# Multi-pulse edge-localized states on quantum graphs

Adilbek Kairzhan  Department of Mathematics, University of Toronto, Toronto,
Ontario, Canada<EMAIL_ADDRESS>and  Dmitry E. Pelinovsky
Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1,
Canada<EMAIL_ADDRESS>
###### Abstract.

Edge-localized stationary states of the focusing nonlinear Schrödinger
equation on a general quantum graph are considered in the limit of large mass.
Compared to the previous works, we include arbitrary multi-pulse positive
states which approach asymptotically to a composition of $N$ solitons, each
sitting on a bounded (pendant, looping, or internal) edge. Not only we prove
that such states exist in the limit of large mass, but also we compute the
precise Morse index (the number of negative eigenvalues in the corresponding
linearized operator). In the case of the edge-localized $N$-soliton states on
the pendant and looping edges, we prove that the Morse index is exactly $N$.
The technical novelty of this work is achieved by avoiding elliptic functions
(and related exponentially small scalings) and closing the existence arguments
in terms of the Dirichlet-to-Neumann maps for relevant parts of the given
graph.

##  1\. Introduction

We address standing waves of the focusing NLS (nonlinear Schrödinger) equation
posed on a quantum graph $\Gamma=\\{\mathcal{E},\mathcal{V}\\}$, where
$\mathcal{E}$ is the set of edges and $\mathcal{V}$ is the set of vertices
(see [21, 22] for review). The evolution system can be written in the
normalized form:

$i\Psi_{t}+\Delta\Psi+2|\Psi|^{2}\Psi=0,$ (1.1)

where the Laplacian $\Delta$ and the nonlinear term are defined componentwise
on edges $\mathcal{E}$ subject to proper boundary conditions on the vertices
$\mathcal{V}$ (see [6, 15] for introduction to linear differential equations
on quantum graphs).

The quantum graph $\Gamma=\\{\mathcal{E},\mathcal{V}\\}$ is assumed to consist
of a finite number $|\mathcal{E}|$ of bounded and unbounded edges. Enumerating
every edge in $\Gamma$ uniquely gives the set
$\mathcal{E}=\\{e_{1},e_{2},\dots,e_{|\mathcal{E}|}\\}$. The function $\Psi$
on $\Gamma$ can be represented as a vector with $|\mathcal{E}|$ components,

$\Psi=\\{\psi_{1},\psi_{2},\dots,\psi_{|\mathcal{E}|}\\},$ (1.2)

where $\psi_{j}$ is defined on the edge $e_{j}$ only. The function $\Psi$ can
be defined in the Hilbert space of square-integrable functions
$L^{2}(\Gamma)=\bigoplus_{e\in\mathcal{E}}L^{2}(e)$.

Weak (resp. strong) solutions of the NLS time flow (1.1) are well defined in
the $L^{2}$-based Sobolev spaces $H^{1}(\Gamma)$ (resp. $H^{2}(\Gamma)$),
where $H^{1,2}(\Gamma)=\bigoplus_{e\in\mathcal{E}}H^{1,2}(e)$, provided the
boundary conditions on $\mathcal{V}$ are symmetric. Since
$|\mathcal{E}|<\infty$, the NLS time flow (1.1) is essentially the evolution
problem in one spatial dimension, which is globally well-posed both in
$H^{1}(\Gamma)$ and $H^{2}(\Gamma)$ due to the cubic ($L^{2}$-subcritical)
nonlinearity.

We consider the natural Neumann–Kirchhoff (NK) boundary conditions at each
vertex $v\in\mathcal{V}$ given by

$\left\\{\begin{array}[]{l}\Psi\text{ is continuous on }\Gamma,\\\ \sum_{e\sim
v}\partial\Psi(v)=0\text{ for every vertex
}v\in\mathcal{V},\end{array}\right.$ (1.3)

where the derivatives $\partial$ are directed away from the vertex
$v\in\mathcal{V}$ and $e\sim v$ denotes the edges $e\in\mathcal{E}$ adjacent
to the vertex $v\in\mathcal{V}$.

Consistent with the boundary conditions (1.3), weak solutions of the NLS
equation (1.1) are defined in the energy space
$H^{1}_{C}(\Gamma):=H^{1}(\Gamma)\cap C^{0}(\Gamma)$, where $C^{0}(\Gamma)$
denotes the space of functions continuous on all edges in $\mathcal{E}$ and
across all vertex points in $\mathcal{V}$. These weak solutions conserve the
energy and mass functionals given respectively by

$E(\Psi)=\|\nabla\Psi\|^{2}_{L^{2}(\Gamma)}-\|\Psi\|^{4}_{L^{4}(\Gamma)},\quad
Q(\Psi)=\|\Psi\|^{2}_{L^{2}(\Gamma)}.$ (1.4)

Standing waves of the NLS equation (1.1) are given by the solutions of the
form $\Psi(t,x)=\Phi(x)e^{-i\omega t}$, where $\Phi\in H^{1}_{C}(\Gamma)$ is a
weak solution of the stationary NLS equation

$\omega\Phi=-\Delta\Phi-2|\Phi|^{2}\Phi,$ (1.5)

for a given $\omega\in\mathbb{R}$. By bootstrapping arguments, every weak
solution of the stationary NLS equation (1.5) is also a strong solution
satisfying the natural NK conditions (1.3). Hence, we can use the vector
representation $\Phi=(\phi_{1},\phi_{2},\dots,\phi_{|\mathcal{E}|})$ and
rewrite the stationary NLS equation (1.5) on every edge $e_{j}$ in
$\mathcal{E}$ as a collection of differential equations:

$\omega\phi_{j}(x)=-\phi_{j}^{\prime\prime}(x)-2|\phi_{j}(x)|^{2}\phi_{j}(x),\quad
x\in e_{j},$ (1.6)

which satisfy the boundary conditions (1.3) on every vertex $v\in\mathcal{V}$.
We write $\Phi\in H^{2}_{\rm NK}(\Gamma)$ if $\Phi\in H^{2}(\Gamma)$ satisfies
the NK conditions (1.3). Only real-valued solutions of the stationary NLS
equation are considered but we write the modulus sign for easy
generalizations.

Among all possible real-valued solutions of the stationary NLS equation (1.5),
we are particularly interested in the positive edge-localized states which
satisfy the following conditions:

  * •

$\Phi(x)>0$ for every $x\in\Gamma$;

  * •

on each bounded edge $e_{j}\in\mathcal{E}$, there is at most one local
critical point of $\phi_{j}$ (either maximum or minimum) inside the edge;

  * •

on each unbounded edge $e_{j}\in\mathcal{E}$, the function $\phi_{j}$ is
monotonically decreasing and has exponential decay to $0$ at infinity.

Depending on the topological properties of the quantum graph $\Gamma$, the
positive edge-localized states could become the ground state, the state of the
least energy $E(\Psi)$ at fixed mass $Q(\Psi)$ [1, 2, 3].

Although the set of conditions on $\Phi$ seems to be restrictive, the positive
edge-localized states exist in the limit of large mass [4, 7, 10, 18]. This
limit for large $\mu=Q(\Phi)$ can be recast as the limit of large negative
$\omega$ in the stationary NLS equation (1.5) for the cubic
($L^{2}$-subcritical) nonlinearity. Existence of such edge-localized states
was confirmed analytically and/or numerically for the tadpole graph [2, 8, 9,
23, 24], the dumbbell graph [12, 20], and the flower graph [16]. The
importance of the positive edge-localized states is motivated by their
(possible) orbital stability in the NLS time flow.

Figure 1. Schematic illustration of a pendant edge (left), a looping edge
(middle), and an internal edge (right).

Bounded edges of $\mathcal{E}$ are divided into pendant, looping, and internal
edges according to the following classification (see illustrations on Fig. 1):

  * •

A pendant edge of length $\ell$ is associated with the segment $[0,\ell]$,
where the left end is isolated from the rest of $\Gamma$ subject to the
Neumann boundary condition and the right end is connected with the rest of
$\Gamma$ at a vertex $v$.

  * •

A looping edge of length $2\ell$ is associated with the segment
$[-\ell,\ell]$, where both ends at connected to the rest of $\Gamma$ at a
single vertex $v$, hence contributing twice to the derivative condition in
(1.3).

  * •

An internal edge of length $2\ell$ is associated with the segment
$[-\ell,\ell]$, where different ends at connected to the rest of $\Gamma$ at
two different vertices $v_{-}$ and $v_{+}$.

Each unbounded edge is associated with the half-line $[0,\infty)$. The edge-
localized states considered in [7] consist of a single large-amplitude
component $\phi_{e}$ on a fixed looping or internal edge $e$ of length $2\ell$
such that $\phi_{e}$ has a single local maximum inside $e$, monotone from its
maximum to the vertices of $e$, and concentrated on $e$ in the following sense

$\frac{\left\|\Phi\right\|_{L^{2}(e)}}{\left\|\Phi\right\|_{L^{2}(\Gamma)}}\geq
1-Ce^{-2\epsilon\ell},$ (1.7)

where $\epsilon:=\sqrt{|\omega|}$ is a large parameter and the constant $C$ is
independent of $\epsilon$. Moreover, $\Phi$ has no internal maxima on the
remainder of graph $\Gamma\backslash\\{e\\}$. The same result also holds for
the pendant edge $e$ except that $\phi_{e}$ has a single local maximum at the
terminal vertex. The edge-localized states considered in [7] are illustrated
on Fig. 2.

Figure 2. Schematic illustration of edge-localized states on a pendant (left),
a looping edge (middle), and an internal edge (right).

The construction of the edge-localized states in [7] relies on the properties
of elliptic functions and on careful rescaling of exponentially small terms
for the elliptic modulus. In addition, the states of lowest energy at fixed
mass were analyzed in [7] by comparing the exponentially small terms in the
expansion of $\mu:=Q(\Phi)$ in $\epsilon:=\sqrt{|\omega|}$.

A similar result on the edge-localized states in the large-mass limit was
obtained independently in [4] using variational methods (for unbounded graphs
only). Since the edge-localized states were identified in [4] as local energy
minimizers in the restricted space of functions in $H^{1}_{C}(\Gamma)$ that
attain their maximum on a given edge $e$, the Morse index for such states is
exactly one, where the Morse index is the number of negative eigenvalues of
the linearized operator $\mathcal{L}:H^{2}_{\rm NK}(\Gamma)\subset
L^{2}(\Gamma)\to L^{2}(\Gamma)$ given by

$\mathcal{L}=-\Delta-\omega-6|\Phi|^{2}.$ (1.8)

If the graph $\Gamma$ is unbounded, $\Phi$ is bounded and decays to $0$ at
infinity exponentially fast, and $\omega<0$, then the linearized operator
$\mathcal{L}$ is self-adjoint and the absolutely continuous part of the
spectrum is strictly positive and bounded away from zero by $|\omega|$ by
Weyl’s theorem on essential spectrum. Therefore, the Morse index is well-
defined as the finite number of negative eigenvalues of $\mathcal{L}$
accounting for their finite multiplicity. We denote this number by
$n(\mathcal{L})$. We also denote the multiplicity of the zero eigenvalue of
$\mathcal{L}$ by $z(\mathcal{L})$.

If the graph $\Gamma$ is bounded, then the spectrum of $\mathcal{L}$ is purely
discrete and the numbers $n(\mathcal{L})$ and $z(\mathcal{L})$ are again well-
defined since $\mathcal{L}$ is bounded from below.

Additional results related to the edge-localized states in the limit of large
energy were obtained in [10], where bounded graphs with the pendant edges were
considered and convergence of the edge-localized states to the half-solitons
was proven. Multi-pulse states were also studied in [10], all pulses localize
at the terminal vertices. Edge-localized states were considered in [18] by
recasting the existence problem to the semi-classical limit of an elliptic
problem. It was proven in [18] that the location of the edge-localized state
with a single maximum as the ground state of energy at fixed mass is
determined by the longest pendant edge of a bounded graph (or the longest
internal edge if no pendant and looping edges are present). These results are
included in those in [7] but they are obtained in a more general setting in
[18].

The main result of this work concerns the construction of the multi-pulse
edge-localized states in the limit of large mass (large negative $\omega$). As
$\omega\to-\infty$, the multi-pulse states approach asymptotically a
composition of $N$ solitons, each sitting on a bounded (pendant, looping, or
internal) edge. The edges of the graph, where the multi-pulse state is
localized, must satisfy the following non-degeneracy assumptions.

######  Assumption 1.

Let $\mathcal{E}_{N}:=\\{e_{1},e_{2},\dots,e_{N}\\}$ be the set of $N$ edges
connected to $\Gamma\backslash\mathcal{E}_{N}$ at $|B|$ boundary vertices
$\mathcal{V}_{B}=\\{v_{1},v_{2},\dots,v_{|B|}\\}$. Let $\ell_{\rm min}$ be the
length of the shortest edge in $\Gamma\backslash\mathcal{E}_{N}$ and
$\ell_{j,{\rm min}}$ be the minimal half-length of the looping and internal
edges or the minimal length of the pendant edges adjacent to the boundary
vertex $v_{j}\in\mathcal{V}_{B}$ from $\mathcal{E}_{N}$, $1\leq j\leq|B|$. The
edge lengths must satisfy the constraints:

$\ell_{\rm min}+\min_{1\leq i\leq|B|}\ell_{i,{\rm min}}>\ell_{j,{\rm
min}}\quad 1\leq j\leq|B|$ (1.9)

and

$3\min_{1\leq i\leq|B|}\ell_{i,{\rm min}}>\ell_{j,{\rm min}},\quad 1\leq
j\leq|B|.$ (1.10)

######  Assumption 2.

Each internal edge has no common boundary vertices with other internal edges
in $\mathcal{E}_{N}$ and its half-length is strictly minimal among the half-
lengths of the looping edges and the lengths of the pendant edges in
$\mathcal{E}_{N}$ adjacent to its two boundary vertices.

######  Remark 1.

In the definition of the edge lengths, we assume that each vertex has degree
$3$ or higher, except for the terminal vertices of the pendant edges which
have degree $1$. In other words, we do not allow fake vertices of degree $2$
in the graph $\Gamma$.

The following theorem gives the existence result.

######  Theorem 1.

Let $\Gamma=\\{\mathcal{E},\mathcal{V}\\}$ be a graph with finitely many edges
and satisfying NK conditions at its vertices. For any $N$ edges
$\mathcal{E}_{N}:=\\{e_{1},e_{2},\dots,e_{N}\\}$ of finite lengths satisfying
Assumptions 1 and 2 and for large enough $\epsilon:=\sqrt{-\omega}$ there
exists a positive edge-localized state $\Phi$ with the following properties:

  1. (1)

$\Phi|_{e_{j}}$ has a single local maximum and is monotone from its maximum to
the end vertices of $e_{j}\in\mathcal{E}_{N}$, for $1\leq j\leq N$;

  2. (2)

$\Phi$ has no internal maxima on the remainder of the graph
$\Gamma\backslash\mathcal{E}_{N}$;

  3. (3)

$\Phi$ is concentrated on $\mathcal{E}_{N}$ in the following sense

$\frac{\left\|\Phi\right\|_{L^{2}(\Gamma\backslash\mathcal{E}_{N})}}{\left\|\Phi\right\|_{L^{2}(\mathcal{E}_{N})}}\leq
Ce^{-\epsilon\ell_{N}},$ (1.11)

where the constant $C$ is independent of $\epsilon$ and
$\ell_{N}=\min\\{\ell_{1,{\rm min}},\ell_{2,{\rm min}},\dots,\ell_{|B|,{\rm
min}}\\}$.

######  Remark 2.

For the pendant edge $e_{j}\in\mathcal{E}_{N}$, the maximum of $\Phi|_{e_{j}}$
occurs at the left edge with the Neumann boundary conditions. For the looping
edge $e_{j}\in\mathcal{E}_{N}$, the maximum of $\Phi|_{e_{j}}$ occurs exactly
at the middle point. For the internal edge $e_{j}\in\mathcal{E}_{N}$, the
maximum of $\Phi|_{e_{j}}$ is generally shifted from the middle point. The
non-degeneracy condition in Assumption 2 allows us to control the shift of the
maximum of $\Phi|_{e_{j}}$ at the internal edge. See Fig. 2 for illustration
of edge-localized states on each bounded edge in $\mathcal{E}_{N}$.

######  Remark 3.

Condition (1.11) is equivalent to (1.7) in the case $N=1$ and
$\mathcal{E}_{N}=\\{e\\}$. Indeed, it follows from (1.11) with
$\ell_{N=1}=\ell$ that

$\|\Phi\|^{2}_{L^{2}(\Gamma)}=\|\Phi\|^{2}_{L^{2}(e)}+\|\Phi\|^{2}_{L^{2}(\Gamma\backslash\\{e\\})}\leq(1+C^{2}e^{-2\epsilon\ell})\|\Phi\|^{2}_{L^{2}(e)},$
(1.12)

which yields (1.7). It follows from Theorems 3.1, 3.3, and 3.5 in [7] that

$|\|\Phi\|^{2}_{L^{2}(e)}-\epsilon|\leq C\epsilon^{2}e^{-2\epsilon\ell}$
(1.13)

for the pendant edge and

$|\|\Phi\|^{2}_{L^{2}(e)}-2\epsilon|\leq C\epsilon^{2}e^{-2\epsilon\ell}$
(1.14)

for the looping and internal edges, where the constant $C$ is independent of
$\epsilon$. In the case of $N$ edges, we have

$\|\Phi\|^{2}_{L^{2}(\mathcal{E}_{N})}=\sum_{j=1}^{N}\|\Phi\|^{2}_{L^{2}(e_{j})}$
(1.15)

with similar estimates for $\|\Phi\|^{2}_{L^{2}(e_{j})}$, where
$e_{j}\in\mathcal{E}_{N}$.

######  Remark 4.

Compared to the work in [7], we do not use elliptic functions and
exponentially small scalings, which makes our results more general and the
proofs simpler. We partition the graph $\Gamma$ into $\mathcal{E}_{N}$ and
$\Gamma\backslash\mathcal{E}_{N}$ and reduce the existence problem to a system
of equations for Dirichlet data on the boundary vertices in $\mathcal{V}_{B}$.
Then, we show that all these equations can be solved independently of each
other under the constraints (1.9) and (1.10) on the lengths of edges in
Assumption 1. These constraints provide compatability of asymptotic solutions
for large $\epsilon$ and allow us to ignore the Dirichlet data at other
boundary vertices.

Our second main result is the precise characterization of the Morse index
$n(\mathcal{L})$ and degeneracy index $z(\mathcal{L})$ of the multi-pulse
edge-localized states in the limit of large mass (large negative $\omega$).
This characterization was not provided in [7] even in the case of $N=1$. The
following theorem gives the result when the set $\mathcal{E}_{N}$ does not
include internal edges.

######  Theorem 2.

Let $\Phi$ be the positive $N$-pulse edge-localized state of Theorem 1 for
large enough $\epsilon$ and assume that the set $\mathcal{E}_{N}$ contains
only pendant and looping edges. Then, $n(\mathcal{L})=N$ and
$z(\mathcal{L})=0$, where $\mathcal{L}$ is the linearized operator in (1.8).

######  Remark 5.

To obtain the exact count of the Morse index and the multiplicity of the zero
eigenvalue of $\mathcal{L}$, we use a homotopy argument relating the non-
positive spectra of the operators $\mathcal{L}:H_{\rm NK}^{2}(\Gamma)\subset
L^{2}(\Gamma)\to L^{2}(\Gamma)$ and $\mathcal{L}_{D}:H_{\rm
D}^{2}(\Gamma)\subset L^{2}(\Gamma)\to L^{2}(\Gamma)$ where $\mathcal{L}_{D}$
has the same differential representation as $\mathcal{L}$ and $H_{\rm
D}^{2}(\Gamma)$ differs from $H_{\rm NK}^{2}(\Gamma)$ by the Dirichlet
conditions at the boundary vertices in $\mathcal{V}_{B}$ instead of the NK
conditions. The technique resembles the surgery principle widely used in the
spectral analysis of differential operators on graphs [5, 19].

######  Remark 6.

The implication of Theorem 2 to the time evolution of perturbations to the
positive $N$-pulse edge-localized states of Theorem 1 is that these states are
orbitally unstable under the NLS time flow (1.1) if $N\geq 2$. This follows
from an easy application of the main results of [13] and [25]. In agreement
with the variational characterization of the single-pulse states on unbounded
graphs $\Gamma$ in [4], the positive edge-localized states with $N=1$ are
orbitally stable. This follows via the standard orbital stability theory [14]
from the monotonicity of the mapping

$\epsilon\mapsto\left\|\Phi\right\|^{2}_{L^{2}(\mathcal{E}_{N})}$ (1.16)

for large $\epsilon$ and the exponential smallness of
$\left\|\Phi\right\|_{L^{2}(\mathcal{E}_{N})}$ due to the decomposition (1.15)
with bounds (1.11), (1.13) and (1.14). Note that it was proven in [7] that the
mapping (1.16) is $C^{1}$.

######  Remark 7.

If an internal edge is present in the set $\mathcal{E}_{N}$, then the explicit
count of the Morse index in Theorem 2 is no longer applicable, as shown in
Example 3. Therefore, characterization of the Morse index and the multiplicity
of the zero eigenvalue of $\mathcal{L}$ is still an open problem in the case
of multi-pulse states localized on internal edges.

The paper is organized as follows. Section 2 reviews preliminary results from
[7] and [16]. Section 3 and 4 give the proof of Theorems 1 and 2 respectively.
Section 5 contains examples of flower, dumbbell, and single-interval graphs
where Theorems 1 and 2 are applicable and discuss counter-examples when
assumptions of Theorem 2 are violated.

##  2\. Preliminary results

Since $\epsilon:=\sqrt{-\omega}$ is considered to be large, we rescale the
stationary NLS equation (1.5) with the transformation $\Phi(x)=\epsilon
U(\epsilon x)$, $x\in\Gamma$. As a result, the graph
$\Gamma=\\{\mathcal{E},\mathcal{V}\\}$ is transformed to the $\epsilon$-scaled
graph $\Gamma_{\epsilon}=\\{\mathcal{E}_{\epsilon},\mathcal{V}\\}$ for which
every bounded edge $e\in\mathcal{E}$ of length $\ell_{e}$ transforms to the
edge $e_{\epsilon}\in\mathcal{E}_{\epsilon}$ of length $\epsilon\ell_{e}$ but
the unbounded edge $e\in\mathcal{E}$ remains the same as
$e\in\mathcal{E}_{\epsilon}$. The stationary NLS equation (1.5) is rewritten
in the parameter-free form

$(1-\Delta)U-2|U|^{2}U=0.$ (2.1)

Writing $U=(u_{1},u_{2},\ldots,u_{|\mathcal{E}|})$ gives a collection of
differential equations

$-u_{j}^{\prime\prime}(z)+u_{j}(z)-2|u_{j}(z)|^{2}u_{j}(z)=0,\quad z\in
e_{j,\epsilon},$ (2.2)

subject to the boundary conditions (1.3) at vertices $v\in\mathcal{V}$.

We select $N$ edges $\mathcal{E}_{N}:=\\{e_{1},e_{2},\ldots,e_{N}\\}$ with
$N\leq|\mathcal{E}|$ in the original graph $\Gamma$ and partition the graph
into two parts $\mathcal{E}_{N}$ and $\Gamma\backslash\mathcal{E}_{N}$. The
two graphs intersect at the vertices in the set
$\mathcal{V}_{B}:=\\{v_{1},v_{2},\ldots,v_{|B|}\\}$, which are referred to as
boundary vertices. After rescaling of $\Gamma$ to $\Gamma_{\epsilon}$, we
obtain $\mathcal{E}_{N,\epsilon}$ and
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$.

We introduce the Dirichlet data on the boundary vertices
$\vec{p}=(p_{1},p_{2},\ldots,p_{|B|})$ and the Neumann data
$\vec{q}=(q_{1},q_{2},\ldots,q_{|B|})$ for the graph
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ with

$p_{j}:=u_{e\sim v_{j}}(v_{j}),\quad q_{j}:=\sum_{e\sim v_{j}}\partial
u_{e}(v_{j}),\quad v_{j}\in\mathcal{V}_{B},$ (2.3)

where the derivatives $\partial$ are directed away from
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ and $e\sim v_{j}$ lists
all edges $e\in\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ incident
to the vertex $v_{j}$. We are looking for solutions of the differential
equations (2.2) such that

$\sup_{z\in e_{j,\epsilon}}|u_{j}(z)|<\frac{1}{\sqrt{2}},\quad
e_{j,\epsilon}\in\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon},$ (2.4)

and

$\sup_{z\in e_{j,\epsilon}}|u_{j}(z)|>\frac{1}{\sqrt{2}},\quad
e_{j,\epsilon}\in\mathcal{E}_{N,\epsilon}$ (2.5)

where $\frac{1}{\sqrt{2}}$ is the constant solution of the differential
equations in (2.2).

The following two lemmas were proven in [7] (Theorem 2.9 and Lemma 2.12).

######  Lemma 1.

There exist $C_{0}>0$, $p_{0}>0$, and $\epsilon_{0}>0$ such that for every
$\vec{p}$ with $\|\vec{p}\|<p_{0}$ and every $\epsilon>\epsilon_{0}$, there
exists a solution $U\in H^{2}_{\rm
NK}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})$ to the stationary
NLS equation (2.1) on $\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$
subject to the Dirichlet data on $\mathcal{V}_{B}$ which is unique among
functions satisfying (2.4). The solution satisfies the estimate

$\|U\|_{H^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}\leq
C_{0}\|\vec{p}\|,$ (2.6)

while its Neumann data satisfies

$|q_{j}-D_{j}p_{j}|\leq C_{0}\left(\|\vec{p}\|e^{-\epsilon\ell_{\rm
min}}+\|\vec{p}\|^{3}\right),\qquad 1\leq j\leq|B|,$ (2.7)

where $D_{j}$ is the degree of the $j$-th boundary vertex in
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ and $\ell_{\rm min}$ is
the length of the shortest edge in $\Gamma\backslash\mathcal{E}_{N}$.
Furthermore, if $p_{j}\geq 0$ for every $j$, then $U(z)\geq 0$ for all
$z\in\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ and $U$ has no
internal local maxima in
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$.

######  Lemma 2.

There exist $C_{0}>0$, $p_{0}>0$, and $\epsilon_{0}>0$ such that for every
$p\in(0,p_{0})$ and every $\epsilon>\epsilon_{0}$, there exists a real
solution $u\in H^{2}(0,\epsilon\ell)$ to the differential equation
$-u^{\prime\prime}+u-2u^{3}=0$ satisfying $u^{\prime}(0)=0$ and
$u(\epsilon\ell)=p$, which is unique among positive and decreasing functions
satisfying (2.5). The solution satisfies $u^{\prime}(\epsilon\ell)<0$ and

$\left|u^{\prime}(\epsilon\ell)-u(\epsilon\ell)+4e^{-\epsilon\ell}\right|\leq
C_{0}\epsilon e^{-3\epsilon\ell}.$ (2.8)

######  Remark 8.

The $C^{1}$ property of Neumann data $\vec{q}$ and $u^{\prime}(\epsilon\ell)$
with respect to parameter $\epsilon$ and Dirichlet data $\vec{p}$ and
$u(\epsilon\ell)=p$ in Lemmas 1 and 2 respectively was established in [7] with
similar exponentially small estimates. We will not write this property
expalicitly but will use the $C^{1}$ property in the application of the
implicit function theorem.

######  Remark 9.

The solution $u(z)$ in Lemma 2 was represented by elliptic functions and its
dependence on the elliptic modulus was also studied in Lemma 2.12 of [7]. This
information is not used in the present work.

For the solution $u\in H^{2}(0,\epsilon\ell)$ to the differential equation
$-u^{\prime\prime}+u-2u^{3}=0$ in Lemma 2, we write the boundary conditions as

$p_{+}:=u(0),\quad 0=u^{\prime}(0),\quad p:=u(\epsilon\ell),\quad
q:=-u^{\prime}(\epsilon\ell),$ (2.9)

where $p_{+}\in(p,1)$, $p>0$, and $q>0$. Denoting $v(z):=u^{\prime}(z)$, the
pair $(u(z),v(z))$ for all $z\in(0,\epsilon\ell)$ stays on the invariant curve

$E_{\beta}:=\\{(u,v):\;\;v^{2}-u^{2}+u^{4}=\beta\\}$ (2.10)

with some constant $\beta>-\frac{1}{4}$. In particular, in the view of (2.9),
the constant $\beta$ satisfies $\beta=q^{2}-p^{2}+p^{4}=-p_{+}^{2}+p_{+}^{4}$.
Fig. 3 shows the $(u,v)$ phase plane and the invariant curve between the
points $(p_{+},0)$ and $(p,-q)$. The period function $T_{+}(p,q)$ defined by

$T_{+}(p,q):=\int_{p}^{p_{+}}\frac{du}{\sqrt{\beta+u^{2}-u^{4}}}$ (2.11)

gives the $z$-length of the solution $u(z)$ obtained along the invariant curve
$E_{\beta}$ between points $(p_{+},0)$ and $(p,-q)$. For the solution $u\in
H^{2}(0,\epsilon\ell)$ in Lemma 2, the relations (2.9) can be written as

$T_{+}(p,q)=\epsilon\ell,\quad u(T_{+}(p,q))=p,\quad
u^{\prime}(T_{+}(p,q))=-q.$

Compared to the standard terminology, see, e.g., [11], where the period
function is introduced for the fundamental period of the periodic function
$u(z)$ along the integral curve $E_{\beta}$, the period function $T_{+}(p,q)$
given by (2.11) corresponds to a part of the integral curve $E_{\beta}$.

Figure 3. Phase plane $(u,v)$ for the differential equation
$-u^{\prime\prime}+u-2u^{3}=0$ showing the homoclinic loop for $\beta=0$, the
integral curve $E_{\beta}$ for $\beta\in(-\frac{1}{4},0)$, and the part of the
integral curve between points $(p_{+},0)$ and $(p,-q)$.

The following two lemmas partially proven in [16] describe properties of the
period function $T_{+}(p,q)$ and the solution $u(z)$ in Lemma 2.

######  Lemma 3.

There is small $\delta>0$ such that

$\partial_{p}T_{+}(p,q)<0\quad\mbox{\rm and}\quad\partial_{q}T_{+}(p,q)<0$
(2.12)

for every $p\in(0,\delta)$ and $q\in(0,\delta)$.

###### Proof.

It follows from the bound (2.8) that

$T_{+}(p,q)=-\ln\left(\frac{p+q}{4}\right)+\mathcal{O}\left((p^{2}+q^{2})(|\ln
p|+|\ln q|)\right)\quad\mbox{\rm as}\quad p,q\to 0.$ (2.13)

Taking derivatives of (2.13) in $p$ and $q$ justifies the inequalities in
(2.12). ∎

######  Remark 10.

The second property in (2.12) is also proven in Lemma 3.6 of [16] by using
more complicated analysis of the integrals in (2.11).

######  Lemma 4.

Let $u$ be a real, positive, and decreasing solution to the nonlinear equation
$-u^{\prime\prime}+u-2u^{3}=0$ on $(0,T_{+}(p,q))$ satisfying

$u(0)=p_{+},\quad u^{\prime}(0)=0,\quad u(T_{+}(p,q))=p,\quad
u^{\prime}(T_{+}(p,q))=-q$ (2.14)

with some $p>0$ and $q>0$, where $T_{+}(p,q)$ and $p_{+}$ are defined in
(2.11). Then, the general solution of the linearized equation
$-w^{\prime\prime}(z)+w(z)-6u^{2}(z)w(z)=0$ is given by

$w(z)=Au^{\prime}(z)+Bs(z),$ (2.15)

where $A$ and $B$ are arbitrary parameters and $s(z)$ satisfies $s(0)\neq 0$,
$s^{\prime}(0)=0$. Moreover, for sufficiently small $p$ and $q$, $u^{\prime}$
is positive on $(0,T_{+}(p,q))$, $s$ vanishes at exactly one point on the
interval $(0,T_{+}(p,q))$ and both functions satisfy

$\frac{u^{\prime\prime}(T_{+}(p,q))}{u^{\prime}(T_{+}(p,q))}>0,\quad\frac{s^{\prime}(T_{+}(p,q))}{s(T_{+}(p,q))}>0.$
(2.16)

###### Proof.

It follows from properties of $u(z)$ in Lemma 2 that $u^{\prime}(z)>0$ for
$z\in(0,T_{+}(p,q))$. The first inequality in (2.16) follows from

$\frac{u^{\prime\prime}(T_{+}(p,q))}{u^{\prime}(T_{+}(p,q))}=\frac{p(1-2p^{2})}{q}>0,$
(2.17)

for $p\in(0,\frac{1}{\sqrt{2}})$ and $q>0$.

In order to study the decomposition (2.15), we introduce
$s(z):=\partial_{q}u(z)$ and $r(z):=\partial_{p}u(z)$. The two functions
satisfy the following boundary conditions obtained after differentiating
(2.14) with respect to $p$ and $q$:

$\left\\{\begin{array}[]{l}s(T_{+}(p,q))=q\partial_{q}T_{+},\\\
s^{\prime}(T_{+}(p,q))=-1-p(1-2p^{2})\partial_{q}T_{+}\end{array}\right.$

and

$\left\\{\begin{array}[]{l}r(T_{+}(p,q))=1+q\partial_{p}T_{+},\\\
r^{\prime}(T_{+}(p,q))=-p(1-2p^{2})\partial_{p}T_{+}.\end{array}\right.$

Since $s(0)=\frac{-q}{p_{+}(1-2p_{+}^{2})}$,
$r(0)=\frac{p(1-2p^{2})}{p_{+}(1-2p_{+}^{2})}$, and
$s^{\prime}(0)=r^{\prime}(0)=0$, uniqueness of solutions of the differential
equations implies that the two functions are related by

$s(z)=-\frac{q}{p(1-2p^{2})}r(z).$ (2.18)

It follows from the boundary conditions at $z=T_{+}(p,q)$ and relation (2.18)
that

$\left\\{\begin{array}[]{l}s(T_{+}(p,q))=q\partial_{q}T_{+},\\\
s^{\prime}(T_{+}(p,q))=q\partial_{p}T_{+},\end{array}\right.$ (2.19)

so that

$\frac{s^{\prime}(T_{+}(p,q))}{s(T_{+}(p,q))}=\frac{\partial_{p}T_{+}(p,q)}{\partial_{q}T_{+}(p,q)}>0,$

where positivity follows from (2.12) for sufficiently small $p$ and $q$.

In order to prove that $s$ vanishes at exactly one point on $(0,T_{+}(p,q))$,
we consider the invariant curve $E_{\beta}$ given by (2.10) and the period
function $T_{+}(p,q)$ given by (2.11). Every $z\in(0,T_{+}(p,q))$ can be
represented by $z=T_{+}(P,Q)$ for some point $(P,Q)\in E_{\beta}$ with
$P\in(p,p_{+})$. By Lemma 3.8 in [16], we have $s(z)=0$ if and only if
$\partial_{Q}T_{+}(P,Q)=0$, which also follows from the first equation in
system (2.19). Then, by Lemmas 3.6, 3.7, 3.9, 3.10 in [16], for sufficiently
small $p$ and $q$ there is exactly one point $(P,Q)\in E_{\beta}$ with
$P\in(p,p_{+})$, where $s(z)=s(T_{+}(P,Q))=0$. ∎

##  3\. Existence of multi-pulse loop-localized states

Here we explain the asymptotic construction of the multi-pulse loop-localized
states and give the proof of Theorem 1.

For each boundary vertex $v_{j}\in\mathcal{V}_{B}$ with $1\leq j\leq|B|$, we
use the Dirichlet data $p_{j}$ as the unknown variable and write the flux
boundary condition to determine $p_{j}$. The main advantage of this method is
that the value of $p_{j}$ can be found independently from the conditions at
the other boundary vertices.

Fix $j$ in $1\leq j\leq|B|$. Assume that the boundary vertex
$v_{j}\in\mathcal{V}_{B}$ is in contact with $K_{j}$ pendants, $L_{j}$ loops,
and $M_{j}$ internal edges from the set $\mathcal{E}_{N,\epsilon}$. We denote
the corresponding sets by $\mathcal{E}_{j,{\rm pend}}$, $\mathcal{E}_{j,{\rm
loop}}$, and $\mathcal{E}_{j,{\rm int}}$ respectively. Furthermore, we divide
$\mathcal{E}_{j,{\rm int}}$ into $\mathcal{E}^{-}_{j,{\rm int}}$ and
$\mathcal{E}^{+}_{j,{\rm int}}$ depending on whether $v_{j}$ is the left or
right vertex of the internal edge $e\in\mathcal{E}_{j,{\rm int}}$,
respectively. The large solutions of Lemma 2 are centered at $0$ on the
pendants and loops but centered at an unknown point
$a_{e}\in(-\epsilon\ell_{e},\epsilon\ell_{e})$ for $e\in\mathcal{E}_{j,{\rm
int}}$. Assume a priori that for all large $\epsilon$, we have

$\max_{e\in\mathcal{E}_{j,{\rm int}}}|a_{\epsilon}|\leq a_{j,{\rm max}},$
(3.1)

where the constant $a_{j,{\rm max}}>0$ is $\epsilon$-independent.

By Lemma 1, the Neumann data at $v_{j}$ directed away from
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ is

$q_{j}^{(1)}=D_{j}p_{j}+\mathcal{O}(\|\vec{p}\|e^{-\epsilon\ell_{\rm
min}}+\|\vec{p}\|^{3}),$ (3.2)

where $\mathcal{O}$ denotes the error terms in the bound (2.7).

By Lemma 2, the Neumann data at $v_{j}$ directed away from
$\mathcal{E}_{N,\epsilon}$ is

$\displaystyle q_{j}^{(2)}$ $\displaystyle=$
$\displaystyle(K_{j}+2L_{j}+M_{j})p_{j}-4\sum_{e\in\mathcal{E}_{j,{\rm
pend}}}e^{-\epsilon\ell_{e}}-8\sum_{e\in\mathcal{E}_{j,{\rm
loop}}}e^{-\epsilon\ell_{e}}$ (3.3)
$\displaystyle-4\sum_{e\in\mathcal{E}^{-}_{j,{\rm
int}}}e^{-\epsilon\ell_{e}-a_{e}}-4\sum_{e\in\mathcal{E}^{+}_{j,{\rm
int}}}e^{-\epsilon\ell_{e}+a_{e}}+\mathcal{O}(\epsilon e^{-3\epsilon\ell_{\rm
j,min}}),$

where

$\ell_{j,{\rm min}}:=\min\\{\min_{e\in\mathcal{E}_{j,{\rm
pend}}}\\{\ell_{e}\\},\min_{e\in\mathcal{E}_{j,{\rm
loop}}}\\{\ell_{e}\\},\min_{e\in\mathcal{E}_{j,{\rm int}}}\\{\ell_{e}\\}\\}$

and $\mathcal{O}$ denotes the error terms in the bound (2.8). Note that the
summation in $q_{j}^{(2)}$ includes two contributions from the looping edges
in $\mathcal{E}_{j,{\rm loop}}$ due to the two ends of the looping edge
incident to the vertex $v_{j}$.

The flux boundary condition in (1.3) gives $q_{j}^{(1)}+q_{j}^{(2)}=0$, which
becomes the implicit equation on $p_{j}$. Since $C^{1}$ property of the error
terms is proven in [7] (Remark 8), the implicit equation is immediately solved
with

$\displaystyle p_{j}$ $\displaystyle=$
$\displaystyle\frac{4}{Z_{j}}\left(\sum_{e\in\mathcal{E}_{j,{\rm
pend}}}e^{-\epsilon\ell_{e}}+2\sum_{e\in\mathcal{E}_{j,{\rm
loop}}}e^{-\epsilon\ell_{e}}+\sum_{e\in\mathcal{E}^{-}_{j,{\rm
int}}}e^{-\epsilon\ell_{e}-a_{e}}+\sum_{e\in\mathcal{E}^{+}_{j,{\rm
int}}}e^{-\epsilon\ell_{e}+a_{e}}\right)$ (3.4)
$\displaystyle+\mathcal{O}\left(\|\vec{p}\|e^{-\epsilon\ell_{\rm
min}}++\|\vec{p}\|^{3}+\epsilon e^{-3\epsilon\ell_{\rm j,min}}\right),$

where $Z_{j}:=D_{j}+K_{j}+2L_{j}+M_{j}$ is the total degree of the vertex
$v_{j}$ in $\Gamma_{\epsilon}$.

######  Remark 11.

Although the boundary conditions (1.3) are satisfied for all vertices if
$p_{j}$ is defined by (3.4) for $1\leq j\leq|B|$, there are two consistency
conditions to be verified. One condition arises from the fact that
$\|\vec{p}\|$ in the error terms in (3.4) include all Dirichlet data
$\vec{p}=(p_{1},p_{2},\dots,p_{|B|})$. Therefore, the error terms in (3.4)
must be smaller than the leading-order terms in (3.4) as $\epsilon\to\infty$
for each $j$. The second condition arises from the fact that the values
$a_{e}$ for $e\in\mathcal{E}_{j,{\rm int}}$ must be uniquely defined and be
bounded as $\epsilon\to\infty$ according to the a priori assumption (3.1).

In order to complete the proof of Theorem 1, it remains to verify the two
consistency conditions in Remark 11 from the two technical assumptions,
Assumptions 1 and 2.

The first consistency condition is satisfied if

$\|\vec{p}\|e^{-\epsilon\ell_{\rm min}}\ll e^{-\epsilon\ell_{j,{\rm
min}}},\quad\|\vec{p}\|^{3}\ll e^{-\epsilon\ell_{j,{\rm min}}},\quad 1\leq
j\leq|B|,$

where $e^{-\epsilon\ell_{j,{\rm min}}}$ defines the size of the leading-order
terms in (3.4). This leads to the constraints (1.9) and (1.10) in Assumption
1.

The second consistency condition can be formulated in terms of the invariant
curve (2.10). The same level $\beta$ for the large solution of Lemma 2 on a
given internal edge must correspond to two segments extending to two different
boundary vertices.

Let $e_{0}\in\mathcal{E}_{N,\epsilon}$ be an internal edge connecting
$v_{j}\in\mathcal{V}_{B}$ at the left end and $v_{k}\in\mathcal{V}_{B}$ at the
right end, so that $e_{0}\in\mathcal{E}^{-}_{j,{\rm int}}$ and
$e_{0}\in\mathcal{E}^{+}_{k,{\rm int}}$ and $j\neq k$. Then, $a_{e_{0}}$ must
be found from the following equation

$q_{e_{0}\sim v_{j}}^{2}-p_{j}^{2}+p_{j}^{4}=q_{e_{0}\sim
v_{k}}^{2}-p_{k}^{2}+p_{k}^{4},$ (3.5)

where $q_{e_{0}\sim v_{j}}$ and $q_{e_{0}\sim v_{k}}$ are given by the
expansion

$\left\\{\begin{array}[]{l}q_{e_{0}\sim
v_{j}}=4e^{-\epsilon\ell_{e_{0}}-a_{e_{0}}}-p_{j}+\mathcal{O}(\epsilon
e^{-3\epsilon\ell_{e_{0}}}),\\\ q_{e_{0}\sim
v_{k}}=-4e^{-\epsilon\ell_{e_{0}}+a_{e_{0}}}+p_{k}+\mathcal{O}(\epsilon
e^{-3\epsilon\ell_{e_{0}}}),\end{array}\right.$ (3.6)

due to the bounds (2.8).

Let us now assume that the internal edge $e_{0}\in\mathcal{E}_{N,\epsilon}$
does not have common boundary vertices with other internal edges in
$\mathcal{E}_{N,\epsilon}$ as in Assumption 2. This means that
$\mathcal{E}_{j,{\rm int}}=\\{e_{0}\\}$ and $\mathcal{E}_{k,{\rm
int}}=\\{e_{0}\\}$. Substituting (3.4) and (3.6) into the two sides of
equation (3.5) gives the expansions:

$\displaystyle q_{e_{0}\sim
v_{j}}^{2}-p_{j}^{2}+p_{j}^{4}=\frac{16}{Z_{j}}\left[(Z_{j}-2)e^{-2\epsilon\ell_{e_{0}}-2a_{e_{0}}}-2e^{-\epsilon\ell_{e_{0}}-a_{e_{0}}}\left(\sum_{e\in\mathcal{E}_{j,{\rm
pend}}}e^{-\epsilon\ell_{e}}+2\sum_{e\in\mathcal{E}_{j,{\rm
loop}}}e^{-\epsilon\ell_{e}}\right)\right]$
$\displaystyle+\mathcal{O}\left(\|\vec{p}\|e^{-\epsilon\ell_{\rm
min}-\epsilon\ell_{0}}++\|\vec{p}\|^{3}e^{-\epsilon\ell_{0}}+\epsilon
e^{-3\epsilon\ell_{j,{\rm
min}}-\epsilon\ell_{e_{0}}}+e^{-4\epsilon\ell_{j,{\rm min}}}+\epsilon
e^{-3\epsilon\ell_{e_{0}}-\epsilon\ell_{j,{\rm min}}}+\epsilon
e^{-4\epsilon\ell_{e_{0}}}\right)$

and

$\displaystyle q_{e_{0}\sim
v_{k}}^{2}-p_{k}^{2}+p_{k}^{4}=\frac{16}{Z_{k}}\left[(Z_{k}-2)e^{-2\epsilon\ell_{e_{0}}+2a_{e_{0}}}-2e^{-\epsilon\ell_{e_{0}}+a_{e_{0}}}\left(\sum_{e\in\mathcal{E}_{k,{\rm
pend}}}e^{-\epsilon\ell_{e}}+2\sum_{e\in\mathcal{E}_{k,{\rm
loop}}}e^{-\epsilon\ell_{e}}\right)\right]$
$\displaystyle+\mathcal{O}\left(\|\vec{p}\|e^{-\epsilon\ell_{\rm
min}-\epsilon\ell_{0}}++\|\vec{p}\|^{3}e^{-\epsilon\ell_{0}}+\epsilon
e^{-3\epsilon\ell_{k,{\rm
min}}-\epsilon\ell_{e_{0}}}+e^{-4\epsilon\ell_{k,{\rm min}}}+\epsilon
e^{-3\epsilon\ell_{e_{0}}-\epsilon\ell_{k,{\rm min}}}+\epsilon
e^{-4\epsilon\ell_{e_{0}}}\right).$

By Remark 1, we have $Z_{j}\geq 3$ and $Z_{k}\geq 3$ for the internal edge
$e_{0}$. Equating the two sides in (3.5) and dividing by
$e^{-2\epsilon\ell{e_{0}}}$ gives the unique solution of the implicit
equation:

$\displaystyle e^{4a_{e_{0}}}=\frac{(Z_{j}-2)Z_{k}}{(Z_{k}-2)Z_{j}}+E_{0},$
(3.7)

where the error terms in $E_{0}$ are given by

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle\frac{2}{Z_{k}-2}e^{3a_{e_{0}}+\epsilon\ell_{e_{0}}}\left(\sum_{e\in\mathcal{E}_{k,{\rm
pend}}}e^{-\epsilon\ell_{e}}+2\sum_{e\in\mathcal{E}_{k,{\rm
loop}}}e^{-\epsilon\ell_{e}}\right)$
$\displaystyle-\frac{2Z_{k}}{Z_{j}(Z_{k}-2)}e^{a_{e_{0}}+\epsilon\ell_{e_{0}}}\left(\sum_{e\in\mathcal{E}_{j,{\rm
pend}}}e^{-\epsilon\ell_{e}}+2\sum_{e\in\mathcal{E}_{j,{\rm
loop}}}e^{-\epsilon\ell_{e}}\right)$
$\displaystyle+\mathcal{O}\left(\|\vec{p}\|e^{\epsilon\ell_{e_{0}}-\epsilon\ell_{\rm
min}}++\|\vec{p}\|^{3}e^{\epsilon\ell_{e_{0}}}+\epsilon
e^{\epsilon\ell_{e_{0}}-3\epsilon\ell_{j,{\rm min}}}+\epsilon
e^{\epsilon\ell_{e_{0}}-3\epsilon\ell_{k,{\rm min}}}\right)$
$\displaystyle+\mathcal{O}\left(e^{2\epsilon\ell_{e_{0}}-4\epsilon\ell_{j,{\rm
min}}}+e^{2\epsilon\ell_{e_{0}}-4\epsilon\ell_{k,{\rm min}}}+\epsilon
e^{-\epsilon\ell_{e_{0}}-\epsilon\ell_{j,{\rm min}}}+\epsilon
e^{-\epsilon\ell_{e_{0}}-\epsilon\ell_{k,{\rm min}}}+\epsilon
e^{-2\epsilon\ell_{e_{0}}}\right).$

The error terms beyond the first (leading-order) term in (3.7) are small if

$\displaystyle\|\vec{p}\|e^{-\epsilon\ell_{\rm min}}\ll
e^{-\epsilon\ell_{e_{0}}},$ $\displaystyle\|\vec{p}\|^{3}\ll
e^{-\epsilon\ell_{e_{0}}},$ $\displaystyle\epsilon e^{-3\epsilon\ell_{j,{\rm
min}}}\ll e^{-\epsilon\ell_{e_{0}}},$ $\displaystyle\epsilon
e^{-3\epsilon\ell_{k,{\rm min}}}\ll e^{-\epsilon\ell_{e_{0}}},$ $\displaystyle
e^{-\epsilon\ell_{e}}\ll e^{-\epsilon\ell_{e_{0}}},\quad\forall
e\in\mathcal{E}_{j,{\rm pend}}\cup\mathcal{E}_{j,{\rm loop}},$ $\displaystyle
e^{-\epsilon\ell_{e}}\ll e^{-\epsilon\ell_{e_{0}}},\quad\forall
e\in\mathcal{E}_{k,{\rm pend}}\cup\mathcal{E}_{k,{\rm loop}}.$

The first two conditions coincide with the consistency conditions satisfied by
Assumption 1. The other four conditions are satisfied if $\ell_{e_{0}}$ is
strictly minimal among the half-lengths of the looping edges and the lenghts
of the pendant edges as in Assumption 2. This means that

$\ell_{e_{0}}=\ell_{j,{\rm min}}=\ell_{k,{\rm min}}<\ell_{e},\quad\forall
e\in\mathcal{E}_{j,{\rm pend}}\cup\mathcal{E}_{j,{\rm
loop}}\cup\mathcal{E}_{k,{\rm pend}}\cup\mathcal{E}_{k,{\rm loop}}.$

Since the first (leading-order) term in (3.7) is independent of $\epsilon$,
the a priori assumption (3.1) on the admissible values of $a_{e_{0}}$ is
satisfied by the solution (3.7).

The proof of Theorem 1 is complete. The bound (1.11) follows from the bound
(2.24) in Theorem 2.9 in [7].

######  Remark 12.

If two or more internal edges in $\mathcal{E}_{N}$ are connected to the same
boundary vertex, a system of two or more equations (3.5) is set up and the
solution for $a_{e_{0}}$ is no longer available in the simple form (3.7). It
is generally hard to obtain the solution for $a_{e_{0}}$ on each internal edge
in the system of nonlinear equations.

##  4\. Morse index for multi-pulse edge-localized states

Here we count the Morse index for multi-pulse edge-localized states and give
the proof of Theorem 2.

Let $\Phi$ be the positive $N$-pulse edge-localized state of Theorem 1 such
that the set $\mathcal{E}_{N}$ only contains pendant and looping edges. Let
$\mathcal{L}:H^{2}_{\rm NK}(\Gamma)\subset L^{2}(\Gamma)\to L^{2}(\Gamma)$ be
the linearized operator with the differential expression given by (1.8). The
number of its negative eigenvalues (the Morse index) is denoted by
$n(\mathcal{L})$ and the multiplicity of its zero eigenvalue is denoted by
$z(\mathcal{L})$. We will prove that $n(\mathcal{L})=N$ and $z(\mathcal{L})=0$
for large enough $\epsilon$.

Applying the scaling transformation $\Phi(x)=\epsilon U(z)$ with $z=\epsilon
x$ and $\epsilon:=\sqrt{-\omega}$, we rewrite the spectral problem for
$\mathcal{L}$ as a collection of differential equations

$-w_{j}^{\prime\prime}(z)+w_{j}(z)-6u_{j}(z)^{2}w_{j}(z)=\lambda
w_{j}(z),\quad z\in e_{j,\epsilon},$ (4.1)

subject to the boundary conditions (1.3) at vertices $v\in\mathcal{V}$. Here
$W=(w_{1},w_{2},\dots,w_{|\mathcal{E}|})$ is a rescaled eigenvector in
$H^{2}_{\rm NL}(\Gamma_{\epsilon})$ and $\lambda$ is a rescaled eigenvalue of
the linearized operator $\mathcal{L}$. Abusing notations, we still refer to
the linearized operator in the rescaled variables as to $\mathcal{L}$.

###  4.1. Idea of the proof

Recall that $\mathcal{V}_{B}$ is the set of boundary vertices separating
$\mathcal{E}_{N,\epsilon}$ and
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$. Then for every
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{|B|})$, let the space
$H^{2}_{\bm{\alpha}}(\Gamma_{\epsilon})$ be defined by the modified boundary
conditions:

$\left\\{\begin{array}[]{l}W\text{ is continuous on }\Gamma_{\epsilon},\\\
\sum_{e\sim v}\partial W(v)=0\text{ for every vertex
}v\in\mathcal{V}\backslash\mathcal{V}_{B},\\\ \sum_{e\sim v_{j}}\partial
W(v_{j})=\alpha_{j}W(v_{j})\text{ for every vertex
}v_{j}\in\mathcal{V}_{B},\end{array}\right.$ (4.2)

where the derivatives $\partial$ are directed away from the vertex.

We use $\bm{\alpha}=0$ to denote $\bm{\alpha}=(0,0,\dots,0)$ and
$\bm{\alpha}=\infty$ to denote $\bm{\alpha}=(+\infty,+\infty,\dots,+\infty)$.
Then, we have $H^{2}_{\rm NK}(\Gamma_{\epsilon})\equiv
H^{2}_{\bm{\alpha}=0}(\Gamma_{\epsilon})$ and $H^{2}_{\rm
D}(\Gamma_{\epsilon})\equiv H^{2}_{\bm{\alpha}=\infty}(\Gamma_{\epsilon})$,
where $H^{2}_{\rm D}(\Gamma_{\epsilon})$ stands for the domain with the
Dirichlet conditions at the boundary vertices:

$\left\\{\begin{array}[]{l}W\text{ is continuous on }\Gamma_{\epsilon},\\\
\sum_{e\sim v}\partial W(v)=0\text{ for every vertex
}v\in\mathcal{V}\backslash\mathcal{V}_{B},\\\ W(v_{j})=0\text{ for every
vertex }v_{j}\in\mathcal{V}_{B}.\end{array}\right.$ (4.3)

We also introduce the linearized operator $\mathcal{L}_{\bm{\alpha}}$, such
that the spectral problem $\mathcal{L}_{\bm{\alpha}}W=\lambda W$ is still
represented by the differential equations (4.1), but the domain of
$\mathcal{L}_{\bm{\alpha}}$ is $H^{2}_{\bm{\alpha}}(\Gamma_{\epsilon})\subset
L^{2}(\Gamma_{\epsilon})$.

The proof of Theorem 2 relies on the continuity argument from
$\bm{\alpha}=\infty$ to $\bm{\alpha}=0$ given by the following proposition.

######  Proposition 1.

Assume that $\mathcal{L}_{\bm{\alpha}}W=0$ admits no solutions $W\in
H^{2}_{\bm{\alpha}}(\Gamma_{\epsilon})$ for every
$\bm{\alpha}\in[0,\infty)^{|B|}$. Then,

$n(\mathcal{L}_{\bm{\alpha}=0})=n(\mathcal{L}_{\bm{\alpha}=\infty}),\qquad
z(\mathcal{L}_{\bm{\alpha}=0})=z(\mathcal{L}_{\bm{\alpha}=\infty}).$

###### Proof.

The proof relies on the standard perturbation theory of linear operators given
in Chapter 7 of [17]. In short, since $\mathcal{L}_{\bm{\alpha}}$ is a
holomorphic family of self-adjoint operators, each isolated eigenvalue of
$\mathcal{L}_{\bm{\alpha}}$ depends continuously on $\alpha$. Therefore, if no
eigenvalues of $\mathcal{L}_{\bm{\alpha}}$ cross zero when $\bm{\alpha}$
traverses from $\bm{\alpha}=\infty$ to $\bm{\alpha}=0$, then the number of
negative eigenvalues as well as the multiplicity of the zero eigenvalue
remains the same. ∎

By using Proposition 1, Theorem 2 holds if we can verify the following claims
for large $\epsilon$:

$\displaystyle\text{\bf Claim 1:}\qquad
n(\mathcal{L}_{\bm{\alpha}=\infty})=N\text{ and
}z(\mathcal{L}_{\bm{\alpha}=\infty})=0,$ $\displaystyle\text{\bf Claim
2:}\qquad\mathcal{L}_{\bm{\alpha}}W=0\text{ admits no solutions }W\in
H^{2}_{\bm{\alpha}}(\Gamma_{\epsilon})\text{ with }\alpha\in[0,\infty)^{|B|}.$

Below we give proofs of these claims.

###  4.2. Proof of Claim 1

The proof is based on the Sturm’s Comparison theorem (see Section 5.5 in
[26]). For simplicity, we assume that $\mathcal{E}_{N}$ contains only looping
edges. Due to the symmetry of solutions in the looping edges, the proof
extends to the pendant edges with minor modifications.

The spectral problem for the operator $\mathcal{L}_{\bm{\alpha}=\infty}$ can
be expressed by the second-order differential equations (4.1) equipped with
the Dirichlet conditions at the boundary vertices given by (4.3). The spectral
problem for $\mathcal{L}_{\bm{\alpha}=\infty}$ is given by $N$ uncoupled
problems on the $N$ edges in $\mathcal{E}_{N,\epsilon}$ and another uncoupled
problem in $\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$. On each edge
$e_{j,\epsilon}\in\mathcal{E}_{N,\epsilon}$, we have

$\left\\{\begin{array}[]{l}-w_{j}^{\prime\prime}(z)+w_{j}(z)-6u_{j}(z)^{2}w_{j}(z)=\lambda
w_{j}(z),\quad z\in(-\epsilon\ell_{j},\epsilon\ell_{j}),\\\
w_{j}(-\epsilon\ell_{j})=w_{j}(\epsilon\ell_{j})=0,\end{array}\right.$ (4.4)

where $u_{j}(z)$ satisfies the condition (2.5). Since $\epsilon$ is large,
$p:=u(\epsilon\ell_{j})$ and $q:=-u^{\prime}(\epsilon\ell_{j})$ are small and
positive by Lemma 2. For $\lambda=0$, the homogeneous equation (4.4) is solved
by the superposition formula (2.15) in Lemma 4, where the even solution $s(z)$
has exactly one nodal point on $(0,\epsilon\ell_{j})$ and the odd solution
$u^{\prime}(z)$ has no nodal points on $(0,\epsilon\ell_{j})$. By Sturm’s
Comparison theorem, the spectral problem (4.4) admits exactly one simple
negative eigenvalue and no zero eigenvalue. Since eigenvalues of all $N$
spectral problems (4.4) also appear in the spectrum of the operator
$\mathcal{L}_{\bm{\alpha}=\infty}$, we have
$n(\mathcal{L}_{\bm{\alpha}=\infty})\geq N$.

######  Remark 13.

If $e_{j,\epsilon}\in\mathcal{E}_{N,\epsilon}$ is a pendant edge, the Neumann
condition at one end implies that the spectral problem (4.4) is solved in the
space of even functions, where it still admits exactly one simple negative
eigenvalue and no zero eigenvalue.

Next, we show that the uncoupled problem in
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ given by

$-W^{\prime\prime}+W-6U^{2}W=\lambda W,\quad
z\in\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon},$ (4.5)

subject to the boundary conditions (4.3) has a strictly positive spectrum.
Since $U$ on $\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ satisfies
the bound (2.6) with small $\vec{p}$ given by (3.4), we obtain from (4.5)
after multiplication by $W$ and integration over
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ with the boundary
conditions (4.3):

$\lambda\|W\|_{L^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}=\|W\|_{L^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}-6\|UW\|_{L^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}+\|W^{\prime}\|_{L^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}\geq
C\|W\|^{2}_{L^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})},$

for some $C>0$. Hence, $\lambda\geq C>0$ for every solution $W\in H^{2}_{\rm
D}((\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})$, and the spectral
problem (4.5) does not contribute into the nonpositive part of the spectrum of
the operator $\mathcal{L}_{\bm{\alpha}=\infty}$. This completes the proof of
$n(\mathcal{L}_{\bm{\alpha}=\infty})=N$ and
$z(\mathcal{L}_{\bm{\alpha}=\infty})=0$.

###  4.3. Proof of Claim 2

Let $W$ be a solution of $\mathcal{L}_{\bm{\alpha}}W=0$ in
$H^{2}(\Gamma_{\epsilon})$ with large enough $\epsilon$. For every looping
edge $e_{j,\epsilon}\in\mathcal{E}_{N,\epsilon}$ associated with the segment
$[-\epsilon\ell_{j},\epsilon\ell_{j}]$, the restriction of $W$ to the edge
$e_{j,\epsilon}$ denoted by $w_{j}$ solves the second-order differential
equation

$-w_{j}^{\prime\prime}(z)+w_{j}(z)-6u_{j}(z)^{2}w_{j}(z)=0,\quad
z\in(-\epsilon\ell_{j},\epsilon\ell_{j}).$ (4.6)

where $u_{j}(z)$ satisfies the condition (2.5). The following lemma computes
the contributions of the solution of the differential equation (4.6) to the
last boundary condition in (4.2).

######  Lemma 5.

If $e_{j,\epsilon}\in\mathcal{E}_{N,\epsilon}$ is a looping edge with
$\epsilon$ large enough, then either $w_{j}\equiv 0$ on the entire edge or

$\alpha_{\pm}(e_{j,\epsilon}):=\mp\frac{w_{j}^{\prime}(\pm\epsilon\ell_{j})}{w_{j}(\pm\epsilon\ell_{j})}<0.$
(4.7)

###### Proof.

By Lemma 4, the general solution is given by (2.15) rewritten again as

$w_{j}(z)=Au_{j}^{\prime}(z)+Bs_{j}(z)$ (4.8)

for some parameters $A$ and $B$. Since $u_{j}^{\prime}(z)$ is odd, $s_{j}(z)$
is even, and $u_{j}^{\prime}(\pm\epsilon\ell_{j})\neq 0$, the continuity
condition $w_{j}(-\epsilon\ell_{j})=w_{j}(\epsilon\ell_{j})$ necessarily
implies $A=0$.

If $B=0$, then $w_{j}\equiv 0$ on the entire edge. If $B\neq 0$, then
$w_{j}(\pm\epsilon\ell_{j})\neq 0$ and $\alpha_{\pm}(e_{j,\epsilon})$ in (4.7)
is defined by $s_{j}(\pm\epsilon\ell_{j})$ and
$s_{j}^{\prime}(\pm\epsilon\ell_{j})$. It follows from the second inequality
in (2.16) and the symmetry of $s_{j}(z)$ that
$\alpha_{\pm}(e_{j,\epsilon})<0$. ∎

######  Remark 14.

If $e_{j,\epsilon}\in\mathcal{E}_{N,\epsilon}$ is a pendant edge, the Neumann
condition at one end still implies that $A=0$, after which the proof of Lemma
5 holds and gives $\alpha_{+}(e_{j,\epsilon})<0$.

We denote the value of $W$ at each boundary point $v_{j}\in\mathcal{V}_{B}$ as
$\mathfrak{p}_{j}:=W(v_{j})$. Then on the subgraph
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$, $W$ solves the
homogeneous equation

$-W^{\prime\prime}+W-6U^{2}W=0,\quad
z\in\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon},$ (4.9)

subject to the nonhomogeneous Dirichlet boundary conditions:

$\left\\{\begin{array}[]{l}W\text{ is continuous on
}\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon},\\\ \sum_{e\sim
v}\partial W(v)=0\text{ for every vertex
}v\in\mathcal{V}\backslash\mathcal{V}_{B},\\\ W(v_{j})=\mathfrak{p}_{j}\text{
for every vertex }v_{j}\in\mathcal{V}_{B}.\end{array}\right.$ (4.10)

The following lemma computes the contributions of the solution of the
boundary-value problem (4.9)–(4.10) to the last boundary condition in (4.2).

######  Lemma 6.

There are $C_{0}>0$ and $\epsilon_{0}>0$ such that for every
$\vec{\mathfrak{p}}=(\mathfrak{p}_{1},\mathfrak{p}_{2},\dots,\mathfrak{p}_{|B|})$
and every $\epsilon>\epsilon_{0}$, the unique solution $W\in
H^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})$ of the boundary-
value problem (4.9)–(4.10) satisfies

$\|W\|_{H^{2}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}\leq
C_{0}\|\vec{\mathfrak{p}}\|$ (4.11)

and

$|\mathfrak{q}_{j}-D_{j}\mathfrak{p}_{j}|\leq
C_{0}\|\vec{\mathfrak{p}}\|e^{-\epsilon\ell_{\rm min}},$ (4.12)

where $\mathfrak{q}_{j}$ is the Neumann data directed away from
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$, $D_{j}$ is the degree
of the $j$-th boundary vertex $v_{j}\in\mathcal{V}_{B}$ in
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$, and $\ell_{\rm min}$ is
the length of the shortest edge in $\Gamma\backslash\mathcal{E}_{N}$.

###### Proof.

Since
$\|U\|_{L^{\infty}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}$ is
small for large enough $\epsilon$ by the bound (2.6) with $\vec{p}$ given by
(3.4), the boundary-value problem (4.9)–(4.10) can be analyzed with Theorem
2.1 and Lemma 2.3 in [7] for $g=0$. The bound (4.12) follows from the bound
(2.4) in [7] due to the smallness of
$\|U\|_{L^{\infty}(\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon})}$. ∎

######  Remark 15.

The Neumann data $\mathfrak{q}_{j}$ in (4.12) is directed in the opposite
direction compared to $\partial\Psi(v_{j})$ in (4.2). In other words,

$\sum_{e\sim
v_{j},e\in\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}}\partial\Psi(v_{j})=-\mathfrak{q}_{j}=-D_{j}\mathfrak{p}_{j}+\mathcal{O}(\|\vec{\mathfrak{p}}\|e^{\epsilon\ell_{\rm
min}}).$ (4.13)

We are now ready to show that no solutions $W\in H^{2}(\Gamma_{\epsilon})$ of
$\mathcal{L}_{\bm{\alpha}}W=0$ satisfy the last boundary condition in (4.2)
with $\alpha_{j}$ being nonnegative.

First, we consider the case when $\mathfrak{p}_{j}=0$ for all
$v_{j}\in\mathcal{V}_{B}$. Then, the right hand side of the last boundary
conditions in (4.2) vanishes and $W$ satisfies the Neumann-Kirchhoff (NK)
boundary conditions. Moreover, by Lemma 5, the
$\mathcal{E}_{N,\epsilon}$-components of the solution $W$ are entirely zero,
and make no contribution into the last condition in (4.2). In what follows,
the restriction of $W$ to
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ solves the second-order
differential equation (4.9) subject to the Dirichlet conditions at the
boundary vertices. By Lemma 6 due to the bound (4.11), the only solution on
$\Gamma_{\epsilon}\backslash\mathcal{E}_{N,\epsilon}$ is zero, hence $W=0$ on
$\Gamma_{\epsilon}$.

Therefore, there exists at least one boundary vertex $v_{j}\in\mathcal{V}_{B}$
such that the corresponding $\mathfrak{p}_{j}\neq 0$. Up to multiplication by
a constant, we may assume that $\mathfrak{p}_{j}=1$. Using Lemmas 5 and 6 and
the expansion (4.13), the last condition in (4.2) for large enough $\epsilon$
becomes

$\alpha_{j}=\sum_{e\in\mathcal{E}_{j,\epsilon}}\alpha_{-}(e)+\sum_{e\in\mathcal{E}_{j,\epsilon}}\alpha_{+}(e)-\mathfrak{q}_{j}<0,$
(4.14)

where $\mathcal{E}_{j,\epsilon}\subset\mathcal{E}_{N,\epsilon}$ is the set of
edges in $\mathcal{E}_{N,\epsilon}$ adjacent to the vertex
$v_{j}\in\mathcal{V}_{B}$. This contradicts to the assumption that
$\alpha_{j}\in[0,\infty)$. Hence no $W\in
H^{2}_{\bm{\alpha}}(\Gamma_{\epsilon})$ exists such that
$\mathcal{L}_{\bm{\alpha}}W=0$ for $\alpha\in[0,\infty)^{|B|}$.

##  5\. Examples

We end this paper with three examples of quantum graphs, where the assumptions
of Theorems 1 and 2 can be checked. We also discuss limitations of results of
Theorems 1 and 2 to cover all edge-localized states on these graphs.

Figure 4. A flower graph with three loops as in Example 1. Each loop $j$ with
$j=1,2,3$ is parametrized by a segment $[-\ell_{j},\ell_{j}]$ of length
$2\ell_{j}$, and the unbounded edge is parametrized by a half-line
$[0,\infty)$. The direction of the axis associated with each of the segments
is specified by the arrows.

######  Example 1.

Consider a flower graph consisting of $L$ loops of the lengths
$\\{2\ell_{j}\\}_{j=1}^{L}$ and one half-line connected at a single common
vertex, see Figure 4 for $L=3$. Assumption 1 is satisfied because $\ell_{\rm
min}=\infty$ in (1.9) and $|B|=1$ in (1.10). Assumption 2 is satisfied because
no internal edges exist. Theorem 1 states existence of the $N$-pulse edge-
localized states for every $1\leq N\leq L$ and Theorem 2 states that Morse
index of this $N$-pulse state is $N$. This coincides with the main results of
[16] (Theorems 1, 2, and 3) in the limit $\omega\to-\infty$ obtained with long
analysis of the period function in the case of loops of the same normalized
length.

######  Example 2.

Consider a dumbbell graph consisting of two loops of lengths $2\ell_{-}$ and
$2\ell_{+}$ connected by an internal edge of length $2\ell_{0}$ at two
vertices, see Figure 5. Assume that $\ell_{-}\leq\ell_{+}$ for convenience. We
can construct several edge-localized states as follows.

  * •

By Theorem 1, assumptions of which are always satisfied if $N=1$, the
$1$-pulse state can be placed at any of the three edges in the limit
$\omega\to-\infty$ in agreement with the main results of [4] and [7]. By
Theorem 2, Morse index of the $1$-pulse state placed at the loop is $1$. Morse
index for the $1$-pulse state placed at the internal edge is not defined by
Theorem 2. By the variational theory in [4], the Morse index of this $1$-pulse
state is also $1$. It was also proven explicitly in Lemma 4.10 in [20] for the
symmetric dumbbell graph with $\ell_{-}=\ell_{+}$ that the Morse index of this
$1$-pulse state is $1$.

  * •

By Theorem 1, the $2$-pulse state can be placed at the two loops if
$\ell_{+}<2\ell_{0}+\ell_{-}$ and $\ell_{+}<3\ell_{-}$ from the conditions
(1.9) and (1.10) respectively. The constraints are satisfied if
$\ell_{-}=\ell_{+}$. By Theorem 2, the Morse index of this $2$-pulse state is
$2$.

  * •

By Theorem 1, the $2$-pulse state can also be placed at one loop and the
internal edge if $\ell_{0}<\ell_{-}$ due to Assumption 2. The conditions (1.9)
and (1.10) of Assumption 1 are satisfied if $\ell_{0}<\ell_{-}\leq\ell_{+}$.
The Morse index of this $2$-pulse state is not defined by Theorem 2.

  * •

By Theorem 1, the $3$-pulse state can be placed at all three edges if
$\ell_{0}<\ell_{-}\leq\ell_{+}$. The Morse index of this $3$-pulse state is
not defined by Theorem 2.

Figure 5. A dumbbell graph as in Example 2. The left loop is parametrized by a
segment $[-\ell_{-},\ell_{-}]$ of length $2\ell_{-}$, and the right loop is
parametrized by a segment $[-\ell_{+},\ell_{+}]$ of length $2\ell_{+}$. The
internal edge is parametrized by a segment $[-\ell_{0},\ell_{0}]$ of length
$2\ell_{0}$. The direction of the axis associated with each of the segments is
specified by the arrows.

######  Example 3.

Consider a graph $\Gamma$ which is given in Figure 6. It formally consists of
three edges, where $e_{1}=[0,\ell_{1}]$ and $e_{3}=[0,\ell_{3}]$ are pendant
edges and $e_{2}=[-\ell_{2},\ell_{2}]$ is the internal edge. However, $v_{2}$
and $v_{3}$ have are fake vertices of degree $2$, hence the same graph is
mapped into a single interval $[-\ell_{1}-\ell_{2},\ell_{2}+\ell_{3}]$ subject
to Neumann boundary conditions at the end points. Assume that
$\ell_{1}\leq\ell_{3}$ for convenience.

Figure 6. A segment graph $\Gamma$ in Example 3.

All possible solutions of the Neumann boundary-value problem for the
stationary equation $-u^{\prime\prime}+u-2u^{3}=0$ are given by the integral
curves shown on Fig. 7. Due to the Neumann boundary conditions, the positive
solution starts and ends at one of the two points $(p_{-},0)$ and $(p_{+},0)$
shown in red. Besides two simplest states loalized at one of the two pendant
edges from a half-loop on Fig. 7, we can construct three additional edge-
localized states as follows.

Figure 7. Phase plane $(u,v)$ of the differential equation
$-u^{\prime\prime}+u-2u^{3}=0$ showing the homoclinic loop for $\beta=0$ and
the integral curve $E_{\beta}$ for $\beta\in(-\frac{1}{4},0)$.

  * •

Let us take a single loop on the integral curve of Fig. 7 with the boundary
conditions

$u(-\epsilon(\ell_{1}+\ell_{2}))=u(\epsilon(\ell_{2}+\ell_{3}))=p_{-}.$

Due to the spatial symmetry, the $1$-pulse state concentrates at the middle
point $z=\frac{1}{2}\epsilon(\ell_{3}-\ell_{1})$. Since $\ell_{1}\leq\ell_{3}$
implies $\ell_{1}<2\ell_{2}+\ell_{3}$, the middle point belongs to $e_{2}$,
hence the edge-localized state in the limit of large $\epsilon$ (when
$p_{-}\to 0$) corresponds to the choice $\mathcal{E}_{1}:=\\{e_{2}\\}$ for the
graph $\Gamma$ on Fig. 6. This $1$-pulse state is illustrated on the left
panel of Fig. 8. The right panel shows the derivative $u^{\prime}(z)$ which
satisfies the homogeneous equation $-w^{\prime\prime}(z)+w(z)-6u(z)^{2}w(z)=0$
and Dirichlet conditions at the end points of the interval
$[-\epsilon(\ell_{1}+\ell_{2}),\epsilon(\ell_{2}+\ell_{3})]$. Using the
Courant’s nodal theorem (see Section 5.5 in [26]), since $u^{\prime}(z)$ has
two nodal domains on the interval, the zero eigenvalue is the second
eigenvalue of the Dirichlet boundary-value problem (denoted as
$\lambda_{j}^{D}$) interlacing with the eigenvalues of the Neumann boundary-
value problem (denoted as $\lambda_{j}^{N}$) as follows:

$\lambda_{1}^{N}<\lambda_{1}^{D}\leq\lambda^{N}_{2}<\lambda^{D}_{2}=0\leq\lambda_{3}^{N}<\lambda_{3}^{D}\leq\lambda_{4}^{N}<\dots$
(5.1)

Here the inequality between $\lambda^{N}_{2}$ and $\lambda^{D}_{2}$ is strict
because the even potential $-6u(z)^{2}$ of the Schrödinger operator
$\partial_{z}^{2}+1-6u(z)^{2}$ is extended periodically on $\mathbb{R}$ and it
is a well-known fact that the anti-periodic eigenfunctions for
$\lambda_{1}^{D}$ and $\lambda_{2}^{N}$ can not exist at the same eigenvalues
as the periodic eigenfunctions for $\lambda^{D}_{2}$ and $\lambda_{3}^{N}$.
Similarly, the inequality between $\lambda_{3}^{N}$ and $\lambda_{3}^{D}$ is
strict. It follows from (5.1) that the Morse index of the $1$-pulse state is
$n(\mathcal{L})=2$. This is a counter-example to conclusion of Theorem 2 with
$N=1$ which is the reason why the internal edges are excluded from the set
$\mathcal{E}_{N}$ in Theorem 2.

Figure 8. Left: the $1$-pulse state on the internal edge $e_{2}$. Right: its
derivative satisfying Dirichlet conditions on the interval.

  * •

Let us take a single loop on the integral curve of Fig. 7 with the boundary
conditions

$u(-\epsilon(\ell_{1}+\ell_{2}))=u(\epsilon(\ell_{2}+\ell_{3}))=p_{+}.$

In the limit of large $\epsilon$ (when $p_{+}\to 1$), this $2$-pulse state
corresponds to the choice $\mathcal{E}_{2}:=\\{e_{1},e_{3}\\}$ for the graph
$\Gamma$ on Fig. 6. Assumption 1 is satisfied if $\ell_{+}<2\ell_{0}+\ell_{-}$
and $\ell_{+}<3\ell_{-}$. The $2$-pulse state exists for sufficiently large
$\epsilon$ by Theorem 1. This solution is illustrated on the left panel of
Fig. 9. The right panel shows the derivative $u^{\prime}(z)$ with two nodal
domains, so that the ordering (5.1) suggests that $n(\mathcal{L})=2$. This
agrees with the count of Theorem 2 with $N=2$ since the internal edge $e_{2}$
is not in the set $\mathcal{E}_{N}$ and the fake vertices $v_{2}$ and $v_{3}$
are not obstruction to the construction of edge-localized states on the
pendant edges.

  * •

Let us take two loops on the integral curve of Fig. 7 with the boundary
conditions

$u(-\epsilon(\ell_{1}+\ell_{2}))=u(\epsilon(\ell_{2}+\ell_{3}))=p_{-},$

as shown on Fig. 10. Although $p_{-}\to 0$ as $\epsilon\to 0$, this $2$-pulse
state does not correspond to the choice $\mathcal{E}_{2}:=\\{e_{1},e_{3}\\}$
in Theorem 1 because the edge-localized states at the pendants must be
centered at the ends under the Neumann conditions. Consequently, the
derivative $u^{\prime}(z)$ also shown on Fig. 10 has four nodal domains
suggesting by the Courant’s nodal theorem that $n(\mathcal{L})=4$ which would
contradict Theorem 2 with $N=2$. However, as we explained, the main results
are not applicable to the $2$-soliton state which is not localized at the ends
of the pendants.

Figure 9. Left: the $2$-pulse state on the two pendants $e_{1}$ and $e_{3}$.
Right: its derivative satisfying Dirichlet conditions on the interval.

Figure 10. Left: the $2$-pulse state centered in the middle of the two
pendants $e_{1}$ and $e_{2}$. Right: its derivative satisfying Dirichlet
conditions on the interval.

Acknowledgements: A. Kairzhan thanks the Fields Institute for Research in
Mathematical Sciences for its support and hospitality during the thematic
program on Mathematical Hydrodynamics in July–December, 2020. D.E. Pelinovsky
acknowledges the support from the NSERC Discovery grant.

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