# On solitary waves for the Korteweg–de Vries equation on metric star graphs

Delio Mugnolo  0000-0001-9405-0874 FernUniversität in Hagen, Lehrgebiet
Analysis, 58084 Hagen, Germany<EMAIL_ADDRESS>,  Diego Noja
0000-0003-1949-9369 Università degli Studi di Milano-Bicocca, Dipartimento di
Matematica e Applicazioni, Via R.Cozzi 55, 20125, Milano, Italy
<EMAIL_ADDRESS>and  Christian Seifert  0000-0001-9182-8687 Technische
Universität Hamburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, 21073
Hamburg, Germany<EMAIL_ADDRESS>
###### Abstract.

We study the Korteweg–de Vries equation on a metric star graph and investigate
existence of solitary waves on the metric graph in terms of the coefficients
of the equation on each edge, the coupling condition at the central vertex of
the star and the speeds of the travelling wave. We show that, with a
continuity condition at the vertex, solitary waves can occur exactly when the
parameters are chosen in a fairly special manner. We also consider coupling
conditions beyond continuity.

###### Key words and phrases:

Korteweg–de Vries equation, metric graphs, solitary waves

###### 2020 Mathematics Subject Classification:

35Q53, 47E05, 47Dxx

Declarations of interest: none. Data Availability Statement: Data sharing not
applicable to this article as no datasets were generated or analysed during
the current study.
This article is based upon work from COST Action 18232 MAT-DYN-NET, supported
by COST (European Cooperation in Science and Technology), www.cost.eu

##  1\. Introduction

The Korteweg–de Vries equation is a spatially one-dimensional model to
describe shallow water waves in a narrow channel. Its investigation started in
1834 with experiments by Russell and was formally derived by Korteweg and de
Vries in 1895 [KdV95]; however, Boussinesq already introduced the equation in
the 1870’s.

Let $u$ be the function describing the height displacement of the water
surface from equilibrium height, i.e. $u(t,x)$ is the deviation from
equilibrium at time $t$ and position $x$. Then the Korteweg–de Vries (KdV)
equation in dimensionless variables is given by

$\partial_{t}u=-\partial_{x}^{3}-6u\partial_{x}u.$

However, note that there are various versions of this equation with different
coefficients as well as first order terms $\partial_{x}u$ on the right-hand
side, depending on rescaling and translation in the derivation process (the
rescaling and translation can be well performed if the spatial domain is the
whole real line).

In 1965, Zabusky and Kruskal [ZK65] numerically discovered that solutions of
the KdV equation admit solitary waves.

The linear part of the KdV equation, which also appears as the small amplitude
long-wave approximation $u\partial_{x}u\approx 0$ gives rise to the so-called
Airy equation (again, possibly with a first order term $\partial_{x}u$ on the
right-hand side). This was studied by Airy as well as by Stokes [Sto47] in
order to understand the solitary waves observed by Russell.

The spatial domain for the KdV equation is given by an interval (or a halfline
or the real line). When modelling branching channels, one thus needs to work
with a bunch of intervals modelling the single channels as well as suitable
coupling conditions at the junctions. Such geometric objects are called metric
graphs and differential equations on metric graphs have been studied
extensively during the last decades, see e.g. [BK13, Mug14, Kur24] and
references therein for an overview on the topic. The linear part of KdV, i.e.
the Airy equation, has been studied on metric graphs recently [SAKJ15, SAU15,
SUA15, DSS16, MNS18, Sei18]; in fact, making use of the theory developed in
[SSVW15], in [MNS18, Sei18] a characterisation of all coupling conditions
giving rise to unitary or contractive dynamics for the Airy equation on metric
(star) graphs were given.

As for the KdV equation on metric graphs, there are some well-posedness
results in [Cav18, APC24] as well as stability results [CMn19, APC21, CMn23].

In this short note we consider the KdV equation on a metric star graph with
semi-infinite edges, i.e. on a bunch of halflines coupled at a single vertex,
which we call $0$. For simplicity, we work with finitely many edges. We will
consider general coefficients on the edges and include first-order terms, i.e.
we consider

$\partial_{t}u_{e}=-\alpha_{e}\partial_{x}^{3}u+\beta_{e}\partial_{x}u+\gamma_{e}u_{e}\partial_{x}u_{e}$

for each edge $e$ (where each single edge corresponds to a halfline). We
investigate existence of solitary waves for this equation on the metric star
graph. In order to fix the model, we have to specify the coupling condition at
the junction $0$. Here, we will work with linear coupling conditions coming
from the linear part of the KdV equation, i.e. the Airy equation. One
particular class of couplings stems from continuity of $u$ along the junction,
but we will also consider couplings giving rise to jumps of $u$ at the
junction (those are inspired by branching channels where there may be a large
gradient at the junction, which can be ideally modelled by a jump). As it
turns out, existence of solitary waves on the metric star graph is a rather
rare situation; the coefficients $\alpha_{e},\beta_{e},\gamma_{e}$ as well as
the wave speeds $c_{e}$ have to satisfy strong compatibility conditions in
order to allow for solitary waves.

Let us outline the remaining part of the paper. In Section 2 we review the KdV
equation (with general coefficients) on halflines and the real line, also
explaining the solitary wave solution for the equation. Then in Section 3 we
move on the the linear part, the Airy equation, on metric star graphs and
collect the necessary results for the coupling conditions at the junction. We
consider the KdV equation on metric star graphs in Section 4 and characterize
existence of solitary waves by means of compatibility conditions for the
coefficients.

##  2\. Solitary waves for Korteweg–de Vries equation on halflines and the
real line

The results stated in this section are essentially contained in [KdV95].
However, for self-consistency, for the reader’s convenience and to fix
notation we include full statements and proofs.

Let
$\Omega\in\mathopen{\mathchoice{\hbox{$\displaystyle\left\\{\hbox{}\right.$}}{\hbox{$\textstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\\{\hbox{}\right.$}}}(-\infty,0),(0,\infty),\mathbb{R}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\\}$}}{\hbox{$\textstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\\}$}}}$,
$\alpha,\beta,\gamma\in\mathbb{R}$ with $\alpha>0$ and $\gamma\neq 0$. Then
the _Korteweg–de Vries (KdV) equation_ (on $\Omega$) is given by

(1) $\partial_{t}u=-\alpha\partial_{x}^{3}u+\beta\partial_{x}u+\gamma
u\partial_{x}u\quad\text{on}\quad\mathbb{R}\times\Omega,$

where the standard choice of coefficients is $\alpha=1$, $\beta=0$, and
$\gamma=-6$.

######  Definition 2.1.

We say that $u\colon\mathbb{R}\times\Omega\to\mathbb{R}$ is a _travelling
wave_ for the KdV equation (1) provided there exist $\varphi\in
C^{3}(\mathbb{R})$ and $c\in\mathbb{R}\setminus\\{0\\}$ such that
$u\colon\mathbb{R}\times\Omega\to\mathbb{R}$ defined by
$u(t,x):=\varphi(x-ct)$ for all $t\in\mathbb{R}$ and $x\in\Omega$ solves (1).
Moreover, if $\varphi$ admits at most one local extremum and the limits
$\lim_{y\to\pm\infty}\varphi(y)$ exist then $u$ determined by $\varphi$ is
called _solitary wave_.

We want to find $\varphi\colon\mathbb{R}\to\mathbb{R}$ such that
$u\colon\mathbb{R}\times\Omega\to\mathbb{R}$ defined by

$u(t,x):=\varphi(x-ct)\quad(x\in\Omega,t\in\mathbb{R}),$

where $c\in\mathbb{R}$, is a solitary wave.

Plugging the description of $u$ into the equation we obtain

$-\alpha\varphi^{\prime\prime\prime}+(\beta+c)\varphi^{\prime}+\gamma\varphi\varphi^{\prime}=0.$

Integrating this equation, we get

$-\alpha\varphi^{\prime\prime}+(\beta+c)\varphi+\frac{\gamma}{2}\varphi^{2}=A$

for some $A\in\mathbb{R}$. The corresponding first order system reads, with
$\psi:=\varphi^{\prime}$,

(2) $\displaystyle\begin{split}\varphi^{\prime}&=\psi,\\\
\psi^{\prime}&=\frac{-A+(\beta+c)\varphi+\frac{\gamma}{2}\varphi^{2}}{\alpha}.\end{split}$

Define

$H(\varphi,\psi):=\frac{\psi^{2}}{2}-\frac{1}{\alpha}\bigl{(}-A\varphi+\frac{\beta+c}{2}\varphi^{2}+\frac{\gamma}{6}\varphi^{3}\bigr{)}.$

Then $H$ is constant along solutions of the first order system.

Let us compute the stationary points of the first order system, i.e.

$\displaystyle\varphi^{\prime}$ $\displaystyle=0,$
$\displaystyle\psi^{\prime}$ $\displaystyle=0.$

Thus, $\psi=0$ and $-A+(\beta+c)\varphi+\frac{\gamma}{2}\varphi^{2}=0$. Since
$\gamma\neq 0$, we obtain

(3) $\varphi^{2}+\frac{2(\beta+c)}{\gamma}\varphi-\frac{2A}{\gamma}=0.$

######  Lemma 2.2.

If $(\beta+c)^{2}+2A\gamma<0$ then there is no stationary point and all
trajectories of solutions of (2) are unbounded.

###### Proof.

Without loss of generality, let $\gamma>0$. Then the minimal value of
$\varphi\mapsto\frac{-A+(\beta+c)\varphi+\frac{\gamma}{2}\varphi^{2}}{\alpha}$
is given for $\varphi=-\frac{\beta+c}{\gamma}$ with value

$\frac{-A-\frac{1}{2\gamma}(\beta+c)^{2}}{\alpha}=\frac{1}{2\alpha\gamma}\bigl{(}-2A\gamma-(\beta+c)^{2}\bigr{)}.$

In view of (2), $\psi^{\prime}$ is bounded from below by a positive constant,
so $\psi$ will be unbounded and therefore also $\varphi$ will be unbounded. ∎

######  Lemma 2.3.

If $(\beta+c)^{2}+2A\gamma=0$ then there is exactly one stationary point

$p=\Bigl{(}-\frac{\beta+c}{\gamma},0\Bigr{)}.$

###### Proof.

Under the stated condition there is exactly one real solution of the quadratic
equation (3), namely $\varphi=-\frac{\beta+c}{\gamma}$, which corresponds to
the stationary point $p=\Bigl{(}-\frac{\beta+c}{\gamma},0\Bigr{)}$. ∎

######  Lemma 2.4.

If $(\beta+c)^{2}+2A\gamma>0$ then there are two stationary points

$\displaystyle p_{-}$
$\displaystyle=\Bigl{(}\frac{-(\beta+c)-\sqrt{(\beta+c)^{2}+2A\gamma}}{\gamma},0\Bigr{)},$
$\displaystyle p_{+}$
$\displaystyle=\Bigl{(}\frac{-(\beta+c)+\sqrt{(\beta+c)^{2}+2A\gamma}}{\gamma},0\Bigr{)}.$

Moreover, $p_{-}$ is a center an $p_{+}$ is a saddle point.

###### Proof.

Under the stated condition the quadratic equation (3) for $\varphi$ has two
real solutions, namely

$\varphi_{1,2}=-\frac{\beta+c}{\gamma}\pm\sqrt{\bigl{(}\frac{\beta+c}{\gamma}\bigr{)}^{2}+\frac{2A}{\gamma}}=\frac{-(\beta+c)\pm\sqrt{(\beta+c)^{2}+2A\gamma}}{\gamma}.$

Since $\psi=0$, we obtain the two stationary points.

Note that the right-hand side of the first order system (2) is given by
$f(\varphi,\psi):=\begin{pmatrix}\psi\\\
\frac{-A+(\beta+c)\varphi+\frac{\gamma}{2}\varphi^{2}}{\alpha}\end{pmatrix}$.
The Jacobian of $f$ at $p_{-}$ has the eigenvalues $\lambda$ determined by

$\lambda^{2}=-\frac{1}{\alpha}\sqrt{(\beta+c)^{2}+2A\gamma}<0.$

Thus, there are two conjugate complex eigenvalues on the imaginary axis, and
so $p_{-}$ is a center. The Jacobian of $f$ at $p_{+}$ has the eigenvalues
$\lambda$ determined by

$\lambda^{2}=+\frac{1}{\alpha}\sqrt{(\beta+c)^{2}+2A\gamma}>0.$

Thus, there is one positive and one negative eigenvalue and $p_{+}$ is a
saddle point. ∎

######  Lemma 2.5.

Let $(\beta+c)^{2}+2A\gamma>0$. Then there exists a unique solution
$(\varphi,\psi)$ of the first order system (2) such that

$\lim_{y\to\pm\infty}\varphi(y)=\frac{-(\beta+c)+\sqrt{(\beta+c)^{2}+2A\gamma}}{\gamma},\qquad\lim_{y\to\pm\infty}\psi(y)=0.$

###### Proof.

By Lemma 2.4, there exists a unique homoclinic orbit (for $p_{+}$) which
corresponds to the unique solution $y\mapsto(\varphi(y),\psi(y))$
approximating $p_{+}$ at $\pm\infty$, see also Figure 1. ∎

$\varphi$$\psi$ Figure 1. Phase portrait for (2).

In view of Lemma 2.5, in order to obtain a solitary wave we have to assume
$A=0$ and $\beta+c>0$; we therefore specialize ot this case.

######  Lemma 2.6.

Let $A=0$ and $\beta+c>0$. Let $(\varphi,\psi)$ be the unique solution of the
first order system according to Lemma 2.5. Then:

  1. (a)

$(\varphi,\psi)$ is given by

$\psi^{2}=\frac{\beta+c}{\alpha}\varphi^{2}+\frac{\gamma}{3\alpha}\varphi^{3}$

and

$\lim_{y\to\pm\infty}\varphi(y)=\lim_{y\to\pm\infty}\psi(y)=0.$

  2. (b)

There exists a unique $y_{0}\in\mathbb{R}$ such that
$\varphi(y_{0})=-\frac{3}{\gamma}(\beta+c)$ and $\psi(y_{0})=0$.

  3. (c)

Let $\gamma>0$. Then $\varphi(y)<0$ for all $y\in\mathbb{R}$, $\varphi$ is
decreasing on $(-\infty,y_{0})$ and $\varphi$ is increasing on
$(y_{0},\infty)$.

  4. (d)

Let $\gamma<0$. Then $\varphi(y)>0$ for all $y\in\mathbb{R}$, $\varphi$ is
increasing on $(-\infty,y_{0})$ and $\varphi$ is decreasing on
$(y_{0},\infty)$.

###### Proof.

(a) By Lemma 2.5, there exists a unique solution $(\varphi,\psi)$, and
$\varphi$ and $\psi$ have the stated limits at $\pm\infty$. Since $A=0$ we
have $p_{+}=\bigl{(}0,0\bigr{)}$. Moreover, as $H$ is constant along
trajectories, for the unique solution $(\varphi,\psi)$ describing the
homoclinic orbit we obtain $H(\varphi,\psi)=0$ and therefore

$\psi^{2}=\frac{\beta+c}{\alpha}\varphi^{2}+\frac{\gamma}{3\alpha}\varphi^{3}.$

(b) Let $y\in\mathbb{R}$ such that $\psi(y)=0$. Then

$0=\frac{\beta+c}{\alpha}\varphi(y)^{2}+\frac{\gamma}{3\alpha}\varphi(y)^{3},$

and therefore $\varphi(y)=0$ or $\varphi(y)=-\frac{3}{\gamma}(\beta+c)$.
Assume there are $y_{1},y_{2}\in\mathbb{R}$, $y_{1}<y_{2}$ such that
$\varphi(y_{1})=\varphi(y_{2})=-\frac{3}{\gamma}(\beta+c)$ and
$\psi(y_{1})=\psi(y_{2})=0$. Since $\lim_{y\to\pm\infty}\varphi(y)=0$ and the
first order system (2) is uniquely solvable given the values $\varphi(y)$ and
$\psi(y)$ of $\varphi$ and $\psi$ at $y\in\mathbb{R}$, then necessarily
$\varphi(y)=-\frac{3}{\gamma}(\beta+c)$ and $\psi(y)=0$ for all
$y\in(y_{1},y_{2})$ (otherwise we would have a periodic orbit, contradicting
the limit as $y\to\pm\infty$). Exploiting the first order system (2) then
yields

$\varphi^{\prime}(y)=0,\qquad\psi^{\prime}(y)=\frac{3(\beta+c)^{2}}{2\alpha\gamma}\neq
0\quad(y\in(y_{1},y_{2})),$

a contradiction.

(c) Let $\gamma>0$. Then the first component of $p_{-}$ is negative. The
unstable direction for the Jacobian of $f$ at $p_{+}$ is given by
$\operatorname{span}\begin{pmatrix}1\\\
\sqrt{\frac{1}{\alpha}(\beta+c)}\end{pmatrix}$. Therefore, the homoclinic
orbit describing the unique solution leaves $p_{+}$ in direction
$\begin{pmatrix}-1\\\ -\sqrt{\frac{1}{\alpha}(\beta+c)}\end{pmatrix}$. Hence,
$\varphi(y)<0$ for all $y\in(-\infty,y_{0})$ and since $\psi=\varphi^{\prime}$
is negative on $(-\infty,y_{0})$, $\varphi$ is decreasing there. Analogously,
the stable direction for the Jacobian of $f$ at $p_{+}$ is given by
$\operatorname{span}\begin{pmatrix}1\\\
-\sqrt{\frac{1}{\alpha}(\beta+c)}\end{pmatrix}$. Therefore, the homoclinic
orbit describing the unique solution enters $p_{+}$ in direction
$\begin{pmatrix}-1\\\ \sqrt{\frac{1}{\alpha}(\beta+c)}\end{pmatrix}$. Thus,
$\varphi(y)>0$ for all $y\in(y_{0},\infty)$ and since $\psi=\varphi^{\prime}$
is positive on $(y_{0},\infty)$, $\varphi$ is increasing there.

(d) The case $\gamma<0$ is proved analogously to the case $\gamma>0$. ∎

For the next theorem note that
$\operatorname{sech}(\cdot)=\frac{1}{\cosh(\cdot)}$ by definition.

######  Theorem 2.7.

Let $A=0$ and $\beta+c>0$. Then (1) admits a solitary wave solution determined
by

$\varphi(y)=-\frac{3(\beta+c)}{\gamma}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta+c}{\alpha}}}{2}(y-y_{0})\Bigr{)}\qquad(y\in\mathbb{R}).$

###### Proof.

Let $\gamma>0$. By Lemma 2.6, for $y<y_{0}$ we have $\varphi(y)<0$ and
$\psi(y)<0$, and moreover
$\psi(y)=-\sqrt{\frac{\beta+c}{\alpha}\varphi(y)^{2}+\frac{\gamma}{3\alpha}\varphi(y)^{3}}$.
Taking into account the facts that $\psi=\varphi^{\prime}$ and
$\varphi(y_{0})=-\frac{3}{\gamma}(\beta+c)$, applying separation of variables
and integrating, we observe

$\int_{\varphi(y)}^{-\frac{3}{\gamma}(\beta+c)}-\sqrt{\frac{1}{\frac{\beta+c}{\alpha}\varphi^{2}+\frac{\gamma}{3\alpha}\varphi^{3}}}\,d\varphi=\int_{y}^{y_{0}}\,dy=y_{0}-y.$

Integrating and solving for $\varphi(y)$ yields

$\varphi(y)=-\frac{3(\beta+c)}{\gamma}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta+c}{\alpha}}}{2}(y_{0}-y)\Bigr{)}\qquad(y<y_{0}).$

Analogously as above, Lemma 2.6 yields for $y>y_{0}$ that $\varphi(y)<0$ and
$\psi(y)>0$, and moreover
$\psi(y)=\sqrt{\frac{\beta+c}{\alpha}\varphi(y)^{2}+\frac{\gamma}{3\alpha}\varphi(y)^{3}}$.
Since $\psi=\varphi^{\prime}$ and $\varphi(y_{0})=-\frac{3}{\gamma}(\beta+c)$,
separation of variables yields

$\int_{-\frac{3}{\gamma}(\beta+c)}^{\varphi(y)}\sqrt{\frac{1}{\frac{\beta+c}{\alpha}\varphi^{2}+\frac{\gamma}{3\alpha}\varphi^{3}}}\,d\varphi=\int_{y_{0}}^{y}\,dy=y-y_{0}.$

Integrating and solving for $\varphi(y)$ we obtain

$\varphi(y)=-\frac{3(\beta+c)}{\gamma}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta+c}{\alpha}}}{2}(y-y_{0})\Bigr{)}\qquad(y>y_{0}).$

Since $\operatorname{sech}(x)=\operatorname{sech}(-x)$ for all
$x\in\mathbb{R}$, $\operatorname{sech}(0)=1$ and
$\varphi(y_{0})=-\frac{3(\beta+c)}{\gamma}$ we thus obtain

$\varphi(y)=-\frac{3(\beta+c)}{\gamma}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta+c}{\alpha}}}{2}(y-y_{0})\Bigr{)}\qquad(y\in\mathbb{R}).$

Now, let $\gamma<0$. Then analogously we obtain

$\varphi(y)=-\frac{3(\beta+c)}{\gamma}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta+c}{\alpha}}}{2}(y-y_{0})\Bigr{)}\qquad(y\in\mathbb{R}).\qed$

##  3\. The Airy equation on metric star graphs

In this section we consider the linear part of the Korteweg–de Vries equation
on metric star graphs with semi-infinite edges.

Consider star graph with halfline edges attached at the vertex $0$. Let
$E=E_{-}\cup E_{+}$ be finite, where $e\in E_{-}$ is an edge parametrized by
the interval $(-\infty,0)$, and $e\in E_{+}$ is an edge parametrized
$(0,\infty)$. For $e\in E$ let $\alpha_{e}>0$ and $\beta_{e}\in\mathbb{R}$.

$0$ Figure 2. A metric star graph with $|E_{-}|=5$ and $|E_{+}|=4$.

We consider the _Airy equation_

$\partial_{t}u_{e}=-\alpha_{e}\partial_{x}^{3}u_{e}+\beta_{e}\partial_{x}u_{e}\quad\text{on
$\mathbb{R}\times\Omega_{e}$},$

where
$\Omega\in\mathopen{\mathchoice{\hbox{$\displaystyle\left\\{\hbox{}\right.$}}{\hbox{$\textstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\\{\hbox{}\right.$}}}(-\infty,0),(0,\infty)\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\\}$}}{\hbox{$\textstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\\}$}}}$
corresponds to the parametrization of $e$ and $e\in E$. Note that the Airy
equation can (also) be deduced from the KdV equation by setting $\gamma=0$
(which was formally not allowed above) or linearizing the equation around the
solution $u=0$. In any case, the Airy equation yields the linear part of the
KdV equation.

We follow the arguments in [MNS18, Sei18] (note that we changed the sign in
front of the $\alpha_{e}$’s compared to the references). All vector spaces
appearing are over the field $\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$.

Define the minimal operator $A_{0}$ in $\mathcal{H}:=\bigoplus_{e\in
E_{-}}L_{2}(-\infty,0)\oplus\bigoplus_{e\in E_{+}}L_{2}(0,\infty)$ by

$\displaystyle D(A_{0})$ $\displaystyle:=\bigoplus_{e\in
E_{-}}C_{c}^{\infty}(-\infty,0)\oplus\bigoplus_{e\in
E_{+}}C_{c}^{\infty}(0,\infty),$ $\displaystyle A_{0}u$
$\displaystyle:=(-\alpha_{e}u_{e}^{\prime\prime\prime}+\beta_{e}u_{e}^{\prime})_{e\in
E}\quad(u\in D(A_{0})).$

Then an easy computation yields

$\displaystyle D(A_{0}^{*})$ $\displaystyle=\bigoplus_{e\in
E_{-}}H^{3}(-\infty,0)\oplus\bigoplus_{e\in E_{+}}H^{3}(0,\infty),$
$\displaystyle A_{0}^{*}u$
$\displaystyle:=(\alpha_{e}u_{e}^{\prime\prime\prime}-\beta_{e}u_{e}^{\prime})_{e\in
E}\quad(u\in D(A_{0})),$

and hence $A_{0}$ is skew-symmetric, i.e. densely defined and $A_{0}\subseteq-
A_{0}^{*}$.

We search for operators $A_{0}\subseteq A\subseteq-A_{0}^{*}$ such that the
abstract Cauchy problem

$\displaystyle u^{\prime}(t)$ $\displaystyle=Au(t)\quad(t>0),$ $\displaystyle
u(0)$ $\displaystyle=u_{0}\in\mathcal{H}$

is well-posed, i.e. $A$ is the generator of a $C_{0}$-semigroup. For this we
will have to impose coupling conditions at the vertex $0$. Note that this
abstract Cauchy problem is a functional analytic description of the Airy
equation on each edge $e\in E$ including the coupling conditions at $0$.

For $u\in D(A_{0}^{*})$ let us abbreviate

$u^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}):=\bigl{(}u_{e}^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\bigr{)}_{e\in
E_{-}},\quad
u^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}):=\bigl{(}u_{e}^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\bigr{)}_{e\in
E_{+}}\quad(k\in\mathopen{\mathchoice{\hbox{$\displaystyle\left\\{\hbox{}\right.$}}{\hbox{$\textstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\\{\hbox{}\right.$}}}0,1,2\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\\}$}}{\hbox{$\textstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\\}$}}}).$

Moreover, let us denote by $\alpha_{\pm}$ and $\beta_{\pm}$ the multiplication
operators on $\mathbb{K}^{E_{\pm}}$ defined by

$\alpha_{\pm}x:=(\alpha_{e}x_{e})_{e\in
E_{\pm}},\quad\beta_{\pm}x:=(\beta_{e}x_{e})_{e\in
E_{\pm}}\quad(x\in\mathbb{K}^{E_{\pm}}).$

Define

$\mathcal{G}_{\pm}:=\mathbb{K}^{E_{\pm}}\oplus\mathbb{K}^{E_{\pm}}\oplus\mathbb{K}^{E_{\pm}},$

and the block operator matrices $B_{\pm}$ on $\mathcal{G}_{\pm}$ by

$B_{\pm}=\begin{pmatrix}-\beta_{\pm}&0&\alpha_{\pm}\\\ 0&-\alpha_{\pm}&0\\\
\alpha_{\pm}&0&0\end{pmatrix}.$

Since $(\alpha_{e})_{e\in E}$, $(1/\alpha_{e})_{e\in E}$ and
$(\beta_{e})_{e\in E}$ are bounded we have
$B_{\pm}\in\mathcal{L}(\mathcal{G}_{\pm})$, $B_{\pm}$ is injective and
$B_{\pm}^{-1}\in\mathcal{L}(\mathcal{G}_{\pm})$. Define an indefinite inner
product $\left\langle\cdot\mkern 1.5mu,\mkern
1.5mu\cdot\right\rangle_{\pm}\colon\mathcal{G}_{\pm}\times\mathcal{G}_{\pm}\to\mathbb{K}$
by

$\left\langle x\mkern 1.5mu,\mkern
1.5muy\right\rangle_{\pm}:=\left(B_{\pm}x\,\middle|\,y\right)_{\mathcal{G}_{\pm}}\quad(x,y\in\mathcal{G}_{\pm}).$

Then $\mathcal{K}_{\pm}:=\bigl{(}\mathcal{G}_{\pm},\left\langle\cdot\mkern
1.5mu,\mkern 1.5mu\cdot\right\rangle_{\pm}\bigr{)}$ are Krein spaces.

For a linear operator $L\colon\mathcal{K}_{-}\to\mathcal{K}_{+}$ we define

$\displaystyle D(A_{L})$
$\displaystyle:=\mathopen{\mathchoice{\hbox{$\displaystyle\left\\{\hbox{}\right.$}}{\hbox{$\textstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\\{\hbox{}\right.$}}}u\in
D(A_{0}^{*});\;L\bigl{(}u(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}),u^{\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}),u^{\prime\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\bigr{)}=\bigl{(}u(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}),u^{\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}),u^{\prime\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\bigr{)}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\\}$}}{\hbox{$\textstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\\}$}}},$
$\displaystyle A_{L}u$ $\displaystyle:=-A_{0}^{*}u\quad(u\in D(A_{L})).$

######  Proposition 3.1 (see [MNS18, Theorem 3.9 and Theorem 3.16]).

Let $L\colon\mathcal{K}_{-}\to\mathcal{K}_{+}$ be a linear operator. Then:

  1. (a)

$A_{L}$ generates a unitary $C_{0}$-group if and only if $L$ is
$(\mathcal{K}_{-},\mathcal{K}_{+})$-unitary, i.e.

$\left\langle Lx\mkern 1.5mu,\mkern 1.5muLy\right\rangle_{+}=\left\langle
x\mkern 1.5mu,\mkern 1.5muy\right\rangle_{-}\quad(x,y\in\mathcal{K}_{-}).$

  2. (b)

$A_{L}$ generates a contractive $C_{0}$-semigroup if and only if $L$ is
$(\mathcal{K}_{-},\mathcal{K}_{+})$-contractive, i.e.

$\left\langle Lx\mkern 1.5mu,\mkern 1.5muLx\right\rangle_{+}\leq\left\langle
x\mkern 1.5mu,\mkern 1.5mux\right\rangle_{-}\quad(x\in\mathcal{K}_{-}),$

and $L^{\sharp}$ is $(\mathcal{K}_{+},\mathcal{K}_{-})$-contractive, i.e.

$\left\langle L^{\sharp}x\mkern 1.5mu,\mkern
1.5muL^{\sharp}x\right\rangle_{-}\leq\left\langle x\mkern 1.5mu,\mkern
1.5mux\right\rangle_{+}\quad(x\in\mathcal{K}_{+}),$

where $L^{\sharp}\colon\mathcal{K}_{+}\to\mathcal{K}_{-}$ is the
$(\mathcal{K}_{-},\mathcal{K}_{+})$-adjoint of $L$ given by

$\left\langle Lx\mkern 1.5mu,\mkern 1.5muy\right\rangle_{+}=\left\langle
x\mkern 1.5mu,\mkern
1.5muL^{\sharp}y\right\rangle_{-}\quad(x\in\mathcal{K}_{-},y\in\mathcal{K}_{+}).$

A special case of coupling conditions at $0$ stems from separating the
boundary values of the first derivatives. In order to formulate this, let
$Y\subseteq\mathbb{K}^{E_{-}}\oplus\mathbb{K}^{E_{+}}$ be a subspace and
$U\colon(\mathbb{K}^{E_{+}},(\alpha_{e})_{e\in
E_{+}})\to(\mathbb{K}^{E_{-}},(\alpha_{e})_{e\in E_{-}})$ be linear. We define

$\displaystyle D(A_{Y,U})$ $\displaystyle:=\Bigl{\\{}u\in
D(A_{0}^{*});\;\bigl{(}u(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}),u(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\bigr{)}\in
Y,\,$ $\displaystyle\qquad\qquad\qquad\qquad\begin{pmatrix}-\alpha_{-}&0\\\
0&\alpha_{+}\end{pmatrix}\begin{pmatrix}u^{\prime\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
u^{\prime\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}+\begin{pmatrix}\frac{\beta_{-}}{2}&0\\\
0&-\frac{\beta_{+}}{2}\end{pmatrix}\begin{pmatrix}u(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
u(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}\in
Y^{\bot},$ $\displaystyle\qquad\qquad\qquad\qquad
u^{\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})=Uu^{\prime}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\Bigr{\\}},$
$\displaystyle A_{Y,U}u$ $\displaystyle:=-A_{0}^{*}u\quad(u\in D(A_{Y,U})).$

######  Proposition 3.2 (compare [MNS18, Corollary 3.20 and Theorem 3.23]).

Let $Y\subseteq\mathbb{K}^{E_{-}}\oplus\mathbb{K}^{E_{+}}$ a subspace and
$U\colon(\mathbb{K}^{E_{+}},(\alpha_{e})_{e\in
E_{+}})\to(\mathbb{K}^{E_{-}},(\alpha_{e})_{e\in E_{-}})$ linear. Then:

  1. (a)

$A_{Y,U}$ generates a unitary $C_{0}$-group if and only if $U$ is unitary.

  2. (b)

$A_{Y,U}$ generates a contractive $C_{0}$-semigroup if and only if $U$ is a
contraction.

##  4\. The KdV equation on metric star graphs

As in Section 3, we consider a metric star graph with halflines attached at
the vertex $0$. Let $E=E_{-}\cup E_{+}$ be finite, where $e\in E_{-}$ is an
edge parametrized by the interval $(-\infty,0)$, and $e\in E_{+}$ is an edge
parametrized $(0,\infty)$. For $e\in E$ let
$\alpha_{e},\beta_{e},\gamma_{e}\in\mathbb{R}$ with $\alpha_{e}>0$ and
$\gamma_{e}\neq 0$.

In this section we consider the KdV equation on the metric star graph, i.e.

(4)
$\partial_{t}u_{e}=-\alpha_{e}\partial_{x}^{3}u_{e}+\beta_{e}\partial_{x}u_{e}+\gamma_{e}u_{e}\partial_{x}u_{e}\quad\text{on
$\mathbb{R}\times\Omega_{e}$}\quad(e\in E),$

where $\Omega_{E}:=(-\infty,0)$ for $e\in E_{-}$ and $\Omega_{e}:=(0,\infty)$
for $e\in E_{+}$.

######  Definition 4.1.

We say that $u=(u_{e})_{e\in E}$ is a _travelling wave_ for the KdV equation
(4) on the star graph provided for $e\in E$ there exist $\varphi_{e}\in
C^{3}(\mathbb{R})$ and $c_{e}\in\mathbb{R}\setminus\\{0\\}$ such that
$u=(u_{e})_{e\in E}$ with $u_{e}\colon\mathbb{R}\times\Omega_{e}\to\mathbb{R}$
defined by $u_{e}(t,x):=\varphi_{e}(x-c_{e}t)$ ($t\in\mathbb{R}$,
$x\in\Omega_{e}$) solves (4). Moreover, if for all $e\in E$ we have that
$\varphi_{e}$ admits at most one local extremum and the limits
$\lim_{y\to\pm\infty}\varphi_{e}(y)$ exist then $u=(u_{e})$ determined by
$(\varphi_{e})$ is called _solitary wave_.

We now combine the results reviewed in Sections 2 and 3, i.e. we look for
solitary waves for (4) satisfying suitable coupling conditions at the vertex
$0$. According to Theorem 2.7, for $e\in E$ let $\varphi_{e}\in
C^{3}(\mathbb{R})$ be defined by

$\varphi_{e}(y):=-\frac{3(\beta_{e}+c_{e})}{\gamma_{e}}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta_{e}+c_{e}}{\alpha_{e}}}}{2}(y-y_{0,e})\Bigr{)}\qquad(y\in\mathbb{R}),$

where $c_{e}\in\mathbb{R}$ and $y_{0,e}\in\mathbb{R}$.

For $u\in\bigoplus_{e\in E_{-}}H^{3}(-\infty,0)\oplus\bigoplus_{e\in
E_{+}}H^{3}(0,\infty)$ define

$u^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}):=\bigl{(}u_{e}^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\bigr{)}_{e\in
E_{-}},\quad
u^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}):=\bigl{(}u_{e}^{(k)}(0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\bigr{)}_{e\in
E_{+}}\quad(k\in\mathopen{\mathchoice{\hbox{$\displaystyle\left\\{\hbox{}\right.$}}{\hbox{$\textstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptstyle\left\\{\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\\{\hbox{}\right.$}}}0,1,2\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\\}$}}{\hbox{$\textstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptstyle\left.\hbox{}\right\\}$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\\}$}}}).$

As above, let us denote by $\alpha_{\pm}$ and $\beta_{\pm}$ the multiplication
operators on $\mathbb{K}^{E_{\pm}}$ defined by

$\alpha_{\pm}x:=(\alpha_{e}x_{e})_{e\in
E_{\pm}},\quad\beta_{\pm}x:=(\beta_{e}x_{e})_{e\in
E_{\pm}}\quad(x\in\mathbb{K}^{E_{\pm}}).$

Note that if we have a travelling wave for (4) with $\varphi_{e}$ as above,
then

$u(t,\cdot)=(u_{e}(t,\cdot))_{e\in E}\in\bigoplus_{e\in
E_{-}}H^{3}(-\infty,0)\oplus\bigoplus_{e\in
E_{+}}H^{3}(0,\infty)\quad(t\in\mathbb{R}).$

We impose coupling conditions in the spirit of Proposition 3.2. Specifically,
we first consider continuity conditions at the vertex $0$, which can be
written as $Y:=\operatorname{span}(\mathbbold{1})$ being the one-dimensional
vector space of vectors having equal entries.

######  Theorem 4.2.

Let
$Y:=\operatorname{span}(\mathbbold{1})\subseteq\mathbb{K}^{E_{-}}\oplus\mathbb{K}^{E_{+}}$,
and $U\colon(\mathbb{K}^{E_{+}},(\alpha_{e})_{e\in
E_{+}})\to(\mathbb{K}^{E_{-}},(\alpha_{e})_{e\in E_{-}})$ contractive.
Consider (4) with coupling conditions

$\displaystyle\bigl{(}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}),u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\bigr{)}$
$\displaystyle\in Y,$ $\displaystyle\begin{pmatrix}-\alpha_{-}&0\\\
0&\alpha_{+}\end{pmatrix}\begin{pmatrix}\partial_{x}^{2}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
\partial_{x}^{2}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}+\begin{pmatrix}\frac{\beta_{-}}{2}&0\\\
0&-\frac{\beta_{+}}{2}\end{pmatrix}\begin{pmatrix}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}$
$\displaystyle\in Y^{\bot},$
$\displaystyle\partial_{x}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})$
$\displaystyle=U\partial_{x}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})$

for all $t\in\mathbb{R}$. Then (4) admits a solitary wave if and only if

$U\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{+}}=\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in E_{-}}$

and for all $e,\tilde{e}\in E$ we have

$\displaystyle\beta_{e}+c_{e}$ $\displaystyle>0,$
$\displaystyle\frac{y_{0,e}}{c_{e}}$
$\displaystyle=\frac{y_{0,\tilde{e}}}{c_{\tilde{e}}},$
$\displaystyle\sqrt{\frac{\beta_{e}+c_{e}}{\alpha_{e}}}c_{e}$
$\displaystyle=\sqrt{\frac{\beta_{\tilde{e}}+c_{\tilde{e}}}{\alpha_{\tilde{e}}}}c_{\tilde{e}},$
$\displaystyle\frac{(\beta_{e}+c_{e})}{\gamma_{e}}$
$\displaystyle=\frac{(\beta_{\tilde{e}}+c_{\tilde{e}})}{\gamma_{\tilde{e}}},$
$\displaystyle\sum_{{e}\in E_{-}}\frac{\alpha_{{e}}}{c_{{e}}^{2}}-\sum_{{e}\in
E_{+}}\frac{\alpha_{{e}}}{c_{{e}}^{2}}$ $\displaystyle=0,$
$\displaystyle\sum_{e\in E_{-}}\beta_{e}-\sum_{e\in E_{+}}\beta_{e}$
$\displaystyle=0.$

The solitary wave is then given by

$\varphi_{e}(y):=-\frac{3(\beta_{e}+c_{e})}{\gamma_{e}}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta_{e}+c_{e}}{\alpha_{e}}}}{2}(y-y_{0,e})\Bigr{)}\qquad(y\in\mathbb{R},e\in
E).$

Note that the condition

$U\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{+}}=\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in E_{-}}$

fits well to the fifth assumption on the coefficients, since

${}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lVert\hbox{}\right.$}}{\hbox{$\textstyle\left\lVert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lVert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lVert\hbox{}\right.$}}}U\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{+}}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rVert$}}{\hbox{$\textstyle\left.\hbox{}\right\rVert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rVert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rVert$}}}_{\alpha_{-}}^{2}={}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lVert\hbox{}\right.$}}{\hbox{$\textstyle\left\lVert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lVert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lVert\hbox{}\right.$}}}\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{-}}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rVert$}}{\hbox{$\textstyle\left.\hbox{}\right\rVert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rVert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rVert$}}}_{\alpha_{-}}^{2}=\sum_{e\in
E_{-}}{}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}\frac{1}{c_{e}}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}^{2}\alpha_{e}=\sum_{e\in
E_{+}}{}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}\frac{1}{c_{e}}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}^{2}\alpha_{e}={}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lVert\hbox{}\right.$}}{\hbox{$\textstyle\left\lVert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lVert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lVert\hbox{}\right.$}}}\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{+}}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rVert$}}{\hbox{$\textstyle\left.\hbox{}\right\rVert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rVert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rVert$}}}_{\alpha_{+}}^{2}.$

So $U$ can indeed be unitary or contractive, and preserves the weighted norm
for inverse wave speeds.

###### Proof.

$\Longrightarrow$: A solitary wave on the metric star graph is also a solitary
wave for each edge $e\in E$ and is thus given by

$\varphi_{e}(y):=-\frac{3(\beta_{e}+c_{e})}{\gamma_{e}}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta_{e}+c_{e}}{\alpha_{e}}}}{2}(y-y_{0,e})\Bigr{)}\qquad(y\in\mathbb{R},e\in
E).$

Using the continuity condition, we obtain that there exist $\varphi\in
C^{3}(\mathbb{R})$ and $k_{e}\in\mathbb{R}$ for all $e\in E$ such that

$\varphi_{e}(t)=\varphi(k_{e}t)\quad(t\in\mathbb{R}).$

We easily see from the continuity condition that
$k_{e}c_{e}=k_{\tilde{e}}c_{\tilde{e}}$ for all $e,\tilde{e}\in E$. Inspecting
the coupling conditions and comparing the coefficients yield the conditions.

$\Longleftarrow$: By the first assumption on the coefficients and Theorem 2.7
we obtain solitary waves described by $\varphi_{e}$ on each edge $e\in E$. It
remains to show that these $\varphi_{e}$ can be coupled at the vertex is a way
that fits to the coupling conditions.

Exploiting the continuity condition, we need to satisfy

$\varphi_{e}(-c_{e}t)=\varphi_{\tilde{e}}(-c_{\tilde{e}}t)\quad(e,\tilde{e}\in
E,t\in\mathbb{R}).$

Put differently, there needs to be $\varphi\in C^{3}(\mathbb{R})$ such that

$\varphi_{e}(t)=\varphi(k_{e}t)\quad(e\in E,t\in\mathbb{R})$

for some $k_{e}\in\mathbb{R}$. In particular
$k_{e}c_{e}=k_{\tilde{e}}c_{\tilde{e}}$ for all $e,\tilde{e}\in E$. Since by
Theorem 2.7 we know how $\varphi_{e}$ looks like, we thus need

$-\frac{3(\beta_{e}+c_{e})}{\gamma_{e}}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta_{e}+c_{e}}{\alpha_{e}}}}{2}(-c_{e}t-y_{0,e})\Bigr{)}=-\frac{3(\beta_{\tilde{e}}+c_{\tilde{e}})}{\gamma_{\tilde{e}}}\operatorname{sech}^{2}\Bigl{(}\frac{\sqrt{\frac{\beta_{\tilde{e}}+c_{\tilde{e}}}{\alpha_{\tilde{e}}}}}{2}(-c_{\tilde{e}}t-y_{0,{\tilde{e}}})\Bigr{)}\quad(e,\tilde{e}\in
E,t\in\mathbb{R}).$

The second, third and forth assumption on the coefficients now yield that this
is indeed true. Hence, the first coupling condition can be achieved.

For the second coupling condition, we note that for $e,\tilde{e}\in E$ we have

$u_{\tilde{e}}(t,x)=\varphi_{\tilde{e}}(x-c_{\tilde{e}}t)=\varphi_{e}\bigl{(}\frac{c_{e}}{c_{\tilde{e}}}x-c_{e}t\bigr{)}\quad(t\in\mathbb{R},x\in\Omega_{\tilde{e}}).$

Hence,

$\partial_{x}^{2}u_{\tilde{e}}(t,0)=\bigl{(}\frac{c_{e}}{c_{\tilde{e}}}\bigr{)}^{2}\varphi_{e}^{\prime\prime}\bigl{(}-c_{e}t\bigr{)}=\frac{1}{c_{\tilde{e}}^{2}}c_{e}^{2}\varphi_{e}^{\prime\prime}(-c_{e}t)\quad(t\in\mathbb{R}).$

Therefore, the second coupling condition requires

$\sum_{\tilde{e}\in
E_{-}}-\alpha_{\tilde{e}}\frac{1}{c_{\tilde{e}}^{2}}c_{e}^{2}\varphi_{e}^{\prime\prime}(-c_{e}t)+\sum_{\tilde{e}\in
E_{+}}\alpha_{e}\frac{1}{c_{\tilde{e}}^{2}}c_{e}^{2}\varphi_{e}^{\prime\prime}(-c_{e}t)+\Bigl{(}\sum_{e\in
E_{-}}\frac{\beta_{e}}{2}\sum_{e\in
E_{+}}-\frac{\beta_{e}}{2}\Bigr{)}\varphi_{e}(-c_{e}t)=0,$

or put differently

$\Bigl{(}\sum_{\tilde{e}\in
E_{-}}-\alpha_{\tilde{e}}\frac{1}{c_{\tilde{e}}^{2}}+\sum_{\tilde{e}\in
E_{+}}\alpha_{\tilde{e}}\frac{1}{c_{\tilde{e}}^{2}}\Bigr{)}c_{e}^{2}\varphi_{e}^{\prime\prime}(-c_{e}t)+\Bigl{(}\sum_{e\in
E_{-}}\frac{\beta_{e}}{2}+\sum_{e\in
E_{+}}-\frac{\beta_{e}}{2}\Bigr{)}\varphi_{e}(-c_{e}t)=0\quad(t\in\mathbb{R}).$

Now, the fifth and sixth assumption on the coefficients yield the assertion.

Let us turn to the third coupling condition. We have

$\partial_{x}u_{e}(t,0{{\scriptstyle{\pm}}})=\varphi_{e}(-c_{e}t)=k_{e}\varphi^{\prime}(-k_{e}c_{e}t)\quad(t\in\mathbb{R},e\in
E_{\pm}).$

Hence, there exists $A\in C^{2}(\mathbb{R})$ such that

$\partial_{x}u_{e}(t,0{{\scriptstyle{\pm}}})=A(t)\frac{1}{c_{e}}\quad(e\in
E_{\pm}).$

By assumption on $U$ we have

$U\partial_{x}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})=A(t)U\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{+}}=A(t)\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{-}}=\partial_{x}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}).\qed$

As a direct consequence we obtain that if the parameter set on any edge is
related by scaling to the parameter set of any other edge, then solitary waves
require equality of the parameter sets over the edges.

######  Corollary 4.3.

For coupling conditions as in the previous theorem, if for all $e_{1},e_{2}\in
E$ there exists $p_{21}\in\mathbb{R}$ such that

$(\alpha_{e_{2}},\beta_{e_{2}},\gamma_{e_{2}},c_{e_{2}})=p_{21}(\alpha_{e_{1}},\beta_{e_{1}},\gamma_{e_{1}},c_{e_{1}})$

and if there exists a solitary wave, then

$(\alpha_{e_{2}},\beta_{e_{2}},\gamma_{e_{2}},c_{e_{2}})=(\alpha_{e_{1}},\beta_{e_{1}},\gamma_{e_{1}},c_{e_{1}})$

for all $e_{1},e_{2}\in E$.

We now fix $(\alpha_{e})_{e\in E}$ and $(\beta_{e})_{e\in E}$ (the parameters
of the linearized part) to be constant, so equal on all edges. Then existence
of solitary waves requires the remaining parameters to be constant and the
metric graph to be _balanced_ , i.e. $|E_{-}|=|E_{+}|$.

######  Corollary 4.4.

Let
$Y:=\operatorname{span}(\mathbbold{1})\subseteq\mathbb{K}^{E_{-}}\oplus\mathbb{K}^{E_{+}}$,
and $U\colon(\mathbb{K}^{E_{+}},(\alpha_{e})_{e\in
E_{+}})\to(\mathbb{K}^{E_{-}},(\alpha_{e})_{e\in E_{+}-})$ contractive. Let
$(\alpha_{e})_{e\in E}$ and $(\beta_{e})_{e\in E}$ be constant. Consider (4)
with coupling conditions

$\displaystyle\bigl{(}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}),u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\bigr{)}$
$\displaystyle\in Y,$ $\displaystyle\begin{pmatrix}-\alpha_{-}&0\\\
0&\alpha_{+}\end{pmatrix}\begin{pmatrix}\partial_{x}^{2}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
\partial_{x}^{2}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}+\begin{pmatrix}\frac{\beta_{-}}{2}&0\\\
0&-\frac{\beta_{+}}{2}\end{pmatrix}\begin{pmatrix}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}$
$\displaystyle\in Y^{\bot},$
$\displaystyle\partial_{x}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})$
$\displaystyle=U\partial_{x}u(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})$

for all $t\in\mathbb{R}$. Then (4) admits a solitary wave with
$c_{e}\geq-\frac{2}{3}\beta_{e}$ for all $e\in E$ if and only if and
$U\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in
E_{-}}=\bigl{(}\frac{1}{c_{e}}\bigr{)}_{e\in E_{+}}$,
${}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}E_{-}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}={}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}E_{+}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}$,
and $(\gamma_{e})_{e\in E}$, $(c_{e})_{e\in E}$ and $(y_{0,e})_{e\in E}$ are
constant.

###### Proof.

Assume we have solitary waves with $c_{e}\geq-\frac{2}{3}\beta_{e}$ for all
$e\in E$. The equality $\sum_{e\in E_{-}}\beta_{e}+\sum_{e\in
E_{+}}\beta_{e}=0$ and constancy of $(\beta_{e})_{e\in E}$ yields
${}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}E_{-}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}={}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}E_{+}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}$.
The equality
$\sqrt{\frac{\beta_{e}+c_{e}}{\alpha_{e}}}c_{e}=\sqrt{\frac{\beta_{\tilde{e}}+c_{\tilde{e}}}{\alpha_{\tilde{e}}}}c_{\tilde{e}}$
for $e,\tilde{e}\in E$ and constancy of $(\alpha_{e})_{e\in E}$ and
$(\beta_{e})_{e\in E}$ yields constancy of $(c_{e})_{e\in E}$ since
$[-\frac{2}{3}\beta,\infty)\ni c\mapsto\sqrt{\beta+c}c$ is injective. Then
$y_{0,e}\frac{1}{c_{e}}=y_{0,\tilde{e}}\frac{1}{c_{\tilde{e}}}$ for
$e,\tilde{e}\in E$ yields constancy of $(y_{0,e})_{e\in E}$, and
$\frac{(\beta_{e}+c_{e})}{\gamma_{e}}=\frac{(\beta_{\tilde{e}}+c_{\tilde{e}})}{\gamma_{\tilde{e}}}$
for $e,\tilde{e}\in E$ yields constancy of $(\gamma_{e})_{e\in E}$.

To show the converse, if the graph is balanced and $(\gamma_{e})_{e\in E}$,
$(c_{e})_{e\in E}$ and $(y_{0,e})_{e\in E}$ are constant, we easily see from
Theorem 2.7 that we then obtain a solitary wave. ∎

The previous corollary actually states that we can view the graph as a bunch
of
${}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}E_{-}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}$
real lines and the solitary wave on the graph is nothing but the same solitary
wave on each of this real lines.

If we drop the continuity assumption at the vertex $0$, we obtain somewhat
non-trivial solitary waves. The proof is analogous to the one of Theorem 2.7.
We demonstrate this by means of a so-called _$Y$ -junction_, i.e. a metric
star graph with three edges.

######  Proposition 4.5.

Let
${}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}E_{-}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}=1$,
${}\mathopen{\mathchoice{\hbox{$\displaystyle\left\lvert\hbox{}\right.$}}{\hbox{$\textstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptstyle\left\lvert\hbox{}\right.$}}{\hbox{$\scriptscriptstyle\left\lvert\hbox{}\right.$}}}E_{+}\mathclose{\mathchoice{\hbox{$\displaystyle\left.\hbox{}\right\rvert$}}{\hbox{$\textstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptstyle\left.\hbox{}\right\rvert$}}{\hbox{$\scriptscriptstyle\left.\hbox{}\right\rvert$}}}=2$,
$a\in\mathbb{R}\setminus\\{0\\}$, $Y:=\operatorname{span}\\{(1,a,a)\\}$, and
$U\colon(\mathbb{K}^{E_{+}},(\alpha_{e})_{e\in
E_{+}})\to(\mathbb{K}^{E_{-}},(\alpha_{e})_{e\in E_{-}})$ contractive. Then
(4) with coupling conditions

$\displaystyle\bigl{(}u_{-}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}),u_{+,1}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}),u_{+,2}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\bigr{)}$
$\displaystyle\in Y,$ $\displaystyle\begin{pmatrix}-\alpha_{-}&0&0\\\
0&\alpha_{+,1}&0\\\
0&0&\alpha_{+,2}\end{pmatrix}\begin{pmatrix}\partial_{x}^{2}u_{-}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
\partial_{x}^{2}u_{+,1}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\\\
\partial_{x}^{2}u_{+,2}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}+\begin{pmatrix}\frac{\beta_{-}}{2}&0&0\\\
0&-\frac{\beta_{+,1}}{2}&0\\\
0&0&-\frac{\beta_{+,2}}{2}\end{pmatrix}\begin{pmatrix}u_{-}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})\\\
u_{+,1}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\\\
u_{+,2}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}$
$\displaystyle\in Y^{\bot},$
$\displaystyle\partial_{x}u_{-}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{-}}$}}}{\vbox{\hbox{$\scriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}}{\vbox{\hbox{$\scriptscriptstyle{-}$}}})$
$\displaystyle=U\begin{pmatrix}\partial_{x}u_{+,1}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\\\
\\\
\partial_{x}u_{+,2}(t,0\mathchoice{\vbox{\hbox{${\scriptstyle{+}}$}}}{\vbox{\hbox{$\scriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}}{\vbox{\hbox{$\scriptscriptstyle{+}$}}})\end{pmatrix}$

for all $t\in\mathbb{R}$ admits a solitary wave if and only if

$U\begin{pmatrix}1/c_{+,1}\\\ 1/c_{+2}\end{pmatrix}=\frac{1}{c_{-}}$

and for all $j\in\\{1,2\\}$ we have

$\displaystyle\beta_{-}+c_{-},\;\beta_{+,j}+c_{+,j}$ $\displaystyle>0,$
$\displaystyle\frac{y_{0,c_{-}}}{c_{-}}$
$\displaystyle=\frac{y_{0,c_{+,j}}}{c_{+,j}},$
$\displaystyle\sqrt{\frac{\beta_{-}+c_{-}}{\alpha_{-}}}c_{-}$
$\displaystyle=\sqrt{\frac{\beta_{+,j}+c_{+,j}}{\alpha_{+,j}}}c_{+,j},$
$\displaystyle a\frac{(\beta_{-}+c_{-})}{\gamma_{-}}$
$\displaystyle=\frac{(\beta_{+,j}+c_{+,j})}{\gamma_{+,j}},$
$\displaystyle\frac{\alpha_{-}}{c_{-}^{2}}$
$\displaystyle=a^{2}\bigl{(}\frac{\alpha_{+,1}}{c_{+,1}^{2}}+\frac{\alpha_{+,2}}{c_{+,2}^{2}}\Bigr{)},$
$\displaystyle\beta_{-}$
$\displaystyle=a^{2}\bigl{(}\beta_{+,1}+\beta_{+,2}\Bigr{)}.$

###### Proof.

The proof is analogous to the proof of Theorem 2.7. ∎

## Acknowledgements

C.S. is grateful to Mitja Röder for useful discussions on the topic.

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11institutetext: 1Istituto Nazionale di Astrofisica, Osservatorio Astronomico
di Brera, via E. Bianchi 46, 23807 Merate (LC), Italy
11email<EMAIL_ADDRESS>
# Rethinking the 67 Hz QPO in GRS 1915+105: type-C QPOs at the innermost
stable circular orbit

S.E. Motta 11 T.M. Belloni, Deceased.11

(Received July 1, 2023; accepted January 30, 2024)

###### Abstract

Context. The study of Quasi-Periodic Oscillations (QPO) at low and high
frequency in the variability of the high-energy emission from black-hole
binaries and their physical interpretation in terms of signatures of General
Relativity in the strong-field regime.

Aims. To understand the nature of the 67 Hz QPOs observed in the X-ray
emission of the peculiar black-hole binary GRS 1915+105 within the general
classification of QPOs, and to determine the spin of the black hole in the
system by applying the Relativistic Precession Model (RPM).

Methods. Within the RPM, the only relativistic frequency that is stable in
time over a large range of accretion rates and can be as low as 67 Hz (for a
black-hole mass as measured dynamically) is the nodal frequency at the
Innermost Stable Circular Orbit (ISCO). In the application of the model, this
corresponds to type-C QPOs. Under this assumption, it is possible to measure
the spin of the black hole by using the mass of the black hole previously
obtained via dynamical measurements. We re-analysed a large number of RossiXTE
observations to check whether other timing features confirm this hypothesis.

Results. The identification of the 67 Hz QPO as the nodal frequency at ISCO
yields a value of 0.706 $\pm$0.034 for the black hole spin. With this spin,
the only two QPO detections at higher frequencies available in the literature
are consistent with being orbital frequencies at a radius outside ISCO. The
high-frequency bumps often observed at frequencies between 10 and 200 Hz
follow the correlation expected for orbital and periastron-precession
frequencies at even larger radii.

###### Key Words.:

X-ray: binaries – accretion, accretion disks – relativistic processes – black
hole physics – stars: black holes

##  1 Introduction

Quasi-periodic oscillations (QPOs) are variability features frequently
observed in the X-ray emissions of compact objects undergoing accretion. These
oscillations are believed to originate from the innermost regions of the
accretion flow. In a power density spectrum (PDS), QPOs manifest as relatively
narrow peaks and their centroid frequency can be linked to dynamic motion
and/or accretion-related time scales. QPOs have been known for several
decades, yet their precise nature remains enigmatic. Numerous models have been
proposed to explain their origin, leading to ongoing debates in the field (see
Ingram & Motta, 2019, for a recent review).

At low frequencies ($<$ 30 Hz), three “flavors” of QPO have been observed in
black-hole (BH) X-ray binaries (e.g., Wijnands & van der Klis, 1999; Remillard
et al., 2002; Casella et al., 2004, 2005). The most common of these types of
QPO - called Type-C QPOs (Casella et al., 2005) \- has a centroid frequency
that can vary over a broad range ($\sim$0.1-30Hz). Although other models have
been proposed (see, e.g, Belloni & Stella, 2014, for a review), type-C QPOs
have been often explained as a manifestation of the nodal (or Lense-
Thirring111The Lense-Thirring precession frequency is a weak-field
approximation of the nodal precession frequency, see Stella & Vietri (1998).)
precession of plasma orbiting around the BH (e.g., Stella & Vietri, 1998;
Ingram et al., 2009). They have also been associated to QPOs observed in
neutron-star (NS) X-ray binaries, called Horizontal Branch Oscillations (HBO),
suggesting that the same process is at work in both classes of systems
(Psaltis et al., 1999; Casella et al., 2005; Motta et al., 2017). In addition,
QPOs with centroid frequencies above hundred Hz (up to $\sim$500 Hz), although
very rarely, have been observed from a small number of BH systems (e.g.,
Belloni et al., 2012), either isolated or, in an even smaller number of cases,
in pairs. Such high-frequency oscillations are referred to as high-frequency
QPOs (HFQPOs).

Within the Theory of General Relativity (GR), a particle in a bound orbit
around a massive object, together with the orbital frequency ($\nu_{\phi}$),
has two additional frequencies associated to it, the vertical and radial
epicyclic frequencies ($\nu_{\theta}$ and $\nu_{\rm r}$, respectively). From
these frequencies, one can easily calculate two additional frequencies
associated to the orbit: the nodal precession frequency ($\nu_{\rm nod}$ =
$\nu_{\phi}$ \- $\nu_{\theta}$), and the periastron precession frequency
($\nu_{\rm per}$ = $\nu_{\phi}$ \- $\nu_{\rm r}$). The gas accreting onto a BH
can be seen as made of orbiting particles and therefore it is possible that
the observed characteristic time scales yielded by the QPOs are the products
of relativistic effects which can be associated to the aforementioned
frequencies. If this association is established, it represents a powerful
diagnostic tool for both GR in the strong-field regime, and the physics of
accretion.

Along these lines, the Relativistic Precession Model (RPM Stella & Vietri,
1998; Stella et al., 1999; Stella & Vietri, 1999) associates the nodal,
periastron precession, and orbital frequency with the type-C QPO and the two
HFQPOs - lower and upper, respectively - observed in the light curves of
accreting BHs and NSs. For BHs, the number of detections of HFQPOs is very low
and they are visible together with a type-C QPO even more rarely. Despite the
scarcity of data, Motta et al. (2014a, b, 2022) showed that the RPM can be
associated to the QPOs and broad noise components observed in the the BH
binaries GRO J1655-40, XTE J1550-564 and XTE J1859+226. These works showed
that the application of the RPM allows the self-consistent estimate of both
spin and mass of a BH from timing features.

The bright BH binary GRS 1915+105 went into outburst in 1992 and is still
active, although in a low-flux state (see, e.g., Motta et al., 2021). This
system is known to be rather peculiar, displaying extreme structured
variability on time scales above the second (see Fender & Belloni, 2004, for a
review). Despite these peculiarities, the PDS from GRS 1915+105 is not unlike
that of other BH binaries, and strong type-C QPOs have been observed in its
harder states (e.g., Markwardt et al., 1997; Ratti et al., 2012). At high
frequencies, at variance with other systems, GRS 1915+105 has shown a number
of features which have been classified as HFQPOs. In particular, a peak around
67 Hz has been consistently detected with RossiXTE for the sixteen years of
operation of the satellite and in the past few years also by Astrosat (see,
e.g., Morgan et al., 1997; Belloni & Altamirano, 2013b; Belloni et al., 2019).
This QPO was never observed together with a type-C QPO and represents a very
stable frequency in the system.

In this paper we show that the features that until now have been classified as
HFQPOs in GRS 1915+105 are instead consistent with being type-C QPOs produced
near the innermost stable circular orbit (ISCO). Based on this assumption, we
infer the spin of the BH hosted in GRS 1915+105, and we show that the inferred
spin, coupled with the dynamical BH mass measurement, can be used to predict
theoretical frequencies that match the data.

##  2 Rethinking the 67 Hz QPO

Belloni & Altamirano (2013b) reported a compilation of all the HFQPOs found in
a systematic search performed on all the available RXTE data on GRS 1915+105.
The centroid frequencies of 49 of the 51 peaks reported in such a work are
found between 58 and 72 Hz (a histogram of such frequencies is shown in Fig.
4). The two remaining peaks, detected at $\sim$134 and 143 Hz, are discussed
in Sec. 3.1.

The frequencies of such 49 QPOs are distributed in a narrow range centred
around 67 Hz, and henceforth we will refer to these QPOs collectively as the
67 Hz QPOs, aware that in reality they span a range of frequencies. Such a
narrow range implies that they are determined by a parameter that - along with
mass and spin - has to remain unchanged despite large swings in accretion
rate. The most obvious candidate is the ISCO, which itself depends only on the
mass and spin of the central black hole. The shape of the histogram in Fig. 4
support the above fact, as it shows a clear drop at frequencies slightly above
67 Hz.

If we calculate the orbital frequency at ISCO (which at such a radius equals
the periastron precession frequency) for a dimensionless spin parameter222The
dimensionless spin parameter is defined as $a^{*}=J/M^{2}$, where $J$ is the
BH angular momentum and M its mass. (hereafter spin) a∗ = 0 and a mass of 14.4
(i.e. the upper limit on the mass based on the values reported by Reid et al.
2014, M = 12.4${}^{+2.0}_{-1.8}$M⊙) we obtain $\sim$ 152 Hz333Note that we
deliberately ignore solutions where the spin is counter rotating, based on the
assumption that the BH spin and the angular momentum of the matter in the
accretion disk are unlikely to be anti-parallel (Motta et al., 2018, for a
discussion).. This is the minimum possible orbital frequency in GRS 1915+105
at ISCO, as any larger BH spin value and any lower mass will yield higher
frequencies. Thus, the 67 Hz QPOs are not consistent with being the result of
the orbital motion of matter at the ISCO.

An intriguing possibility is that the 67 Hz QPO is the result of nodal
precession at the ISCO. This would imply that the oscillations observed around
67 Hz are not HFQPOs, as it has been believed for decades, but are instead
type-C QPOs arising from very close to the ISCO. In this scenario, 67 Hz is
the maximum possible centroid frequency for a type-C QPO for a given BH mass
and spin. Consequently, the 67 Hz QPOs in GRS 1915+105 represent the high-
frequency end of a broad distribution that includes all type-C QPOs in this
system, which are most commonly observed below 10 Hz. The fact that a 67 Hz
QPO has never been concurrently observed with another type-C QPO supports this
hypothesis. This perspective also coherently clarifies why GRS 1915+105,
unlike other BH X-ray binaries, exhibits numerous PDS peaks identified as
HFQPOs: the 67 Hz QPOs would be reclassified not as HFQPOs, which are
relatively rare in BH X-ray binaries, but rather as type-C QPOs, a more
frequent occurrence in these systems.

To the best of our knowledge, current literature lacks efforts to interpret
the $\sim$67 Hz QPOs in GRS 1915+105 as anything other than HFQPOs (Morgan et
al., 1997). Alternative models have been previously proposed (see, e.g.,
Fragile et al., 2016; Ingram & Motta, 2019) that could offer different
interpretations of these QPOs, and which could be tested using the data
considered here (all available in the literature). However, conducting such
tests falls outside the scope of of this work.

In the scenario we have outlined, three facts are worth mentioning. Firstly,
the 67 Hz QPOs do not exhibit characteristics typical of Type-C QPOs.
Secondly, there are additional QPOs identified in GRS 1915+105 with
frequencies close yet inconsistent with the 67 Hz QPOs (Belloni & Altamirano,
2013a, and references therein). Finally, at least one other source has been
observed to exhibit a QPO at approximately 67 Hz. We discuss each of these
points in detail below.

At first glance the 67 Hz QPOs do not show the typical characteristics of
type-C QPOs. While type-C QPOs at lower frequencies are generally superimposed
on broad-band variability, the 67 Hz QPOs distinctively emerge from the
Poisson noise. Additionally, these QPOs exhibit a relatively low rms amplitude
(lower than 1%). These properties should not surprise. Firstly, the broad band
variability typical of BH XRBs which always shows a cut-off between a few
tenths of Hz and at a few Hz). Consequently, as the frequency of a QPO
increases, it becomes increasingly probable for it to arise from Poisson noise
rather than from broad-band variability. A similar situation can be observed
in GRO J1655-40, where Type-C QPOs observed in the soft state are observed at
around 30 Hz in a region of the PDS which is Poisson-noise dominated.
Furthermore, the amplitude of type-C QPOs is anti-correlated with the centroid
frequency, and thus higher frequency oscillations are expected to show lower
amplitudes (see, e.g., Motta et al., 2015).

Several authors reported the discovery of various QPO peaks in the PDS from
GRS 1915+105, at 27 (Belloni et al., 2001), 34 (Belloni & Altamirano, 2013a)
and 41 Hz (Strohmayer, 2001). The 34 Hz QPO can be readily interpreted as
harmonically related to the 68 Hz QPO observed in the same PDS. This harmonic
relationship is a common feature of Type-C QPOs, which makes the above
unsurprising. Even the occurrence of an isolated 34 Hz QPO (i.e., without a
concurrent higher frequency harmonic peak) would not be unexpected. This is
due to the variable amplitude of harmonically-related peaks observed in
several X-ray binaries, particularly in neutron star systems, where the most
prominent, and sometimes the only significant QPO peak does not necessarily
correspond to the fundamental frequency of the modulation underlying the
signal (Motta et al., 2017). Consequently, the detection of an orphan 27 Hz
peak might be interpreted in a similar way, presuming an undetected
fundamental peak at approximately 54 Hz. It is more challenging to provide a
straightforward explanation for the 41 Hz frequency reported in Strohmayer
(2001), which was detected concurrently with the 67 Hz QPO, and yet appears to
be unrelated to it. We hypothesise that these two QPOs, despite being observed
in the same dataset spanning over 15 ks, might not be occurring strictly
simultaneously. Under this assumption, the 41 Hz peak could be a type-C QPO at
a lower frequency, with its frequency increasing to 67 Hz at a different time.
However, this hypothesis requires further analysis for validation or
refutation, a task we reserve for future work.

The 67 Hz QPO is not exclusive to GRS 1915+105 as a similar QPO at 66 Hz has
been detected in the candidate BH X-ray binary IGR J17091-3624 during its
first observed outburst (2011). Notably, this QPO was observed only once
throughout all recorded outbursts of the source, although some excess in the
PDS around 66 Hz and 164 Hz was detected in a number of observations of the
same outburst (Altamirano et al., 2011). While we cannot exclude this
possibility, current evidence does not support the conclusion that this QPO is
of the same type as the 67 Hz QPO observed in GRS 1915+105. Nonetheless, it is
intriguing that among known BH X-ray transients, the one exhibiting a QPO near
67 Hz shares phenomenological resemblances with those observed in GRS 1915+105
(Altamirano et al., 2011; Motta et al., 2021). This similarity warrants
further investigation, particularly should any future data confirm that the 66
Hz modulation in IGR J17091-3624 shares properties with the 67 Hz QPOs in GRS
1915+105.

###  2.1 The spin

Under the intriguing hypothesis that the 67 Hz QPOs arise from nodal
precession in the proximity of ISCO, it is possible to obtain an estimate of
the BH spin as follows. First, we fit the histogram in Fig. 4 with a Gaussian
function, and we take the Gaussian’s peak frequency as the frequency of the
QPO arising at ISCO, and the Gaussian’s FWHM as the 1-sigma error on such a
frequency. This yield<EMAIL_ADDRESS>2.1$ (see Tab. 1).

Next, we use the RPM to obtain an estimate of the spin. As described in
Franchini et al. (2017), by substituting the expression of the radius of ISCO
into the equation of the nodal precession frequency, one can remove the radial
dependency and obtain an equations that only depend on the mass and spin of
the BH. Adopting a BH mass of 12.4${}^{+2.0}_{-1.8}$M⊙ we obtain a spin of a∗
= 0.706 $\pm$0.034. The corresponding graphical solution is shown in Fig. 1.
For this spin the radius of the innermost stable circular orbit radius is
$R_{\rm ISCO}$ = (3.36$\pm$0.15) $R_{\rm g}$ (see Tab. 1).

As noted in Motta et al. (2022), we stress that if the RPM was an exact
description of the behaviour of particles orbiting a BH, the uncertainties on
the derived on the BH spin (and mass, and emission radius) would only come
from the uncertainties on the measured QPO frequencies, which are dominated by
the accuracy of the detecting instrument. However, the exact geometry of the
emitting region is unknown, and so is the exact emission mechanism behind
QPOs. Both the above facts (and likely others) might be sources of
systematics. For instance, in Motta et al. (2022) we gauged the uncertainties
that could be related with the the radial extent of the region originating the
QPOs, and found that such systematics would be of the order 15%. Since the
effective impact of systematics on the spin estimate is hard to determine
accurately, we warn the reader that in the following - for simplicity and to
avoid biasses - we will report and use the uncertainties derived uniquely from
the errors on the QPO centroid frequencies.

Figure 1: Mass versus spin solutions of the RPM obtained assuming that the
nodal precession occurs at ISCO. The solutions corresponding to the frequency
67.712 +/- 1-sigma uncertainties are shown. The horizontal solid lines and the
dotted line are the upper and lower limit to the dynamical mass, and its
central value, respectively (M = 12.4${}^{+2.0}_{-1.8}$M⊙). A version of this
figure spanning a broader mass and spin range is given in Franchini et al.
(2017). Figure 2: Nodal precession frequencies (bottom grey line), periastron
precession frequencies (middle lines), and orbital frequencies (top lines)
plotted as a function of the nodal precession frequency, as predicted by the
RPM using M = (12.4$\pm$0.46) M$\odot$ and a∗ = 0.706 $\pm$0.034. The set of
lines forming grey regions in the figure each correspond to mass-spin pairs
drawn from a bi-variate distribution (see main text ). The lines corresponding
to the Nodal precession Frequency overlap to each other as the Nodal
precession frequency is the independent variable in this plot, hence the only
difference between lines is in the maximum value they reach (i.e. the nodal
precession frequency at ISCO). The dotted lines correspond to the sub-
harmonic, the second and third harmonic of the nodal precession frequency. The
vertical black lines correspond to the predicted nodal frequency at $R_{\rm
ISCO}$ for the same pairs. For clarity we plot only the predicted frequencies
for mass-spin pairs within 1-$\sigma$ from the central mass-spin value. The
QPO doublet formed by a HFQPO at 135Hz and a type-C QPO at 8.7 Hz are
indicated by clear blue diamonds. The 67Hz QPOs are marked by clear blue
squares. All the PDS broad components described in Section 3.2 are marked by
small blue dots with vertical error bars. Figure 3: Example of a PDS from GRS
1915+105 (RTXE observation 40703-01-01-00). The Lorentzian components fitted
to the data are marked with dashed lines. The classifications of the peaks
according to Belloni et al. (2002) is shown.

##  3 Completing the picture

The hypothesis that the 67 Hz QPOs is compatible with being the result of
nodal precession near ISCO can be tested by considering the characteristic
frequencies of additional components in the PDS, such as the high-frequency
broad components often observed with type-C QPOs in BH binaries (Psaltis et
al., 1999; Stella et al., 1999) or, in case they are detected, the frequencies
of actual HFQPOs.

We proceed by predicting the frequencies of the particles orbiting the BH at
any radius larger than $R_{\rm ISCO}$ as was done in, e.g., Motta et al.
(2014a, b, 2022). In order to take into account the uncertainty associates
with the mass and spin of the BH, we simulated a 106 elements mass and spin
normal bi-variate distribution with central value and width equal to the
values and uncertainties given in Tab. 1, and we predict the nodal, periastron
precession, and orbital frequencies as a function of radius, as well as the
value of $R_{\rm ISCO}$, and the corresponding nodal frequency for mass-spin
pairs drawn randomly form the distribution.

In Figure 2 we plot the three set of frequencies as a function of the nodal
frequency (grey lines), as well as the values of the nodal frequency at
$R_{\rm ISCO}$ calculated in each step, following the procedure adopted in
Motta et al. 2022 (black vertical lines). Every set of grey lines corresponds
to a different mass-spin pair drawn from a bi-variate distribution (for
clarity we only plot values for the mass-spin pairs within 1-$\sigma$ from the
central mass-spin value). The 67Hz QPOs are marked by squares.

###  3.1 Bringing in the real HFQPOs

HFQPOs in BH X-ray binaries are classified solely based on their frequency,
which in the majority of systems is of the order 100 Hz. Hence, we classify
the QPOs at $\sim$134 and $\sim$143 Hz in GRS 1915+105 reported in Belloni &
Altamirano (2013b) as HFQPOs. In this work we will refer to these oscillations
as the real HFQPOs to distinguish them from the 67 Hz QPOs, which, as already
noted, are found at frequencies significantly lower than the typical HFQPO in
other BH systems.

The inspection of the two PDS showing peaks at $\sim$134 and $\sim$143 Hz,
respectively, reveals in both cases a clear LFQPO at $\sim$6.5Hz. Such
oscillations can be classified as a type-B QPO, based on their frequency and
morphology (see, e.g., Casella et al., 2005; Motta et al., 2009). A close
inspection of the PDS shows the presence of a weak peak at $\sim$8.54 Hz in
one of the two cases, i.e. observation 50703-01-10-02, which according to
Belloni & Altamirano (2013b) shows $\rho$-type variability). We classify the
weak peak as a type-C QPO because it is not consistent with being the second
harmonic to the modulation at 6.5 Hz, and because no type-B QPO has been
observed above 7 Hz in GRS 1915+105 (see, e.g., Soleri et al., 2008). This PDS
configuration, i.e. a type-B next to a weak type-C QPO, is relatively uncommon
in BH X-ray binaries (Motta et al., 2015), but is typical of neutron star
Z-sources (see, e.g., Motta & Fender, 2019), which like GRS 1915+105 are
believed to accrete near the Eddington limit (see Motta et al., 2017, for the
case of Sco X-1)444We note that the type-B and type-C QPOs that we detected
are found at centroid frequencies in a 4/3 ratio. We believe that the fact
that the two peaks are in a 4/3 ratio is pure coincidence as any slightly
different rebinning of the PDS results in slightly different centroid
frequencies, which are not in a 4/3 ratio..

We fitted the PDS as described in several works in the literature, with a
number of Lorentzian shapes (e.g. Belloni et al., 2002). The best fit yields a
type-C QPO frequency $\nu_{C}$ = 8.71$\pm$ 0.09 and a HFQPO at 135.25$\pm$0.9.
A plot showing the fitted PDS is shown in the appendix (Fig. 5). Belloni &
Altamirano (2013b) already showed that the HFQPO is formally significant after
considering the number of trials. The type-C QPO, instead, is borderline
significant, i.e. approximately 3.1 $\sigma$ single trial based on the QPO
amplitude and its uncertainty. Determining the formal significance of a QPO
super-imposed to a complex continuum (i.e., not simple red noise) is non
trivial (see, e.g., Vaughan, 2005), but for the sake of the argument, we will
assume that the type-C QPO is correctly classified and statistically
significant, and hence we will treat the tentative type-C QPO plus the HFQPO
at $\sim$143 Hz as a QPO “doublet”. Marking this QPO “doublet” on the plot in
Fig. 2 we see that the HFQPO peak is consistent with being a upper HFQPO, and
can be associated with the orbital frequency according to the RPM. By
extension and in virtue of their similar frequency, we classify also the other
high-frequency peak at 143 Hz as a upper HFQPO.

Following the prescription in Ingram & Motta (2014) and by considering the BH
mass for GRS 1915+105 from dynamical measurements, we are able to infer an
additional, independent value of the spin to be compared with the value
obtained above for consistency. We derive a second spin value, a${}_{*}^{D}$ =
0.71$\pm$0.13, which is consistent with the value we obtained above in Sec.
2.1.

###  3.2 Bringing in the PDS broad components

Finally, we consider the broad PDS components that have been associated with
the HFQPOs in BH and NS X-ray binaries thanks to the well-known correlation
discovered by Psaltis et al. (1999), the so-called Psaltis-Belloni-Van der
Klis (PBK) correlation. While these broad-band components frequently appear in
the PDS of BH X-ray binaries in general and in GRS 1915+105 in particular,
their presence is not ubiquitous across all observations. Thus, instead of
inspecting all the 1816 RXTE archival observations of GRS 1915+105, we
focussed on the sample considered by Zhang et al. (2022), constructed based on
the presence of broad-band signal at high frequencies (or a “bump”).

Each observation was reduced following the standard procedures described in
several works by our group (one recent example is Motta et al. 2022), with one
main difference: since GRS 1915+105 is known for being a variable source, and
changes in QPO centroid frequencies can happen on time-scales shorter than the
average RXTE observation (a few thousands seconds spread acorss several
satellite revolutions), we produced PDS employing 64 s long data segments and
generated an average PDS accumulating a fixed number of segments (48) for a
total of approximately 3000 s per average PDS, i.e. long enough to provide a
good signal-to-noise-ratio in each spectrum, but not so long that the
underlying power spectral density distribution varied significantly in the
time interval considered, (thus violating the assumption of the Fourier
analysis).

We generated a total of 480 average PDS using custom software under IDL
(GHATS555http://www.brera.inaf.it/utenti/belloni/GHATS_Package/Home.html),
which we normalised according to Leahy et al. (1983) and we fitted with a
combination of Lorentzian components. For all the PDS we calculated the
characteristic frequency of all the Lorentzians components, defined as
$\nu_{\rm max}=\sqrt{\nu^{2}+(\Delta/2)^{2}}$, where $\nu$ and $\Delta$ are
the frequency and width, respectively (see Belloni et al., 2002). Note that
for narrow features such as QPOs the characteristic frequency is by
construction very similar to the Lorentzian peak frequency.

While the PDS from GRS 1915+105 (and of BH binaries in general) can be
complex, and it is often hard to classify components based on a given scheme
(i.e., not all the components are all present all the time in every PDS), the
majority of the components can be classified as follows, and as it is shown in
Fig. 3. The PDS can include:

  * •

a type-C QPO (LLF) and its harmonic content, i.e., a sub-harmonic (L1/2), a
second harmonic (LII), and sometimes a third or fourth harmonic;

  * •

a broad Lorentzian at low frequencies (Lb);

  * •

a second broad Lorentzian peaking at frequency below the PDS break frequency
(L?);

  * •

a third broad Lorentzian centred at a frequency close to the QPO centroid
frequency (Lh);

  * •

one or two broad Lorentzians at high frequencies (Lu) and/or (Ll, not shown in
the figure).

In the attempt to interpret the PDS components from GRS 1915+105 in an
unbiased way, we plotted all the characteristic frequencies we found in all
the PDS we generated in Fig. 2, together with the frequencies predicted by the
RPM given the dynamical mass from photometry and the spin we calculated in
Sec. 2.1. All frequencies increase with the frequency of the type-C QPO (LLF).
In particular, in the context of the RPM, the Nodal Precession frequency
corresponds to LLF, and the sub-harmonic, second and third harmonic to this
frequency correspond to L1/2, LII and LIII. The periastron precession
frequency and orbital frequency correspond to the broad components detected at
high frequencies, Ll or Lu, respectively, or to HFQPOs (not shown in Fig. 3),
as suggested by Psaltis et al. (1999).

In Fig. 3 several points do not clearly coincide with one of the frequencies
of the RPM, and this is because the RPM aims to explain only three components
of the several in the PDS. What is important to notice here is that several
points are consistent with the frequencies predicted by the RPM for the mass
and spins we derived from the 67 Hz QPOs under the assumption that they are
the result of nodal precession of particles at ISCO. The vast majority of the
points that are inconsistent with the RPM frequencies can be associated with
the Lb, Lh or L? components (see Fig. 3).

###  3.3 Implications

The spin we obtained for the BH in GRS 1915+105 is the fourth one obtained via
X-ray timing. Previous estimates were obtained for the BH X-ray binaries GRO
J1655-40, XTE J1550-564, XTE J1859-226 presented in Motta et al. 2014a, Motta
et al. 2014b, Motta et al. (2022), and the the spins were 0.29, 0.34, and
0.15, respectively. In these cases, the spin values we derived are relatively
low if compared to those typically obtained via other ‘electromagnetic’ spin
measurement methods (continuum fitting, reflection spectroscopy, and
reverberation, Reynolds 2021 for a recent review), which so far yielded spin
distributions peaked at values above 0.9 (especially true in the case of the
reflection-based measurements, see e.g. Draghis et al. 2023). Instead - and
given the many peculiarities of this system this is perhaps unsurprising - the
spin for GRS 1915+105 is relatively high - 0.76 - although still inconsistent
with the value from X-ray spectroscopy (Miller et al., 2013).

While the number of timing-based spin values from the RPM is still low, they
appear to be consistent with the BH spin distribution obtained based on the
entire gravitational waves sample GWTC-3 (The LIGO Scientific Collaboration et
al., 2021). This distribution peaks around spin $\sim$ 0.2 and shows a thin
tail extending to higher values, thus indicating that the vast majority BHs in
binary BH systems are relatively slow rotators. Unlike the case of other
‘electromagnetic’ methods, the results of which indicate that the spin
distributions from binary BHs and BH X-ray binaries may be different (Draghis
et al., 2023), our results seem to suggest that the LIGO/Virgo/KAGRA BHs and
X-ray binary BHs may feature a similar spin distribution, and hence may be
members of strictly related populations (see Belczynski et al., 2021; Fishbach
& Kalogera, 2022).

Table 1: A summary of the quantities measured and derived in this work. Uncertainties are given at a 1-sigma level.  M | = | 12.4${}^{+2.0}_{-1.8}$M⊙ M$\odot$ | Reid et al. (2014)
---|---|---|---
a∗ | = | 0.706$\pm$0.034 | From 67Hz QPOs
a${}_{*}^{D}$ | = | 0.71$\pm$0.13 | From QPO doublet
$R_{\rm ISCO}$ | = | 3.36$\pm$0.15 $R_{\rm g}$ | Derived
$\nu_{\rm nod@ISCO}$ | = | 67.712 $\pm$ 2.056 Hz | Measured
$\nu_{HFQPO}^{\rm upper}$ | = | 135.250 $\pm$ 0.70 | Measured
$\nu_{C}$ | = | 8.72 $\pm$ 0.09 | Measured

##  4 Conclusions

Building on the observation that the 67 Hz QPOs in GRS 1915+105 are confined
to a narrow frequency range, we hypothesized these QPOs to be a consequence of
nodal precession at the ISCO around a spinning black hole.

Using the Relativistic Precession Model and the black hole mass determined
from dynamical measurements (M = 12.4${}^{+2.0}{-1.8}$M${\odot}$), we derived
a moderately high black hole spin (a∗ = 0.706 $\pm$ 0.034). Our predictions of
frequencies around a black hole with this spin and mass nicely matched the
observational data, lending further support to the idea that certain PDS
features in accreting BH X-ray binaries can be explained by matter motion near
a compact object.

We conclude that the 67 Hz QPOs observed in GRS 1915+105, historically
classified as HFQPOs, are more plausibly type-C QPOs originating from the
vicinity of the ISCO of a moderately spinning black hole.

###### Acknowledgements.

The authors acknowledge financial contribution from grant PRIN INAF 2019 n.15.
This work benefited from discussions during Team Meetings of the International
Space Science Institute (Bern), whose support we acknowledge.
SEM heartily thanks Tomaso Belloni, who sadly passed away before this paper
could be completed. Among many other contributions to the field, his work was
crucial for the understanding of GRS 1915+105.
SEM acknowledge the assistance of ChatGPT for proofreading and language
enhancement in the preparation of this manuscript.

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##  Appendix A Additional figures

In this appendix we show additional figures to complement the analysis
presented in the main text. In Fig. 4 we show a histogram including the
centroid frequencies of the 67 Hz QPOs reported in Belloni & Altamirano
(2013b), save for the 2 peaks detected above 100 Hz.

Figure 5 shows the PDS from observation 50703-01-10-02, including a type-B, a
type-C, and a HFQPO. The broad peak at the high-frequency side of the LFQPOs
is centered at $\sim$13.2 Hz, and could be the hint of a second harmonic to
the type-B QPO (albeit significantly broader than what would be expected in
this case), or a Lh component (see Fig. 3).

Figure 4: A Histograms of the centroid frequencies of the QPOs found around 67 Hz in the data from GRS 1915+105, as reported in Belloni & Altamirano (2013b). The black solid line is the best Gaussian fit to the histogram, which yields a centroid frequency of 67.712 Hz and a FWHM of 2.056 Hz.  Figure 5: A PDS from observation 50703-01-10-02, which includes a type-B QPO at 6.53$\pm$0.38 Hz, a type-C QPO at 8.71$\pm$0.09 Hz, and a high frequency QPO at 135.250$\pm$0.70.  Table 2: A summary of the quantities measured and derived in this work. Uncertainties are given at a 1-sigma level. The parameters marked by a $\dagger$ were fixed to the reported values. We note that the HFQPO has been reported in Belloni & Altamirano (2013b) as a significant feature after taking into account the number of trials.  Component | Centroid Frequency | Width | Leahy normalisation | rms [1-300 Hz]
---|---|---|---|---
| [Hz] | [Hz] | [counts/Hz] | [1/Hz]
Broad component 1 | 0.0† | 4.0$\pm$0.3 | 30$\pm$1 | 0.84$\pm$0.03
Type-B QPO | 6.53$\pm$0.05 | 1.2$\pm$0.1 | 6.4$\pm$0.5 | 0.19$\pm$0.01
Type-C QPO | 8.7$\pm$0.1 | 0.6† | 0.6$\pm$0.2 | 0.018$\pm$0.006
Upper HFQPO | 135.1${}^{+0.6}_{-0.8}$ | 1.73112${}^{+3}_{-1}$ | 0.7$\pm$0.2 | 0.20$\pm$0.006
Broad component 2 | 13.2${}^{+0.7}_{-0.8}$ | 10${}^{+3}_{-2}$ | 8.7${}^{+0.9}_{-0.8}$ | 0.157$\pm$${}^{+0.027}_{-0.024}$
$\chi^{2}/\nu$ | 219.90/221 |