlickeiser et al., 2011) and is projected to be studied in the
laboratory (Carter et al., 2015).

Some authors worked out the MHD conservation equations in the case of
anisotropic pressure, and studied the consequences on the shock properties
(Erkaev et al., 2000; Double et al., 2004; Gerbig & Schlickeiser, 2011). Yet,
in these works, while the upstream is assumed isotropic, the downstream degree
of anisotropy is left as a free parameter.

Recently, a self-contained theory of magnetized collisionless shocks has been
developed. By making some assumptions on the kinetic history of the plasma as
it crosses the front, we could compute the downstream degree of anisotropy,
for the parallel and the perpendicular cases, in terms of the magnetic field
strength (Bret & Narayan, 2018, 2019, 2020).

Noteworthily, the theory for parallel shocks described in Bret & Narayan
(2018) has been successfully tested against Particle-In-Cell (PIC) simulations
in Haggerty et al. (2022).

In MHD, several shock solutions exist when the upstream magnetic field is
aligned with the flow. The most common solution is the one where the
downstream field is also aligned with the flow. This is the fully parallel
case, where the fluid and the field are decoupled (Lichnerowicz, 1976;
Majorana & Anile, 1987). Yet, still for the case where the upstream field is
parallel to the flow, MHD offers a second option: the switch-on shocks
(Fitzpatrick, 2014; Kulsrud, 2005; Goedbloed et al., 2010). In such shocks,
while the magnetic field does not have any components along the shock front in
the upstream, it has one in the downstream. Indeed, the MHD conservation
equations only enforce the continuity of the field component perpendicular to
the front, not the continuity of the normal component. Therefore, they allow
for solutions, the switch-on solutions, where the upstream field is normal to
the front while the downstream field is not.

The theory developed in Bret & Narayan (2018) was the collisionless version of
the fully parallel MHD case. The present article deals with the collisionless
version of the MHD switch-on shocks.

As in Bret & Narayan (2018), we consider, for simplicity, pair plasmas for
which both species have the same perpendicular and parallel temperatures to
the field. In Section 2, we remind the MHD results for switch-on shocks. In
Section 3, we explain the method used. It significantly differs from Bret &
Narayan (2018) since we need to account for an oblique downstream field. In
addition, MHD results suggest the obliquity of the downstream field, labelled
$\theta_{2}$ in the sequel, can be as high as $0.56\frac{\pi}{2}$ (see Fig.
2-left). We cannot therefore work out a theory restricted to
$\theta_{2}=\varepsilon$, with $0<\varepsilon\ll 1$. Then, in Sections 4 & 5,
we explain the solutions found for switch-on shocks within our model.

Figure 1: System considered. The upstream has density $n_{1}$ and isotropic
temperature $T_{1}$. Both the upstream field $\mathbf{B}_{1}$ and velocity
$\mathbf{v}_{1}$ are normal to the front. The downstream has density $n_{2}$
and temperatures $T_{2\parallel},T_{2\perp}$, parallel and perpendicular to
the downstream field. The downstream field $\mathbf{B}_{2}$ and velocity
$\mathbf{v}_{2}$ make an angle $\theta_{2}$ and $\xi_{2}$ respectively with
the front normal. The parallel and perpendicular directions are therefore
defined with respect to the local magnetic field.

##  2 MHD results

The system considered is sketched on Figure 1. The upstream field
$\mathbf{B}_{1}$ and velocity $\mathbf{v}_{1}$ are normal to the front, but
the downstream ones $\mathbf{B}_{2}$ and $\mathbf{v}_{2}$ are not. They make
an angle $\theta_{2}$ and $\xi_{2}$ with the shock normal and by default,
$\theta_{2}\neq\xi_{2}$ (even though they will be found equal in the sequel).

We here briefly remind the MHD theory for switch-on shocks. Due to the
complexity of the forthcoming calculations, we treat only the case of a sonic
strong shock, namely upstream temperature $T_{1}=0$, or equivalently, upstream
sonic Mach number $\mathcal{M}_{s1}=\infty$.

For isotropic pressures in the upstream and the downstream, and
$\theta_{1}=\xi_{1}=0$, the MHD conservation equations for strong shock and an
adiabatic index of $\gamma=5/3$ read (see for example Kulsrud (2005), p. 141),

$\displaystyle n_{2}v_{2}\cos\xi_{2}=$ $\displaystyle n_{1}v_{1},$ (1)
$\displaystyle B_{2}\cos\theta_{2}=$ $\displaystyle B_{1},$ (2) $\displaystyle
B_{2}v_{2}\sin\theta_{2}\cos\xi_{2}-B_{2}v_{2}\cos\theta_{2}\sin\xi_{2}=$
$\displaystyle 0,$ (3)
$\displaystyle\frac{B_{2}^{2}\sin^{2}\theta_{2}}{8\pi}+n_{2}k_{B}T_{2}+mn_{2}v_{2}^{2}\cos^{2}\xi_{2}=$
$\displaystyle mn_{1}v_{1}^{2},$ (4) $\displaystyle
mn_{2}v_{2}^{2}\sin\xi_{2}\cos\xi_{2}-\frac{B_{2}^{2}\sin\theta_{2}\cos\theta_{2}}{4\pi}=$
$\displaystyle 0,$ (5) $\displaystyle
mn_{2}v_{2}\cos\xi_{2}\left(\frac{5}{2}\frac{k_{B}T_{2}}{m}+\frac{B_{2}^{2}\sin^{2}\theta_{2}}{4\pi
mn_{2}}+\frac{v_{2}^{2}}{2}\right)-\frac{B_{2}^{2}}{4\pi}v_{2}\sin\theta_{2}\cos\theta_{2}\sin\xi_{2}=$
$\displaystyle\frac{1}{2}mn_{1}v_{1}^{3},$ (6)

where $m$ is the mass of the particles and $k_{B}$ the Boltzmann constant.

Eq. (1) stands for the conservation of mass. Eq. (2) for the conservation of
the magnetic field normal component. Eq. (3) for the vanishing of the $z$
component of the electric field. Eqs. (4,5) come from the conservation of the
momentum flux (see Appendix A), and Eq. (6) from the conservation of energy.

By eliminating $v_{2}$ and $B_{2}$ thanks to Eqs. (1,2), and then eliminating
$T_{2}$ thanks to Eq. (4), the system is amenable to 3 equations,

$\displaystyle\tan\theta_{2}-\tan\xi_{2}$ $\displaystyle=$ $\displaystyle 0,$
(7) $\displaystyle\mathcal{M}_{A1}^{2}\tan\xi_{2}-r\tan\theta_{2}$
$\displaystyle=$ $\displaystyle 0,$ (8) $\displaystyle
2\mathcal{M}_{A1}^{2}\left[(r-5)r+5-\sec^{2}\xi_{2}\right]+r\tan\theta_{2}\left(\tan\theta_{2}+4\tan\xi_{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ (9)

in terms of the dimensionless density ratio $r$ and the Alfvén Mach number
$\mathcal{M}_{A1}$,

$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{n_{2}}{n_{1}},$ (10)
$\displaystyle\mathcal{M}_{A1}^{2}$ $\displaystyle=$
$\displaystyle\frac{mn_{1}v_{1}^{2}}{B_{1}^{2}/4\pi}.$

The first equation imposes $\theta_{2}=\xi_{2}$. Replacing in the last 2
gives,

$\displaystyle\tan\theta_{2}\left(\mathcal{M}_{A1}^{2}-r\right)$
$\displaystyle=$ $\displaystyle 0,$ (11) $\displaystyle
2\mathcal{M}_{A1}^{2}\left[(r-5)r+5-\sec^{2}\theta_{2}\right]+5r\tan^{2}\theta_{2}$
$\displaystyle=$ $\displaystyle 0.$ (12)

Equation (11) clearly defines 2 kinds of shocks,

  * •

The first kind comes from $\tan\theta_{2}=0$, that is, $\theta_{2}=0$.
Inserting it into (12) gives $r=1$ or $r=4$. The first option, $r=1$ is the
continuity solution, where nothing changes between the upstream and the
downstream. The second option is the parallel shock solution, with $r=4$ for a
sonic strong shock and an adiabatic index $\gamma=5/3$.

  * •

Yet, (11) also allows for,

$r=\mathcal{M}_{A1}^{2},$ (13)

which is the MHD switch-on solution. Inserting it into (12) gives,

$\cos^{2}\theta_{2}=\frac{3}{10\mathcal{M}_{A1}^{2}-2\mathcal{M}_{A1}^{4}-5}.$
(14)

The value of $\theta_{2}$ so defined is displayed on Fig. 2-left.
$\theta_{2}\neq 0$ is only permitted within a finite range of Alfvén Mach
numbers defined by $\cos^{2}\theta_{2}<1$, that is, $1<\mathcal{M}_{A1}<2$.
Note that instead of parameterizing $\theta_{2}$ by the Alfvén Mach number
$\mathcal{M}_{A1}$, we choose the variable,

$\sigma=\frac{B_{1}^{2}/4\pi}{mn_{1}v_{1}^{2}}=\frac{1}{\mathcal{M}_{A1}^{2}}.$
(15)

This $\sigma$ parameter allows for a straightforward comparison with PIC
simulations where $\sigma$ is usually used instead of $\mathcal{M}_{A1}$ (see
for example Sironi & Spitkovsky (2011); Bret (2020)). As a function of
$\sigma$, $\theta_{2}\neq 0$ is allowed for $\sigma\in[1/4,1]$.

Figure 2: Left: Value of $\theta_{2}$ from Eq. (14) in terms of
$\sigma=\mathcal{M}_{A1}^{-2}$. Its maximum value is $\arccos\sqrt{2/5}\sim
0.56\frac{\pi}{2}$. Right: MHD density jump $r$ in terms of
$(\sigma,\theta_{1})$ for $\theta_{1}\in[0,\pi/2]$. Only the blue line, which
has $\theta_{1}=\theta_{2}=0$, was considered in Bret & Narayan (2018). The
red line is the switch-on solution (13), with $\theta_{1}=0$ but
$\theta_{2}\neq 0$.

For a finite upstream temperature $T_{1}>0$, MHD switch-on solutions are also
restricted to a range of upstream temperatures via
$\beta_{1}=n_{1}k_{B}T_{1}/B_{1}^{2}<2/\gamma$, where $\gamma$ is the
adiabatic index (Kennel et al., 1989; de Sterck & Poedts, 1999; Delmont &
Keppens, 2011)111See in particular Fig. 3 in de Sterck & Poedts (1999).. Since
the present work is limited to $T_{1}=0$, it cannot explore this dimension of
the switch-on solutions range.

While they have been produced in the laboratory (Craig & Paul, 1973), such
shocks have been rarely detected in space due to the smallness of the
parameter window that allows them. Feng et al. (2009) reported the detection
of a “possible” interplanetary switch-on shock. Also, Farris et al. (1994);
Russell & Farris (1995) reported the detection of one switch-on shock among
the ISEE222International Sun-Earth Explorer, see Ogilvie et al. (1977). data.
The more recent review of Balogh & Treumann (2013) still refers to Farris et
al. (1994) as “the rare case of observation of a switch-on shock” in its
§2.3.6.

Finally, it is interesting to compute the MHD density jump $r$ for any
upstream angle $\theta_{1}$. This can be done solving the “shock adiabatic”
equation given in Fitzpatrick (2014) §7.21, and setting $T_{1}=0$333$V_{S1}=0$
in the notation of Fitzpatrick (2014).. The result is pictured on Fig.
2-right, in terms of $\theta_{1}\in[0,\pi/2]$ and $\sigma$. Only the blue
line, which has $\theta_{1}=\theta_{2}=0$, was considered in Bret & Narayan
(2018). The red line is the switch-on solution (13), with $\theta_{1}=0$ but
$\theta_{2}\neq 0$.

##  3 Method

Our method to determine the downstream anisotropy in terms of the upstream
field relies on a monitoring of the kinetic history of the plasma as it
crosses the front. In this process, the parallel and perpendicular
temperatures of the plasma are changed according to some prescriptions
explained below. The resulting state of the plasma downstream is labelled
“Stage 1”. Stage 1 is generally _not_ isotropic.

Depending on the strength of the downstream field $\mathbf{B}_{2}$, Stage 1
can be stable or not. If it is stable, then Stage 1 is the end state of the
downstream. If it is unstable, then the plasma migrates towards its
instability threshold, namely mirror or firehose stability. This is “Stage 2”.
In such case, Stage 2 is the end state of the downstream.

Stage 1 and 2 are therefore temporal evolving stages of the downstream plasma.
This has been verified for the parallel case by the PIC simulations performed
by Haggerty et al. (2022), where the 2 stages have been clearly identified.

Also, the stability alluded here is not the one of the whole shock structure,
like for example the corrugation instability (Landau & Lifshitz (2013), §90).
It is rather the stability of the downstream plasma as an isolated and
homogenous entity.

This algorithm was applied to the parallel and perpendicular cases in Bret &
Narayan (2018) and Bret & Narayan (2019) respectively. In both cases, the
orientation of $\mathbf{B}_{2}$ makes it simple to set the temperatures of
Stage 1. We will now see that the obliquity of $\mathbf{B}_{2}$ demands
further characterization of Stage 1.

###  3.1 Characterization of Stage 1

If the motion of the plasma through the front were adiabatic, the
corresponding evolution of the parallel and perpendicular temperatures would
be described by the double adiabatic equations of Chew et al. (1956),

$\displaystyle\frac{T_{\parallel}B^{2}}{n^{2}}$ $\displaystyle=$
$\displaystyle cst,$ $\displaystyle\frac{T_{\perp}}{B}$ $\displaystyle=$
$\displaystyle cst.$ (16)

Here, like in the rest of the paper, the parallel and perpendicular are
defined with respect to the local magnetic field.

Now, since we are dealing with shockwaves, the evolution of the plasma from
the upstream to the downstream is _not_ adiabatic. For parallel and
perpendicular shocks, this results in different prescriptions.

  * •

For the parallel shock case treated in Bret & Narayan (2018), we took
$\theta_{1,2}=\xi_{1,2}=0$ and considered the entropy excess goes into the
parallel temperature. Intuitively, this steams from the fact that the transit
of the plasma through the front can be viewed as a compression between 2
converging virtual walls, normal to the flow. These walls by no means exist.
They are simply an analogy of how the entropy gain is realized.

Regarding the perpendicular temperature, Eq. (3.1) simply gives
$T_{\perp}=cst$ in the parallel case, since $B_{2}=B_{1}$ for such a shock.
Such changes of the temperatures have been successfully checked through PIC
simulations in Haggerty et al. (2022).

  * •

For the perpendicular shock case treated in Bret & Narayan (2019), we took
$\theta_{1,2}=\pi/2$ and $\xi_{1,2}=0$. Here the plasma can still be viewed as
compressed between 2 virtual walls normal to the flow. We therefore considered
that the temperature normal to the flow, that is, parallel to the field,
evolves adiabatically.

An additional constraint that must always be satisfied is the equality of the
2 temperatures perpendicular to the field, enforced by the Vlasov equation
(Landau & Lifshitz (1981), §53).

These considerations are summarized in Table 1 which gives the values of
$T_{2\parallel}$ and $T_{2\perp}$ in the parallel and perpendicular cases.

Cases | $T_{2\parallel}$ | $T_{2\perp}$
---|---|---
Parallel, $\theta_{1,2}=0$ |  $T_{1}\left(\frac{n_{2}B_{1}}{n_{1}B_{2}}\right)^{2}$ \+ entropy | $T_{1}\frac{B_{1}}{B_{2}}$
Perpendicular, $\theta_{1,2}=\pi/2$ | $T_{1}\left(\frac{n_{2}B_{1}}{n_{1}B_{2}}\right)^{2}$ |  $T_{1}\frac{B_{1}}{B_{2}}$ \+ entropy
Table 1: Values of $T_{2\parallel}$ and $T_{2\perp}$ in Stage 1 for the
parallel and perpendicular cases.

As already stated in the introduction, MHD suggests that in a switch-on shock,
the obliquity $\theta_{2}$ of the downstream field can be as high as
$0.56\frac{\pi}{2}$. Hence, we need to interpolate between the 2 extremes of
Table 1. We cannot just elaborate from Bret & Narayan (2018) by exploring
$\theta_{2}=\varepsilon$, with $0<\varepsilon\ll 1$.

For intermediate values of $\theta_{2}$, we propose the following
interpolation between the 2 extremes of Table 1,

$\displaystyle T_{2\parallel}=$ $\displaystyle
T_{1}\left(\frac{n_{2}B_{1}}{n_{1}B_{2}}\right)^{2}$
$\displaystyle+~{}T_{e}\cos^{2}\theta_{2},$ $\displaystyle T_{2\perp}=$
$\displaystyle T_{1}\frac{B_{1}}{B_{2}}$
$\displaystyle+~{}\frac{1}{2}T_{e}\sin^{2}\theta_{2}.$ (17)

Our _ansatz_ is therefore that the downstream temperatures are the sum of the
adiabatic ones given by Chew et al. (1956), plus an entropy excess. For the
parallel temperature, the entropy excess is a fraction $\cos^{2}\theta_{2}$ of
a quantity we label $T_{e}$ (subscript “e” for _e_ ntropy). For the
perpendicular temperature, the entropy excess is a fraction
$\frac{1}{2}\sin^{2}\theta_{2}$ of the same $T_{e}$. Here, the factor $1/2$
accounts for the necessary identity of the 2 perpendicular temperatures.
Finally, the 3 temperature excesses sum to $T_{e}$.

The $\cos^{2}\theta_{2}$ and $\sin^{2}\theta_{2}$ functions are the simplest
choice fulfilling these requirements. Further works, notably PIC simulations
(see conclusion), should allow to test their relevance.

Note that $T_{e}$ is not arbitrary but is solved for using the conservation
equations (see Eq. (56) in Appendix B). It represents the heat generated from
the shock entropy.

We now compute the properties of Stage 1 accounting for these extended
prescriptions for Stage 1.

##  4 Properties of Stage 1

Due to the complexity of the calculations, we treat only the case of a sonic
strong shock, namely $T_{1}=0$.

###  4.1 Conservation equations for anisotropic temperatures

The conservation equations for anisotropic temperatures in the downstream are
established in Appendix A. Though with different notations, they can be found
in Hudson (1970); Erkaev et al. (2000). With $T_{1}=0$, they read,

$\displaystyle n_{2}v_{2}\cos\xi_{2}$ $\displaystyle=$ $\displaystyle
n_{1}v_{1},$ (18) $\displaystyle B_{2}\cos\theta_{2}$ $\displaystyle=$
$\displaystyle B_{1},$ (19) $\displaystyle
B_{2}v_{2}\sin\theta_{2}\cos\xi_{2}-B_{2}v_{2}\cos\theta_{2}\sin\xi_{2}$
$\displaystyle=$ $\displaystyle 0,$ (20)
$\displaystyle\cos^{2}\theta_{2}n_{2}k_{B}T_{2\parallel}+\sin^{2}\theta_{2}n_{2}k_{B}T_{2\perp}+mn_{2}v_{2}^{2}\cos^{2}\xi_{2}-\frac{B_{2}^{2}\cos\left(2\theta_{2}\right)}{8\pi}$
$\displaystyle=$ $\displaystyle-\frac{B_{1}^{2}}{8\pi}+mn_{1}v_{1}^{2},$ (21)
$\displaystyle\sin\theta_{2}\cos\theta_{2}n_{2}k_{B}\left(T_{2\parallel}-T_{2\perp}\right)+mn_{2}v_{2}^{2}\sin\xi_{2}\cos\xi_{2}-\frac{B_{2}^{2}\sin\left(2\theta_{2}\right)}{8\pi}$
$\displaystyle=$ $\displaystyle 0,$ (22)
$\displaystyle\left[v\left(\mathcal{A}\cos\xi+\mathcal{B}\cos\xi+\mathcal{C}\sin\xi\right)\right]_{1}^{2}$
$\displaystyle=$ $\displaystyle 0,$ (23)

where,

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}nk_{B}T_{\parallel}+nk_{B}T_{\perp}+\frac{B^{2}}{8\pi}+\frac{1}{2}mnv^{2},$
$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle-\frac{B^{2}\cos\left(2\theta\right)}{8\pi}+\cos^{2}\theta~{}nk_{B}T_{\parallel}+\sin^{2}\theta~{}nk_{B}T_{\perp},$
$\displaystyle\mathcal{C}$ $\displaystyle=$
$\displaystyle\sin\theta\cos\theta~{}nk_{B}\left(T_{\parallel}-T_{\perp}\right)-\frac{B^{2}\sin\left(2\theta\right)}{8\pi}.$

In equation (23), the notation $\left[Q\right]_{1}^{2}$ stands for the
difference of any quantity $Q$ between the upstream and the downstream.

With $T_{1}=0$, prescriptions (3.1) for the downstream temperatures in Stage 1
simply read,

$\displaystyle T_{2\parallel}$ $\displaystyle=$ $\displaystyle
T_{e}\cos^{2}\theta_{2},$ (24) $\displaystyle T_{2\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{2}T_{e}\sin^{2}\theta_{2}.$

###  4.2 Resolution of the system of equations

The resolution of the system (18-24) is lengthy and reported in Appendix B. It
turns out that it is convenient to determine first the angle $\theta_{2}$ as a
function of $\sigma$, and then to compute the density jump $r(\sigma)$.

The algebra unravels 3 $\theta_{2}$-branches for $\theta_{1}=0$,

  * •

One branch is simply $\theta_{2}=0$, with $r=1$ and $r=2$. The first one, with
$r=1$, is the continuity solution. The second one, with $r=2$, is the parallel
strong sonic shock solution for Stage 1, already studied in Bret & Narayan
(2018).

  * •

The other branch defines 2 values of $\theta_{2}(\sigma)$ which correspond to
our switch-on solutions. They are pictured on Figure 3-left. Then the
corresponding density jump $r(\sigma)$ is computed and plotted on Figure
3-right. The green curve pictures the MHD switch-on solution Eq. (13), defined
for $\sigma\in[1/4,1]$. In Stage 1, numerical exploration shows solutions
exists only for $\sigma\in[0.432,1]$.

Therefore, while there is only 1 switch-on solution in MHD, our model offers
2. Figure 3-left shows that both of our branches merge with the MHD result for
$\sigma=1$ as far as $\theta_{2}$ is concerned. Such is only the case for the
lower of our $r$-branch, as can be seen from Figure 3-right.

Figure 3: Left: Values of $\theta_{2}$ arising from $Q=0$ in Eq. (61).
Solutions exist only for $\sigma\in[0.432,1]$. Right: Corresponding values of
the density jump from Eq. (60). The color code refers to the corresponding
$\theta_{2}$-branch. The green curves pertain to the MHD switch-on solution.

Figure 3-left shows that the largest value of $\theta_{2}$ in Stage 1 is
almost as high as its MHD counterpart.

In accordance with the method explained in Section 3, we now study the
stability of Stage 1.

###  4.3 Stability of Stage 1

If unstable, Stage 1 is mirror or firehose unstable. The thresholds for these
instabilities are given by (Gary, 1993; Gary & Karimabadi, 2009),

$\frac{T_{2\perp}}{T_{2\parallel}}\equiv A_{2}=1\pm\frac{1}{\beta_{\parallel
2}},$ (25)

where

$\beta_{\parallel 2}=\frac{n_{2}k_{B}T_{2\parallel}}{B_{2}^{2}/(8\pi)},$ (26)

and the “+” and “-” signs stand for the thresholds of the mirror and firehose
instabilities respectively. From Eq. (24), we obtain the downstream anisotropy
$A_{2}$,

$A_{2}=\frac{1}{2}\tan^{2}\theta_{2}.$ (27)

For $\beta_{\parallel 2}$, we obtain in Appendix C,

$\beta_{\parallel
2}=-2\frac{r\sigma\tan^{2}\theta_{2}-2r+2}{r\sigma(\tan^{4}\theta_{2}+2)}.$
(28)

Figure 4: Left: Thresholds for the mirror (upper gray line) and firehose
(lower gray line) instabilities. The plasma is unstable within the shaded
areas. The loop shows the curves $(\beta_{\parallel 2}(\sigma),A_{2}(\sigma))$
for the 2 $\theta_{2}$-branches defined previously. The 2 branches start from
the same point for $\sigma=0.432$, and both reach $A_{2}=0$ for $\sigma=1$.
Right: Stability analysis of the blue branch for $\sigma\in[0.432,1]$. It is
found firehose unstable for $\sigma\in[0.82,1]$.

In order to assess the stability of Stage 1, we then proceed as follow,

  * •

From Eq. (25), we plot the thresholds for the mirror and firehose
instabilities in the $(\beta_{\parallel 2},A_{2})$ plane.

  * •

Then on the same graph, we plot the curves $(\beta_{\parallel 2},A_{2})$ for
the 2 non-trivial $\theta_{2}$-branches found in Section 4.2.

The result is pictured on Figure 4-left. In Bret & Narayan (2018), Stage 1 had
$A_{2}=0$ for the sonic strong shock case. Here, $A_{2}$ departs from 0 but
remains small.

It turns out that the orange branch pictured on Figure 3-left, namely the one
closest to the MHD solution, is stable for any $\sigma$. Yet, the blue one is
slightly unstable in some $\sigma$ range. For this branch, the quantity
$A_{2}-(1-\beta_{\parallel 2}^{-1})$ is plotted on Figure 4-right. It is
negative for $\sigma\in[0.83,1]$, indicating firehose instability. In this
$\sigma$-range, the downstream will therefore migrate to Stage 2, on the
firehose threshold.

As can be seen on Figure 4-left, the blue branch Stage 1 is only slightly
unstable. Consequently, the corresponding marginally stable Stage 2 is very
close to the unstable states. This is confirmed below in Section 5, where
Stage 2 is analysed.

For now, in order to document the differences between our 2 branches, we
further study Stage 1 by computing its entropy and its Alfvénic downstream
velocity.

###  4.4 Entropy of Stage 1

The 2 branches for Stage 1 cannot be distinguished by their energy since they
both fulfill the energy conservation equation (23), where the upstream energy
is the same in both cases. Their energy densities are therefore identical.
Yet, they can be distinguished on the basis of their entropy.

For a bi-Maxwellian of the form,

$F=\frac{n}{\pi^{3/2}\sqrt{a}b}\exp\left(-\frac{v_{x}^{2}}{a}\right)\exp\left(-\frac{v_{y}^{2}+v_{z}^{2}}{b}\right),$
(29)

where $a=2k_{B}T_{\parallel}/m$ and $b=2k_{B}T_{\perp}/m$, the entropy reads,

$S=-k_{B}\int F\ln Fd^{3}v=\frac{1}{2}k_{B}n\left[3+\ln(\pi^{3}ab^{2})-2\ln
n\right],$ (30)

where $n=\int Fd^{3}v$. Using the subscript “b” for the blue branch on Fig. 3,
and subscript “o” for the orange one, we get for the entropy difference per
particle between the 2 branches,

$\displaystyle\frac{2}{k_{B}}\left(\frac{S_{o}}{n_{o}}-\frac{S_{b}}{n_{b}}\right)\equiv\Delta
s$ $\displaystyle=$
$\displaystyle\ln\left[\frac{a_{o}b_{o}^{2}}{a_{b}b_{b}^{2}}\right]+2\ln\frac{n_{b}}{n_{o}},$
(31) $\displaystyle=$ $\displaystyle\ln\left[\frac{T_{\parallel
2,o}~{}T_{\perp 2,o}^{2}}{T_{\parallel 2,b}~{}T_{\perp
2,b}^{2}}\right]+2\ln\frac{n_{b}}{n_{o}},$ $\displaystyle=$
$\displaystyle\ln\left[\frac{T_{\parallel 2,o}^{3}~{}A_{2,o}^{2}}{T_{\parallel
2,b}^{3}~{}A_{2,b}^{2}}\right]+2\ln\frac{n_{b}}{n_{o}},$ $\displaystyle=$
$\displaystyle\ln\left[\left(\frac{T_{e,o}\cos^{2}\theta_{2,o}}{T_{e,b}\cos^{2}\theta_{2,b}}\right)^{3}\frac{A_{2,o}^{2}}{A_{2,b}^{2}}\right]+2\ln\frac{n_{b}}{n_{o}},$

where we have used $A_{2}=T_{\perp 2}/T_{\parallel 2}$ and then $T_{\parallel
2}=T_{e}\cos^{2}\theta_{2}$ for both branches.

Figure 5: Left: Entropy difference $\Delta s$ as defined by Eq. (31), between
the blue branch on Fig. 3, and the orange one. $\Delta s<0$ implies the orange
branch has lower entropy than the blue one. Right: Value of $\mathcal{M}_{A2}$
in MHD (green) and for the 2 branches of our model.

The numerical evaluation444$T_{e}$ is given by Eq. (56). of this quantity
displayed on Figure 5-left shows that $\Delta s<0$ for any
$\sigma\in[0.432,1]$. Therefore, $S_{o}/n_{o}<S_{b}/n_{b}$ for any $\sigma$.
The orange branch on Fig. 3 has _lower_ entropy than the blue one.

###  4.5 Downstream Alfvénic Mach number of Stage 1

Another difference between the 2 branches lies in their respective Alfvénic
Mach number, namely,

$\mathcal{M}_{A2}^{2}=\frac{mn_{2}v_{2}^{2}}{B_{2}^{2}/4\pi}.$ (32)

From Eq. (19) we get $B_{2}=B_{1}/\cos\theta_{2}$. Then Eq. (18) gives
$v_{2}=n_{1}v_{1}/n_{2}\cos\xi_{2}=n_{1}v_{1}/n_{2}\cos\theta_{2}$, since
$\xi_{2}=\theta_{2}$ in our model as in MHD555See Eq. (7) for MHD and Appendix
B for our model.. We finally obtain, in terms of the dimensionless variables
(10), in both our model and MHD,

$\mathcal{M}_{A2}=\frac{\mathcal{M}_{A1}}{\sqrt{r}}=\frac{1}{\sqrt{\sigma
r}}.$ (33)

  * •

In MHD, Eq. (13) has $r=\mathcal{M}_{A1}^{2}$, so that for the switch-on MHD
shock, $\mathcal{M}_{A2}=1$ (Goedbloed et al. (2010), p. 853).

  * •

In our model, the value of $\mathcal{M}_{A2}$ is pictured on Fig. 5-right for
the 2 branches represented on Fig. 3. Our 2 branches are found slightly sub-
Alfvénic.

##  5 Properties of Stage 2

The firehose instability of the blue branch for $\sigma\in[0.83,1]$ requires
studying the properties of Stage 2 when marginally firehose stable. The
conservation equations are the same. But instead of imposing prescriptions
(3.1) for the temperatures, we now impose firehose marginal stability for the
downstream, namely,

$\frac{T_{2\perp}}{T_{2\parallel}}=1-\frac{1}{\beta_{\parallel 2}}.$ (34)

The resolution of the system follows the same path as that describes in
Appendix B for Stage 1. It yields 3 equations for $\theta_{2},\xi_{2}$ and
$r$,

$\displaystyle\tan\theta_{2}-\tan\xi_{2}$ $\displaystyle=$ $\displaystyle 0,$
(35)
$\displaystyle\frac{2\tan\xi_{2}}{r}-\frac{\tan\theta_{2}}{\mathcal{M}_{A1}^{2}}$
$\displaystyle=$ $\displaystyle 0,$ (36)
$\displaystyle\mathcal{M}_{A1}^{2}\left(-\sec^{2}\xi_{2}+(r-5)r+5\right)+r\tan\theta_{2}\tan\xi_{2}+r$
$\displaystyle=$ $\displaystyle 0.$ (37)

Equation (35) imposes again $\theta_{2}=\xi_{2}$. Replacing in Eq. (36) gives,

$\tan\theta_{2}(r-2\mathcal{M}_{A1}^{2})=0,$ (38)

which leaves 2 options only,

  * •

$\theta_{2}=0$, which pertains to the parallel shock solution. Setting then
$\theta_{2}=0$ in Eq. (37) then gives exactly the Stage 2 solution found in
Bret & Narayan (2018)666See Eq. (3.5) of Bret & Narayan (2018) for
$\chi_{1}=\infty$..

  * •

The other option is,

$r=2\mathcal{M}_{A1}^{2}=2/\sigma.$ (39)

Inserting this result in Eq. (37) and solving for $\theta_{2}$ gives,

$\cos^{2}\theta_{2}=\frac{1}{10\mathcal{M}_{A1}^{2}-4\mathcal{M}_{A1}^{4}-5},$
(40)

reminiscent of Eq. (14) for the MHD case. Solutions can here be found for,

$\displaystyle\mathcal{M}_{A1}$ $\displaystyle\in$
$\displaystyle\left[\frac{\sqrt{5-\sqrt{5}}}{2},\frac{\sqrt{5+\sqrt{5}}}{2}\right]\sim[0.83,1.34],$
$\displaystyle\Leftrightarrow\sigma=\frac{1}{\mathcal{M}_{A1}^{2}}$
$\displaystyle\in$
$\displaystyle\left[\frac{4}{5+\sqrt{5}},\frac{4}{5-\sqrt{5}}\right]\sim[0.55,1.44].$
(41)

The counterpart to switch-on shock is therefore recovered in our model for
Stage 2 as well, still in a limited range of Alfvén Mach numbers.

Figure 6: Same as Figure 3-right but showing how Stages 1 & 2 fit together
when accounting for the firehose instability of the blue branch for
$\sigma\in[0.83,1]$. Stage 2 density jump $r=2/\sigma$ is discontinued from
$\sigma=1$ since Stage 1 has no solution in this range.

Figure 6 is eventually the end result of the present work. Like Figure
3-right, it features the density jump of the MHD switch-on solution, together
with the 2 branches of our model. But here, the way Stages 1 & 2 fit together
in the $\sigma$ unstable range is elucidated. Since the blue branch has been
found firehose unstable for $\sigma\in[0.83,1]$, it is replaced by Stage 2,
namely Eq. (39), in this range. As expected, the corresponding density jump is
very close to the one of Stage 1 since the system is almost marginally stable
in this range, while Stage 2 sits exactly on marginal stability.

On Figure 6, the jump for Stage 2, namely $r=2/\sigma$, is showed in black and
plotted within the full range (41) where it is defined. For $\sigma<0.83$, the
line is dashed because Stage 1 is stable, hence defining the density jump.
Then for $\sigma\in[0.83,1]$, the blue branch is dashed since it pertains to
the unstable Stage 1. There, the jump is now given by Stage 2 through
$r=2/\sigma$. Beyond $\sigma=1$, Stage 1 offers no solutions. Since in our
scenario Stage 1 is the first state of the downstream after crossing the
front, the shock cannot accommodate such values of $\sigma$ in the switch-on
regime. For $\sigma>1$, there is therefore no Stage 1 from where the system
could jump to Stage 2, even though Stage 2 offers solutions. As a consequence,
the black curve is dashed from $\sigma=1$ to $1.44$.

##  6 Conclusion

In a collisionless non-magnetized plasma, the Weibel instability ensures
isotropy, since it makes anisotropies unstable (Weibel, 1959; Silva et al.,
2021). Therefore, for collisionless shocks in such medium, the only source of
departures from MHD should stem from accelerated particles (Haggerty &
Caprioli, 2020; Bret, 2020).

In contrast, a temperature anisotropy can be stabilized in a collisionless
plasma by an external magnetic field. Therefore, if the plasma turns
anisotropic when crossing the front of a collisionless shock, its downstream
anisotropy could be stable, resulting in a departure from MHD.

Several authors studied the conservation equations for a shock accounting for
anisotropic pressures. Yet, the downstream degree of anisotropy is considered
a free parameter in these works (Erkaev et al., 2000; Double et al., 2004;
Gerbig & Schlickeiser, 2011). In the present article, we devised a model
allowing to compute the degree of anisotropy of the downstream, in terms of
the upstream parameters. We focused on the switch-on solutions where the field
is aligned with the flow in the upstream, but not in the downstream.

For such a configuration, MHD allows for one shock solution, the switch-on
solution, for which the density jump is given by Eq. (13). According to our
model, which has been successfully tested against PIC simulations for the
parallel case (Haggerty et al., 2022), there are two collisionless switch-on
solutions for which the angle $\theta_{2}$ and the density jump $r$ are
plotted on Figures 3 & 6.One solution for what we named “Stage 1” is stable
for any $\sigma$ where it is defined. The other is slightly firehose unstable
within a limited $\sigma$-range. Exploring then Stage 2 in this range allows
to correct the computed density jump. Since the Stage 1 that needed to be
corrected was only slightly firehose unstable, the correction found with Stage
2 _marginally_ firehose stable, is small.

The existence of 2 switch-on solutions in our model instead of 1 in MHD could
be explained. We plotted on Figure 2-right the MHD solutions for a cold
upstream and any upstream field obliquity $\theta_{1}$. One can see that the
MHD switch-on solution for $\theta_{1}=0$ splits into 2 different solutions
for $\theta_{1}>0$. These 2 solutions are the intermediate and fast shocks.
They merge for $\theta_{1}=0$, which is why MHD switch-on shocks can be termed
intermediate or fast (Goedbloed et al. (2010), p. 853).

Possibly within our model, these 2 kinds of shocks do not merge for
$\theta_{1}=0$. Future works dealing with the fully oblique case
$\theta_{1}>0$ will explore how the MHD intermediate and fast shocks morph
within our model.

Is one of our 2 branches physically favored? Both pertain to a downstream
plasma with the same energy density since both fulfil the energy conservation
equation (23) where the upstream term is the same. We see from Figs. 3 & 5
that the orange one is the closest to the MHD solution, yet we found in
Section 4.4 that it has lower entropy than the blue branch. Further works,
notably PIC simulations, would be needed to find out if these 2 branches are
just our model’s version of the oblique intermediate and fast shocks in the
limit $\theta_{1}=0$.

In the same way that the theory devised for the parallel case has been tested
through PIC simulation (Bret & Narayan, 2018; Haggerty et al., 2022), it would
be interesting to test the present conclusions through the same means. Yet, to
our knowledge, no PIC simulations of switch-on shocks have been performed to
date (Sironi & Lembège, 2022). An option in this respect would be to reproduce
in PIC the bow shock MHD simulation performed in de Sterck & Poedts (1999).
There, it was found that a portion of the bow shock produced by the simulation
was of the switch-on type. Possibly a PIC counterpart of this work would allow
to produce a switch-on shock and study it at the kinetic scale.

## Acknowledgments

Thanks are due to Lorenzo Sironi and Bertrand Lembège for enriching
discussions.

## Funding

A.B. acknowledges support by grants ENE2016-75703-R from the Spanish
Ministerio de Economía y Competitividad and SBPLY/17/180501/000264 from the
Junta de Comunidades de Castilla-La Mancha. R.N. acknowledges support from the
NSF Grant No. AST- 1816420\. R.N. thanks the Black Hole Initiative at Harvard
University for support. The BHI is funded by grants from the John Templeton
Foundation and the Gordon and Betty Moore Foundation.

## Declaration of Interests

The authors report no conflict of interest.

##  Appendix A Derivation of the conservation equations for anisotropic
temperatures

Equations (18-20) are identical to their MHD counterpart since they do not
involve the pressure. The differences due to anisotropic pressure are rather
to be found in Eqs. (21-23). For Eqs. (21,22), we start from the momentum flux
density tensor equation (Landau & Lifshitz (2013), §7),

$\frac{\partial(\rho v_{i})}{\partial t}=-\frac{\partial\Pi_{ik}}{\partial
x_{k}}.$ (42)

In the shock frame, the left-hand-side vanishes. Using the basis $x,y,z$
represented on Fig. 1, where the shock jump is in the $x$ direction, we obtain
the following jump conditions,

$\left[\Pi_{xx}\right]_{1}^{2}=\left[\Pi_{xy}\right]_{1}^{2}=0,$ (43)

where the notation $\left[Q\right]_{1}^{2}$ stands for the difference of any
quantity $Q$ between the upstream and the downstream.

There are 3 contributions to the tensor $\Pi_{ik}$: ram pressure, magnetic
pressure and thermal pressure:

  * •

The ram pressure part reads,

$\Pi_{ram}=\left(\begin{array}[]{ccc}nmv^{2}\cos^{2}\xi&nmv^{2}\cos\xi\sin\xi&0\\\
nmv^{2}\cos\xi\sin\xi&nmv^{2}\sin^{2}\xi&0\\\ 0&0&0\end{array}\right),$ (44)

where all quantities are to be taken with subscript 1 for the upstream and 2
for the downstream.

  * •

For the magnetic pressure, we start in a basis
$(x^{\prime},y^{\prime},z^{\prime})$ aligned with the field. In such a basis,

$\Pi_{mag}^{\prime}=\left(\begin{array}[]{ccc}-B^{2}/8\pi&0&0\\\
0&B^{2}/8\pi&0\\\ 0&0&B^{2}/8\pi\end{array}\right).$ (45)

We now express this tensor in our basis $(x,y,z)$, where $z=z^{\prime}$ and
$(x^{\prime},y^{\prime})$ are rotated by an angle $\theta$. Hence, we compute
$R^{-1}\Pi_{mag}^{\prime}R$, with

$R=\left(\begin{array}[]{ccc}\cos\theta&\sin\theta&0\\\
-\sin\theta&\cos\theta&0\\\ 0&0&1\\\ \end{array}\right).$ (46)

The result is,

$\Pi_{mag}=\frac{B^{2}}{8\pi}\left(\begin{array}[]{ccc}-\cos 2\theta&-\sin
2\theta&0\\\ -\sin 2\theta&\cos 2\theta&0\\\ 0&0&1\end{array}\right).$ (47)

  * •

The calculation is similar for the thermal pressure. We start in a basis
adapted to the field,

$\Pi_{th}^{\prime}=nk_{B}\left(\begin{array}[]{ccc}T_{\parallel}&0&0\\\
0&T_{\perp}&0\\\ 0&0&T_{\perp}\\\ \end{array}\right),$ (48)

where the directions $\parallel$ and $\perp$ are considered with respect to
the field. Computing $R^{-1}\Pi_{th}^{\prime}R$, where the tensor $R$ is still
given by Eq. (46), gives in our basis $(x,y,z)$,

$\Pi_{th}=nk_{B}\left(\begin{array}[]{ccc}T_{\parallel}\cos^{2}\theta+T_{\perp}\sin^{2}\theta&(T_{\parallel}-T_{\perp})\cos\theta\sin\theta&0\\\
(T_{\parallel}-T_{\perp})\cos\theta\sin\theta&T_{\parallel}\sin^{2}\theta+T_{\perp}\cos^{2}\theta&0\\\
0&0&T_{\perp}\\\ \end{array}\right).$ (49)

When adding the contributions (44,47,49), the conservation equations (43)
yield Eqs. (21,22).

For the last equation, namely Eq. (23), we start from the energy conservation
equation (Landau & Lifshitz (2013), §6),

$\frac{\partial}{\partial
t}\left(\frac{1}{2}nmv^{2}+\varepsilon_{mag}+\varepsilon_{th}\right)=-\frac{\partial}{\partial
x_{k}}\left[v_{k}\left(\frac{1}{2}nmv^{2}+\varepsilon_{mag}+\varepsilon_{th}\right)\right]-\frac{\partial}{\partial
x_{k}}\left[v_{i}\left(\Pi_{ik,mag}+\Pi_{ik,th}\right)\right],$ (50)

where $\varepsilon$ is the internal energy density,

$\displaystyle\varepsilon_{mag}$ $\displaystyle=$
$\displaystyle\frac{B^{2}}{8\pi},$ $\displaystyle\varepsilon_{th}$
$\displaystyle=$
$\displaystyle\frac{1}{2}nk_{B}T_{\parallel}+nk_{B}T_{\perp},$

and $\Pi_{mag},\Pi_{th}$ are given by Eqs. (47,49) respectively. Setting the
left-hand-side of Eq. (50) to 0, and equating the right-hand-side between the
upstream and the downstream, gives Eq. (23).

##  Appendix B Resolution of the system (18-24)

The system (18-24) can be reduced to a system of 3 equations for $r\equiv
n_{2}/n_{1},\theta_{2}$ and $T_{e}$. The pathway to do so is,

  * •

We first notice that equation (20) imposes $\xi_{2}=\theta_{2}$ (as in MHD).
We can therefore set $\xi_{2}=\theta_{2}$ everywhere.

  * •

We then use equation (18) to eliminate $v_{2}$ everywhere.

  * •

Next, we use equation (19) to eliminate $B_{2}$ everywhere.

At this junction we are left with 3 equations which are the updated versions
of (21-23). They read,

$\displaystyle
r^{2}\overline{T_{e}}\sin^{4}\theta_{2}+2r^{2}\overline{T_{e}}\cos^{4}\theta_{2}+r\sigma\tan^{2}\theta_{2}-2r+2$
$\displaystyle=$ $\displaystyle 0,$ (51) $\displaystyle
r^{2}\overline{T_{e}}\left[2\sin(2\theta_{2})+3\sin(4\theta_{2})\right]+16\tan\theta_{2}(1-r\sigma)$
$\displaystyle=$ $\displaystyle 0,$ (52)
$\displaystyle\sec^{2}\theta_{2}+r^{2}\left[\overline{T_{e}}\cos(2\theta_{2})+2\overline{T_{e}}-1\right]$
$\displaystyle=$ $\displaystyle 0,$ (53)

in terms of the density ratio $r$ and the magnetic parameter $\sigma$ defined
in Eqs. (15), plus,

$\overline{T_{e}}=\frac{k_{B}T_{e}}{mv_{1}^{2}}.$ (54)

For further progress, it is convenient to define,

$X_{2}=\arcsin\theta_{2}.$ (55)

This change of variables makes the forthcoming equations polynomial in
$X_{2}$, easy to solve numerically. The value of $\overline{T_{e}}$ can be
extracted from equation (51) and reads,

$\overline{T_{e}}=\frac{r[(\sigma+2)X_{2}^{2}-2]-2X_{2}^{2}+2}{r^{2}(X_{2}^{2}-1)(X_{2}^{2}(3X_{2}^{2}-4)+2)}.$
(56)

Substituting it in Eqs. (52,53) yields the 2 equations,

$\displaystyle
X_{2}\underbrace{\left[r\left\\{\sigma(4+3(X_{2}^{2}-2)X_{2}^{2})-6X_{2}^{4}+10X_{2}^{2}-4\right\\}-2X_{2}^{2}\right]}_{\equiv\Lambda}$
$\displaystyle=$ $\displaystyle 0,$ (57)
$\displaystyle\sum_{k=0}^{3}a_{k}X_{2}^{2k},$ $\displaystyle=$ $\displaystyle
0,$ (58)

with,

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle 2(r-2)(r-1),$
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle r(-6r+3\sigma+10)-6,$
$\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle 7r^{2}-2(\sigma+2)r+1,$
$\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle-3r^{2}.$ (59)

Equation (57) clearly displays 2 branches,

  * •

One branch is $X_{2}=0$, that is, $\theta_{2}=0$. Inserting it into (58) gives
$a_{0}=0$, that is, $r=1$ or $r=2$. The first one, with $r=1$, is the
continuity solution. The second one, with $r=2$, is the parallel strong sonic
shock solution for Stage 1, already studied in Bret & Narayan (2018).

  * •

The second branch pertains to $\Lambda=0$. We can extract the value of $r$
from $\Lambda=0$, namely,

$r=\frac{2X_{2}^{2}}{\sigma[3(X_{2}^{2}-2)X_{2}^{2}+4]-6X_{2}^{4}+10X_{2}^{2}-4},$
(60)

and substitute in (58). This eventually gives a polynomial equation for
$X_{2}$ only, which reads,

$Q(X_{2})=\sum_{k=0}^{4}b_{k}X_{2}^{2k},$ (61)

with,

$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle
32\sigma^{2}-64\sigma+32,$ $\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle-80\sigma^{2}+200\sigma-120,$ $\displaystyle b_{2}$
$\displaystyle=$ $\displaystyle 76\sigma^{2}-224\sigma+160,$ $\displaystyle
b_{3}$ $\displaystyle=$ $\displaystyle-30\sigma^{2}+100\sigma-84,$
$\displaystyle b_{4}$ $\displaystyle=$ $\displaystyle
3\sigma^{2}-12\sigma+12.$ (62)

It can be solved numerically and gives the 2 values of
$\theta_{2}(\sigma)=\arcsin X_{2}(\sigma)$ plotted on Figure 3-left. Solutions
exist only for $\sigma\in[0.432,1]$. Then Eq. (60) allows to compute the
density jump $r$ for each $\theta_{2}$-branch, and plot them on Figure
3-right.

##  Appendix C Calculation of $\beta_{\parallel 2}$

We need to evaluate,

$\beta_{\parallel 2}=\frac{n_{2}k_{B}T_{2\parallel}}{B_{2}^{2}/(8\pi)}.$ (63)

Eq. (19) gives $B_{2}=B_{1}/\cos\theta_{2}$. Also, Eq. (24) gives
$T_{2\parallel}=T_{e}\cos^{2}\theta_{2}$. Finally, Eq. (56) gives $T_{e}$.
Expressing the result in terms of the dimensionless variables $r$ and $\sigma$
yields Eq. (28) for $\beta_{\parallel 2}$.

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#  Fourier spectrum and related characteristics of the fundamental bright
soliton solution

Natanael Karjanto 🖂<EMAIL_ADDRESS>Department of Mathematics, University
College, Natural Science Campus
Sungkyunkwan University, Suwon 16419, Republic of Korea

( Updated )

###### Abstract

We derive exact analytical expressions for the spatial Fourier spectrum of the
fundamental bright soliton solution for the $(1+1)$-dimensional nonlinear
Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for
the hyperbolic secant shape is also shape-preserving. We further confirm that
the fundamental soliton indeed satisfies essential characteristics such as
Parseval’s relation and the stretch-bandwidth reciprocity relationship. The
fundamental bright solitons find rich applications in nonlinear fiber optics
and optical telecommunication systems.

Keywords: fundamental bright soliton, nonlinear Schrödinger equation, spatial
Fourier spectrum, Parseval’s relation, stretch-bandwidth reciprocity
relationship, optical communication systems.

##  1 Introduction

The nonlinear Schrödinger (NLS) equation is a nonlinear evolution equation for
slowly varying wave packet envelopes in dispersive media. It belongs to a
category of completely integrable systems or exactly solvable models of a
nonlinear partial differential equation (PDE), with infinitely many conserved
quantities and explicit analytical solutions. The NLS equation finds rich
applications in mathematical physics, including surface gravity waves,
nonlinear optics, superconductivity, and Bose-Einstein condensates (BEC) [1,
2, 3].

The focus of this article is the one-dimensional focusing type of the NLS
equation. The former is sometimes also written as $(1+1)$-D, where usually,
there is only one dimension each in the spatial and temporal variables.
Higher-dimensional models usually extend the dimension of the dispersive term.
In the canonical form, the focusing $(1+1)$-D NLS equation can be expressed as
follows [6, 4, 5]:

$iq_{t}+q_{xx}+2|q|^{2}q=0,\qquad(x,t)\in\mathbb{R}^{2},\qquad
q(x,t)\in\mathbb{C}.$ (1)

In surface gravity waves, the independent variables $(x,t)$ represent spatial
and temporal quantities, respectively, whereas in nonlinear optics, $t$
denotes the transversal pulse propagation in space and $x$ designates the time
variable. The focusing type is indicated by the positive product of the
dispersive and nonlinear coefficients, which are also known in optical pulse
propagation as group-velocity dispersion and self-phase modulation,
respectively. In this article, we adopt the physical interpretation from
hydrodynamics for the independent variables.

The dependent variable $q(x,t)$ is a complex-valued amplitude. It describes
slowly-varying envelope dynamics of the corresponding weakly nonlinear quasi-
monochromatic wave packet profile. Denoted by $\eta(\bar{x},\bar{t})$, the
usual relationship with the $q$ is given by

$\eta(\bar{x},\bar{t})=\text{Re}\left\\{q(x,t)e^{i(k\bar{x}-\omega\bar{t})}\right\\},$

where $x=\varepsilon(\bar{x}-c_{g}\bar{t})$ and $t=\varepsilon^{2}\bar{t}$,
with $\varepsilon\ll 1$. Here, $c_{g}$ denotes the group velocity, and the
wavenumber $k$ and wave frequency $\omega$ are related by the linear
dispersion relationship for the corresponding medium.

The purpose of this article is to provide a step-by-step explanation for
finding the physical spectrum, i.e., the spatial Fourier spectrum, of the
bright soliton solution of the NLS equation. The term “spectrum” should not be
confused with the one used in the well-known inverse scattering transform
(IST) technique [6, 7, 8]. For working definitions, we adopt the ones from [9,
10].

The article is organized as follows. Section 2 outlines the proof of the
spatial Fourier transform for the fundamental bright soliton. Section 3
discusses some essential characteristics of the soliton, including Parseval’s
theorem and the stretch-bandwidth reciprocity relationship. Section 4
concludes our discussion.

##  2 Spatial Fourier spectrum

For the NLS equation (1), the simplest form of the fundamental soliton
solution is given by [11]

$q(x,t)=e^{it}\,\mathrm{sech}\,x=\frac{e^{it}}{\cosh x}.$

Also called a bright soliton, this exact analytic expression was discovered
one-half century ago using the IST [12]. One may also apply the Darboux
transformation using the seed function $q=0$ to obtain this solution [13, 14].
Without employing the IST, the bright soliton can also be obtained by solving
the NLS equation directly by assuming the existence of a shape-preserving
solution in the form of the phase and time-independent amplitude [11]. The
term _bright_ soliton is often used in the nonlinear optics literature to
distinguish and contrast it with _dark_ soliton. The former exists in the
anomalous dispersion regime, modeled by the focusing NLS equation (1), whereas
the latter occurs under the normal dispersion regime, which is governed by the
defocusing NLS equation [15, 16].

This fundamental bright soliton solution occurs as limiting cases of a family
of stationary periodic wave solutions, another family of periodic solutions,
in which both involve Jacobi elliptic functions, and the Kuznetsov-Ma breather
family [17, 18, 19, 20]. Although the possibility for the formation of the
bright soliton was suggested as early as 1973, it was not until 1980 that its
appearance was observed experimentally in optical fibers [21, 22, 23, 24].
Indeed, this fundamental soliton has a remarkable feature that allures it for
practical application, i.e., if a hyperbolic secant pulse is initiated inside
an ideal lossless fiber, it would propagate without altering its shape for
arbitrarily long-distance [25, 26, 27, 28]. In BEC, the formation of matter-
wave bright solitons was observed in 2002 [29, 30].

For $a$, $k$, and $\lambda\in\mathbb{R}\backslash\\{0\\}$, the following
transformations of the fundamental bright soliton (2) will also satisfy the
NLS equation (1) [31]:

$\displaystyle q_{1}(x,t)$ $\displaystyle=aq(ax,at^{2}),$ $\displaystyle
q_{2}(x,t)$ $\displaystyle=q(x-2kt,t)e^{i\left(kx-k^{2}t+\lambda\right)},$
$\displaystyle q_{3}(x,t)$ $\displaystyle=aq(a(x-2kt),at^{2})e^{i\left(akx-
ak^{2}t^{2}+\lambda\right)}.$

These three arbitrary parameters characterize and are related to the
amplitude, frequency, and phase of the soliton, respectively. The fourth
parameter is absent and might always be re-introduced if one wishes to
indicate the position of the soliton peak. However, this is inessential as we
can always shift it to $x=0$ when $t=0$ [11].

We have the following theorem.

######  Theorem 1.

The spatial Fourier spectrum of the bright NLS soliton in its simplest form
(2) is given by

$\widehat{q}(k,t)=\pi\,\mathrm{sech}\,\left(\frac{\pi}{2}k\right)e^{it}.$

###### Proof.

Observe that if we let

$f(x)=\mathrm{sech}\,x=\frac{1}{\cosh
x}=\frac{2}{e^{x}+e^{-x}}=\frac{2e^{-x}}{1+e^{-2x}},$

then $f$ is an even function and

$\left|f(x)\right|=\frac{2e^{-|x|}}{1+e^{-2|x|}}\leq 2e^{-|x|}\in
L_{1}(\mathbb{R}).$

We are interested in finding the spatial Fourier spectrum of the bright
soliton $q(x,t)$.

$\widehat{q}(k,t)=\int_{-\infty}^{\infty}\mathrm{sech}\,xe^{-i(kx-t)}\,dx=e^{it}\left(\int_{-\infty}^{\infty}\frac{\cos
kx}{\cosh x}\,dx-i\int_{-\infty}^{\infty}\frac{\sin kx}{\cosh x}\,dx\right).$

Since the second term inside the bracket of the last expression vanishes, we
only need to evaluate

$\int_{-\infty}^{\infty}\frac{\cos kx}{\cosh
x}\,dx=\lim\limits_{R\to\infty}\int_{-R}^{R}\frac{2\cos
kx}{e^{x}+e^{-x}}\,dx,\qquad R>0.$

Consider the complex-valued function

$\phi(z)=\frac{2\cos kz}{e^{z}+e^{-z}},\qquad z\in\mathbb{C},$ (2)

and the rectangular contour $C_{R}$ in the complex plane with corners at $\pm
R$ and $\pm R+i\pi$, with $R>0$, as shown in Figure 1.

Re($z$)Im($z$)O$R$$-R$$i\pi$$i\frac{\pi}{2}$$C_{R}$ Figure 1: A rectangular
contour $C_{R}$ with corners $\pm R$ and $\pm R+i\pi$ determined by a closed,
simple, piecewise smooth curve traversed in positive orientation. The function
$\phi$ is holomorphic in an open connected domain containing the interior of
$C_{R}$ and its closure, except at the pole $z=i\frac{\pi}{2}$.

The function $\phi(z)$ (2) has only one pole inside the region bounded by the
rectangular contour $C_{R}$ at $z=i\frac{\pi}{2}$. Hence, according to the
consequence of the Residue Theorem [32, 33, 34], we can calculate that the
contour integral over $C_{R}$ is given by

$\int_{C_{R}}\phi(z)\,dz=2\pi
i\;\text{res}\left(\phi,i\frac{\pi}{2}\right)=2\pi
i\;\frac{2\cos\left(ki\frac{\pi}{2}\right)}{e^{i\frac{\pi}{2}}-e^{-i\frac{\pi}{2}}}=2\pi
i\;\frac{2\cosh\left(\frac{\pi}{2}k\right)}{2i\sin\frac{\pi}{2}}=2\pi\cosh\left(\frac{\pi}{2}k\right).$
(3)

On the other hand, this very same contour integral can also be expressed as
separate four distinct line integrals along each side of the rectangle. Hence,
we have

$\displaystyle\int_{C_{R}}\phi(z)\,dz$ $\displaystyle=\int_{-R}^{R}\frac{2\cos
kx}{e^{x}+e^{-x}}\,dx+\int_{0}^{\pi}\frac{2\cos
k(R+iy)}{e^{R+iy}+e^{-R-iy}}\,i\,dy-\int_{-R}^{R}\frac{2\cos
k(x+\pi\,i)}{e^{x+\pi\,i}+e^{-x-\pi\,i}}\,dx$
$\displaystyle\quad-\int_{0}^{\pi}\frac{2\cos
k(-R+iy)}{e^{-R+iy}+e^{R-iy}}\,i\,dy.$ (4)

Now observe that the integrand of the second integral can be expressed as
follows for sufficiently large $R$:

$\displaystyle\left|\frac{2\cos k(R+iy)}{e^{R+iy}+e^{-R-iy}}\right|$
$\displaystyle=\left|\frac{e^{ikR}\,e^{-ky}+e^{-ikR}\,e^{ky}}{e^{R+iy}+e^{-R-iy}}\right|=\left|\frac{e^{(1+ik)R}\,e^{(-k+i)y}+e^{(1-ik)R}\,e^{(k+i)y}}{e^{2(R+iy)}+1}\right|$
$\displaystyle\leq\left|\frac{e^{(1+ik)R}\,e^{(-k+i)y}}{e^{2(R+iy)}+1}\right|+\left|\frac{e^{(1-ik)R}\,e^{(k+i)y}}{e^{2(R+iy)}+1}\right|\leq\frac{e^{R-ky}}{e^{2R}-1}+\frac{e^{R+ky}}{e^{2R}-1}=\frac{2e^{R}\,\cosh
ky}{e^{2R}-1}$ $\displaystyle\leq 4e^{-R}\cosh ky.$

Thus, the absolute value of the second integral reads

$\displaystyle\left|\int_{0}^{\pi}\frac{2\cos
k(R+iy)}{e^{R+iy}+e^{-R-iy}}\,i\,dy\right|\leq\int_{0}^{\pi}\left|\frac{2\cos
k(R+iy)}{e^{R+iy}+e^{-R-iy}}\,i\right|\,dy\leq\int_{0}^{\pi}4e^{-R}\cosh
ky\,dy=4e^{-R}\sinh k\pi,$

which tends to $0$ as $R\to\infty$. Similarly, for sufficiently large $R$, the
function of the fourth integral satisfies the following relationship:

$\displaystyle\left|\frac{2\cos k(-R+iy)}{e^{-R+iy}+e^{R-iy}}\right|$
$\displaystyle=\left|\frac{e^{-ikR}\,e^{-ky}+e^{ikR}\,e^{ky}}{e^{-R+iy}+e^{R-iy}}\right|=\left|\frac{e^{-(1+ik)R}\,e^{(-k+i)y}+e^{(-1+ik)R}\,e^{(k+i)y}}{1+e^{-2(R-iy)}}\right|$
$\displaystyle\leq\frac{\left|e^{-(1+ik)R}\,e^{(-k+i)y}\right|+\left|+e^{(-1+ik)R}\,e^{(k+i)y}\right|}{\left|1+e^{-2(R-iy)}\right|}\leq\frac{e^{-R-ky}+e^{-R+ky}}{1-e^{-2R}}=\frac{2e^{-R}\cosh
ky}{1-e^{-2R}}$ $\displaystyle\leq 2e^{-R}\cosh ky.$

So, the absolute value of the fourth integral becomes

$\displaystyle\left|\int_{0}^{\pi}\frac{2\cos
k(-R+iy)}{e^{-R+iy}+e^{R-iy}}\,i\,dy\right|\leq\int_{0}^{\pi}\left|\frac{2\cos
k(-R+iy)}{e^{-R+iy}+e^{R-iy}}\,i\right|\,dy\leq\int_{0}^{\pi}2e^{-R}\cosh
ky\,dy=2e^{-R}\sinh k\pi,$

which again tends to $0$ as $R\to\infty$. Considering the third integral, it
follows that

$\displaystyle-\int_{-R}^{R}\frac{2\cos
k(x+\pi\,i)}{e^{x+\pi\,i}+e^{-x-\pi\,i}}\,dx$
$\displaystyle=\int_{-R}^{R}\frac{e^{ikx}e^{-\pi k}+e^{-ikx}e^{\pi
k}}{e^{x}+e^{-x}}\,dx=\int_{-R}^{R}\frac{\cos kx\,\cosh\pi k}{\cosh
x}\,dx-i\int_{-R}^{R}\frac{\sin kx\,\sinh\pi k}{\cosh x}\,dx$
$\displaystyle=\cosh\pi k\int_{-R}^{R}\frac{\cos kx}{\cosh x}\,dx,$

where the imaginary component of the integral vanishes since $\sin(kx)$ is an
odd function. Hence, letting $R\to\infty$, we observe from (3) and (4) that

$\lim\limits_{R\to\infty}\left(1+\cosh\pi k\right)\int_{-R}^{R}\frac{\cos
kx}{\cosh x}\,dx=\left(1+\cosh\pi k\right)\int_{-\infty}^{\infty}\frac{\cos
kx}{\cosh x}\,dx=2\pi\cosh\left(\frac{\pi}{2}k\right).$

Therefore,

$\displaystyle\int_{-\infty}^{\infty}\frac{\cos kx}{\cosh x}\,dx$
$\displaystyle=\frac{2\pi\cosh\left(\frac{\pi}{2}k\right)}{1+\cosh\pi
k}=\frac{2\pi\cosh\left(\frac{\pi}{2}k\right)}{2\cosh^{2}\left(\frac{\pi}{2}k\right)}=\frac{\pi}{\cosh\left(\frac{\pi}{2}k\right)}=\pi\,\mathrm{sech}\,\left(\frac{\pi}{2}k\right),$

and thus, the spatial Fourier transform for the bright soliton reads

$\widehat{q}(k,t)=\pi\,\mathrm{sech}\,\left(\frac{\pi}{2}k\right)e^{it}.$

This completes the proof. ∎

Figure 2: The moduli of the NLS bright soliton (left panel, blue curve) and
its corresponding spectrum (right panel, red curve). Although both profiles
are secant hyperbolic, the former is wider and shorter, whereas the latter is
narrower and taller. Both plots are depicted on the same axis scale.

A similar approach in calculating the bright soliton spectrum yields an
expression of convergent alternating series form, where the integral in the
associated Fourier transform is expressed as Gauss’ hypergeometric function
[35]:

$\widehat{q}(k,t)=4\,e^{it}\sum_{n=0}^{\infty}(-1)^{n}\frac{2n+1}{k^{2}+(2n+1)^{2}}.$

##  3 Soliton characteristics

In this section, we examine some characteristics of the fundamental bright
soliton and its corresponding spatial Fourier spectrum. While the literature
usually considers the temporal-frequency domains relationship, i.e., pulse or
signal (together with its envelope) in the time domain and its temporal
Fourier spectrum in the frequency domain, we focus on the relationship of the
spatial-wavenumber domains. Hence, some terminologies in the definition are
adjusted to the related domains accordingly. We commence with the following
definitions and propositions.

######  Definition 1 (Soliton power and energy spectral density).

The squared-absolute value of the fundamental bright soliton
$\left|q(x,t)\right|^{2}$ is called the _(envelope) soliton power_ , whereas
the squared-modulus of its spatial Fourier spectrum
$\left|\widehat{q}(k,t)\right|^{2}$ is called the _energy spectral density_.

Figure 2 displays the envelope power of the bright soliton (left panel) and
its associated energy spectral density (right panel) for a fixed value of
$t\in\mathbb{R}$. We observe that the Fourier transform preserves the
hyperbolic secant profile albeit with different height and width between the
spatial and wavenumber domains.

######  Proposition 1.

The fundamental bright soliton satisfies Parseval’s Theorem in the spatial-
wavenumber domains [36]:

$\int_{-\infty}^{\infty}\left|q(x,t)\right|^{2}\,dx=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left|\widehat{q}(k,t)\right|^{2}\,dk.$

###### Proof.

The left-hand side reads

$\displaystyle\int_{-\infty}^{\infty}\left|q(x,t)\right|^{2}\,dx$
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{sech}\,^{2}x\,dx=\lim\limits_{c\to\infty}\int_{-c}^{c}\mathrm{sech}\,^{2}x\,dx=\lim\limits_{c\to\infty}\tanh
x\,\Big{|}_{-c}^{c}$ $\displaystyle=\lim\limits_{c\to\infty}\left[\tanh
c-\tanh(-c)\right]=\left[1-(-1)\right]=2.$

The right-hand side calculates

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\left|\widehat{q}(k,t)\right|^{2}\,dk$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\pi^{2}\mathrm{sech}\,^{2}\left(\frac{\pi}{2}k\right)\,dk=\frac{\pi}{2}\lim\limits_{d\to\infty}\int_{-d}^{d}\mathrm{sech}\,^{2}\left(\frac{\pi}{2}k\right)\,dk=\lim\limits_{d\to\infty}\tanh\left(\frac{\pi}{2}k\right)\Bigg{|}_{-d}^{d}$
$\displaystyle=\lim\limits_{d\to\infty}\left[\tanh\left(\frac{\pi}{2}d\right)-\tanh\left(-\frac{\pi}{2}d\right)\right]=\left[1-(-1)\right]=2.$

Since we verify that both sides equal to two, we complete the proof. ∎

######  Definition 2 (Mean/centroid).

Let $q(x,t)$ and $\widehat{q}(k,t)$ be a complex-valued soliton amplitude and
its associated spatial Fourier spectrum, respectively. Then, the _soliton
mean_ or _centroid_ is defined by

$\bar{x}=\frac{\displaystyle\int_{-\infty}^{\infty}x\left|q(x,t)\right|^{2}\,dx}{\displaystyle\int_{-\infty}^{\infty}\left|q(x,t)\right|^{2}\,dx},$

whereas the (spatial) _spectrum centroid_ or _mean_ is defined as

$\bar{k}=\frac{\displaystyle\int_{-\infty}^{\infty}k\left|\widehat{q}(k,t)\right|^{2}\,dk}{\displaystyle\int_{-\infty}^{\infty}\left|\widehat{q}(k,t)\right|^{2}\,dk}.$

######  Definition 3 (Power-root-mean-square soliton width and spectral
bandwidth).

Let $q(x,t)$ and $\widehat{q}(k,t)$ be a complex-valued soliton amplitude and
its associated spatial Fourier spectrum, respectively. The _power-root-mean-
square soliton width_ $\sigma_{x}\geq 0$, also called the _soliton standard
deviation_ or _soliton radius of gyration_ , is defined as

$\sigma_{x}^{2}=\frac{\displaystyle\int_{-\infty}^{\infty}(x-\bar{x})^{2}\left|q(x,t)\right|^{2}\,dx}{\displaystyle\int_{-\infty}^{\infty}\left|q(x,t)\right|^{2}\,dx},$

where as the (spatial) _power-root-mean-square spectral bandwidth_
$\sigma_{k}\geq 0$ is defined by

$\sigma_{k}^{2}=\frac{\displaystyle\int_{-\infty}^{\infty}(k-\bar{k})^{2}\left|\widehat{q}(k,t)\right|^{2}\,dk}{\displaystyle\int_{-\infty}^{\infty}\left|\widehat{q}(k,t)\right|^{2}\,dk}.$

In both instances, $\bar{x}$ and $\bar{k}$ denote the soliton and spectrum
centroids mentioned earlier in Definition 2, respectively.

######  Proposition 2.

The product of the power-root-mean-square soliton width with its associated
spectral bandwidth satisfies the (spatial) _stretch-bandwidth reciprocity
relationship_ , also called the _space-wavenumber uncertainty relationship_
[37, 38]:

$\sigma_{x}\,\sigma_{k}\geq\frac{1}{2}.$

###### Proof.

Since $x\left|q(x,t)\right|^{2}=x\,\mathrm{sech}\,^{2}x$ and
$k\left|\widehat{q}(k,t)\right|^{2}=\pi^{2}\,k\,\mathrm{sech}\,^{2}\left(\frac{\pi}{2}k\right)$
are both odd functions with respect to the length $x$ and wavenumber $k$,
respectively, then the numerators of both the soliton’s and spectrum’s means
vanish, i.e., $\bar{x}=0=\bar{k}$. From Proposition 1, we calculated the
denominators of $\sigma_{x}^{2}$ and $\sigma_{k}^{2}$ as $2$ and $4\pi$,
respectively. It then remains to evaluate their numerators. The former reads

$\displaystyle\int_{-\infty}^{\infty}(x-\bar{x})^{2}\left|q(x,t)\right|^{2}\,dx$
$\displaystyle=\int_{-\infty}^{\infty}x^{2}\,\mathrm{sech}\,^{2}x\,dx=\lim\limits_{c\to\infty}\int_{-c}^{c}x^{2}\,\mathrm{sech}\,^{2}x\,dx$
$\displaystyle=\lim\limits_{c\to\infty}\text{Li}_{2}(-e^{-2x})+x^{2}\left(\tanh
x-1\right)-2x\ln(1+e^{-2x})\,\Big{|}_{-c}^{c}=\frac{\pi^{2}}{6}.$

Thus, the soliton variance $\sigma_{x}^{2}$ and its power-root-mean-square
width $\sigma_{x}$ are given by

$\sigma_{x}^{2}=\frac{\pi^{2}}{12}\qquad\text{and}\qquad\sigma_{x}=\frac{\pi}{2\sqrt{3}}.$

Similarly, we evaluate

$\displaystyle\int_{-\infty}^{\infty}(k-\bar{k})^{2}\left|\widehat{q}(k,t)\right|^{2}\,dk$
$\displaystyle=\int_{-\infty}^{\infty}\pi^{2}\,k^{2}\,\mathrm{sech}\,^{2}\left(\frac{\pi}{2}k\right)\,dk=\pi^{2}\lim\limits_{d\to\infty}\int_{-d}^{d}k^{2}\,\mathrm{sech}\,^{2}\left(\frac{\pi}{2}k\right)\,dk$
$\displaystyle=\frac{1}{\pi}\lim\limits_{d\to\infty}8\text{Li}_{2}(-e^{-\pi
k})+\pi^{2}k^{2}\left[\tanh\left(\frac{\pi}{2}k\right)-2\right]-8\pi
k\ln(1+e^{-\pi k})\,\Big{|}_{-d}^{d}=\frac{4\pi}{3}.$

It follows that the spectrum variance $\sigma_{k}^{2}$ and its power-root-
mean-square spectral bandwidth $\sigma_{k}$ are given as follows:

$\sigma_{k}^{2}=\frac{1}{3}\qquad\text{and}\qquad\sigma_{k}=\frac{1}{\sqrt{3}}.$

In both instances, Li${}_{2}(z)$ denotes the dilogarithm Spence’s function,
defined as

$\text{Li}_{2}(z)=-\int_{0}^{z}\frac{\ln(1-u)}{u}\,du\qquad\text{or}\qquad\text{Li}_{2}(z)=\int_{1}^{z}\frac{\ln
t}{1-t}\,dt,\qquad z\in\mathbb{C}.$

Hence, the product of the two widths yields

$\sigma_{x}\,\sigma_{k}=\frac{\pi}{6}\geq\frac{1}{2},$

which satisfies the stretch-bandwidth reciprocity relationship and completes
the proof. ∎

It can also be verified that the equality in the stretch-bandwidth reciprocity
relationship is satisfied for the Gaussian pulse profile. In other words, the
Gaussian function possesses the minimum permissible value of the stretch-
bandwidth product, i.e., the minimum uncertainty in the context of
Heisenberg’s uncertainty principle in quantum mechanics.

##  4 Conclusion

In this article, we have considered the spatial Fourier spectrum for the
fundamental bright soliton solution of the NLS equation. Deriving the
analytical expression of the spectrum requires performing integration in the
complex plane. Interestingly, the bright soliton has a similar characteristic
of hyperbolic secant profiles in both the spatial and wavenumber domains. The
dynamics of the fundamental bright soliton is a topic of ongoing interest,
thanks to its applications in optical telecommunication systems and nonlinear
fiber optics.

### Acknowledgment

The author gratefully acknowledges E. (Brenny) van Groesen and Andonowati for
the long-lasting guidance and concerted cultivation in thinking and growing up
mathematically.

### Conflict of Interest

The author declares no conflict of interest.

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