# Instability theory of kink and anti-kink profiles for the sine-Gordon
equation on Josephson tricrystal boundaries

Jaime Angulo Pava 1 and Ramón G. Plaza 2

###### Abstract

The aim of this work is to establish a instability study for stationary kink
and antikink/kink profiles solutions for the sine-Gordon equation on a metric
graph with a structure represented by a $\mathcal{Y}$-junction so-called a
Josephson tricrystal junction. By considering boundary conditions at the
graph-vertex of $\delta^{\prime}$-interaction type, it is shown that these
kink-soliton type stationary profiles are linearly (and nonlinearly) unstable.
The extension theory of symmetric operators, Sturm-Liouville oscillation
results and analytic perturbation theory of operators are fundamental
ingredients in the stability analysis. The local well-posedness of the sine-
Gordon model in $H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ is also
established. The theory developed in this investigation has prospects for the
study of the (in)-stability of stationary wave solutions of other
configurations for kink-solitons profiles.

1 Department of Mathematics, IME-USP

Rua do Matão 1010, Cidade Universitária, CEP 05508-090, São Paulo, SP (Brazil)

<EMAIL_ADDRESS>
2 Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas,

Universidad Nacional Autónoma de México, Circuito Escolar s/n,

Ciudad Universitaria, C.P. 04510 Cd. de México (Mexico)

<EMAIL_ADDRESS>
Mathematics Subject Classification (2010). Primary 35Q51, 35Q53, 35J61;
Secondary 47E05.

Key words. sine-Gordon model, Josephson tricrystal junction,
$\delta^{\prime}$-type interaction, kink, anti-kink solitons, perturbation
theory, extension theory, instability.

##  1 Introduction

Nonlinear dispersive models on quantum star-shaped graphs (quantum graphs,
henceforth) arise as simplifications for wave propagation, for instance, in
quasi one-dimensional (e.g. meso- or nano-scaled) systems that look like a
thin neighborhood of a graph. We recall that a star-shaped metric graph,
$\mathcal{G}$, is a structure represented by a finite or countable edges
attached to a common vertex, $\nu=0$, having each edge identified with a copy
of the half-line, $(-\infty,0)$ or $(0,+\infty)$ (see Figure 1 below). Hence,
a quantum star-shaped metric graph, $\mathcal{G}$, is a star-shaped metric
graph with a linear Hamiltonian operator (for example, a Schrödinger-like
operator) suitably defined on functions which are supported on the edges.

Quantum graphs have been used to describe a variety of physical problems and
applications, for instance, condensed matter, $\mathcal{Y}$-Josephson junction
networks, polymers, optics, neuroscience, DNA chains, blood pressure waves in
large arteries, or in shallow water models describing a fluid networks (see
[15, 17, 19, 24, 36] and the many references therein). Recently, they have
attracted much attention in the context of soliton transport in networks and
branched structures since wave dynamics in networks can be modeled by
nonlinear evolution equations (see, e.g., [3, 5, 6, 7, 10, 11, 37, 52, 51]).

The present study focuses on the dynamics of the one-dimensional sine-Gordon
equation,

$u_{tt}-c^{2}u_{xx}+\sin u=0,$ (1.1)

posed on a metric graph. The sine-Gordon model appears in a great variety of
physical and biological models. For example, it has been used to describe the
magnetic flux in a long Josephson line in superconductor theory [12, 13, 49],
mechanical oscillations of a nonlinear pendulum [21, 33] and the dynamics of a
crystal lattice near a dislocation [26]. Recently, soliton solutions to
equation (1.1) have been used as simplified models of scalar gravitational
fields in general relativity theory [18, 25] and of oscillations describing
the dynamics of DNA chains [20, 30] in the context of the _solitons in DNA
hypothesis_ [23]. In addition, the sine-Gordon equation (1.1) underlies many
remarkable mathematical features such as a Hamiltonian structure [53],
complete integrability [1, 2] and the existence of localized solutions
(solitons) [48, 47].

In recent contributions (cf. [10, 11]), we performed the first rigorous
analytical studies of the stability properties of stationary soliton solutions
of kink and/or anti-kink profiles to the sine-Gordon equation _posed on a
$\mathcal{Y}$-junction graph_. There exist two main types of
$\mathcal{Y}$-junctions. A $\mathcal{Y}$-junction of the first type (or type
I) consists of one incoming (or parent) edge, $E_{1}=(-\infty,0)$, meeting at
one single vertex at the origin, $\nu=0$, with other two outgoing (children)
edges, $E_{j}=(0,\infty)$, $j=2,3$. The second type (or $\mathcal{Y}$-junction
of type II) resembles more a starred structure and consists of three identical
edges of the form $E_{j}=(0,\infty)$, $1\leqq j\leqq 3$. See Figure 1 for an
illustration. Junctions of type I are more common in unidirectional fluid flow
models (see, for example, [16]). The junctions of the second type (which
belong to the star graph class) are often referred as _Josephson tricrystal
junctions_ and are common in network of transmission lines (see, for instance,
[52, 34, 46]).

(a) $\mathcal{Y}=(-\infty,0)\cup(0,\infty)\cup(0,\infty)$

(b) $\mathcal{Y}=(0,\infty)\cup(0,\infty)\cup(0,\infty)$

Figure 1: Panel (a) shows a $\mathcal{Y}$-junction of the first type with
$E_{1}=(-\infty,0)$ and $E_{j}=(0,\infty)$, $j=2,3$, whereas panel (b) shows a
$\mathcal{Y}$-junction of the second type (star graph) with
$E_{j}=(0,\infty)$, $1\leqq j\leqq 3$.

Our work focuses on the sine-Gordon model posed on a $\mathcal{Y}$-tricrystal
junction, more preciely, on the equations

$\partial_{t}^{2}u_{j}-c_{j}^{2}\partial_{x}^{2}u_{j}+\sin u_{j}=0,\qquad x\in
E_{j}=(0,\infty),\;\,t>0,\;\,1\leqq j\leqq 3,$ (1.2)

where $u_{j}=u_{j}(x,t)$. It is assumed that the characteristic speed on each
edge $E_{j}$ is constant and positive, $c_{j}>0$, without loss of generality.

Posing the sine-Gordon equation on a metric graph comes out naturally from
practical applications. Indeed, in the context of superconductor theory, the
sine-Gordon equation on a metric graph arises as a model for coupling of two
or more Josephson junctions in a network. A Josephson junction is a quantum
mechanical structure that is made by two superconducting electrodes separated
by a barrier (the junction), thin enough to allow coupling of the wave
functions of electrons for the two superconductors [31]. After appropriate
normalizations, it can be shown that the phase difference $u$ (also known as
order parameter) of the two wave functions satisfies the sine-Gordon equation
(1.1) [31, 12]. Coupling three junctions at one common vertex, the so called
tricrystal junction, can be regarded (and fabricated) as a probe of the order
parameter symmetry of high temperature superconductors (cf. [55, 54]).
Physically coupling three otherwise independent long Josephson junctions,
$\mathcal{Y}=\cup_{j=1}^{3}E_{j}$, together at one common vertex, was first
proposed by Nakajima _et al._ [41, 40] as a prototype for logic circuits.

What is more crucial in the analysis on quantum graphs is the choice of
boundary conditions, mainly because the transition rules at the vertex
completely determine the dynamics of the PDE model on the graph. For the sine-
Gordon equation in $\mathcal{Y}$-junctions, previous studies have basically
(and almost exclusively) considered two types of boundary conditions:
interactions of $\delta$ (continuity of the wave functions plus a law of
Kirchhoff-type for the fluxes at the vertex, see [10, 19, 22] ) and of
$\delta^{\prime}$-type (continuity of the derivatives plus a Kirchhoff law for
the self-induced magnetic flux). Since Josephson models arise in the
description of electromagnetic flux, interactions of $\delta^{\prime}$-type
have received more attention (see, for example, [11, 28, 34, 40, 41, 52]).
Thus, since the surface current density should be the same in all three films
at the vertex, Nakajima _et al._ [41, 40] (see also [28, 34]) impose the
condition

$c_{1}\partial_{x}{u_{1}}_{|x=0}=c_{2}\partial_{x}{u_{2}}_{|x=0}=c_{3}\partial_{x}{u_{3}}_{|x=0},$
(1.3)

expressing that the magnetic field, which is proportional to the derivative of
phase difference, should be continuous at the intersection. Moreover, the
magnetic flux computed along an infinitesimal small contour encircling the
origin (vertex) must vanish, that is, the total change of the gauge invariant
phase difference must be zero [34, 51]. This leads to the Kirchhoff-type of
boundary condition

$\sum_{j=1}^{3}c_{j}u_{j}(0+)=0.$ (1.4)

The interaction conditions (1.3)-(1.4) are known as boundary conditions of
$\delta^{\prime}$-type: they express continuity of the fluxes (derivatives)
plus a Kirchhoff-type rule for the self-induced magnetic flux.

Motivated by physical applications, the purpose of the present paper is to
study the stability of particular stationary solutions to the sine-Gordon
equation seen as a first order system posed on a $\mathcal{Y}$-tricrystal
junction, namely, the system

$\begin{cases}\partial_{t}u_{j}=v_{j}\\\
\partial_{t}v_{j}=c_{j}^{2}\partial_{x}^{2}u_{j}-\sin u_{j},\end{cases}\qquad
x\in E_{j}=(0,\infty),\;\,t>0,\;\,1\leqq j\leqq 3.$ (1.5)

As far as we know, there is no rigorous analytical study of the stability of
stationary solutions of type kink and/or anti-kink to the vectorial sine-
Gordon model (1.5) on a tricrystal junction with boundary conditions of
$\delta^{\prime}$-interaction type available in the literature (see in Angulo
and Plaza [11] the case of $\delta^{\prime}$-interactions on
$\mathcal{Y}$-junction graph of type I). The stability of these static
configurations is an important property from both the mathematical and the
physical points of view. Stability can predict whether a particular state can
be observed in experiments or not. Unstable configurations are rapidly
dominated by dispersion, drift, or by other interactions depending on the
dynamics, and they are practically undetectable in applications.

For tricrystal junctions we will study stationary solutions with a kink type
structure. The latter have the form $u_{j}(x,t)=\varphi_{j}(x)$,
$v_{j}(x,t)=0$, for all $j=1,2,3$, and $x\in E_{j}$, $t>0$, in (1.5), where
each of the profile functions $\varphi_{j}$ satisfies the equation

$-c_{j}^{2}\varphi^{\prime\prime}_{j}+\sin\varphi_{j}=0,$ (1.6)

on each edge $E_{j}=(0,\infty)$ and for all $j$, as well as the boundary
conditions of $\delta^{\prime}$-type, at the vertex $\nu=0$:

$\displaystyle c_{1}\varphi^{\prime}_{1}(0+)=c_{2}\varphi^{\prime}_{2}(0+)$
$\displaystyle=c_{3}\varphi^{\prime}_{3}(0+),$ (1.7)
$\displaystyle\sum_{j=1}^{3}c_{j}\varphi_{j}(0+)$ $\displaystyle=\lambda
c_{1}\varphi^{\prime}_{1}(0+).$

These conditions depend upon the parameter $\lambda$, which ranges along the
whole real line and determines the dynamics of the model. Therefore, the value
$\lambda\in\mathbb{R}$ is part of the physical parameters that determine the
model (such as the speeds $c_{j}$, for instance). Instead of adopting _ad hoc_
boundary conditions, we consider a parametrized family of transition rules
covering a wide range of applications and which, for the particular value
$\lambda=0$, include the Kirchhoff condition (1.4) previously studied in the
literature. Our analysis focuses on two particular class of solutions of the
sine-Gordon equation known as _kink_ and _anti-kink_ (also referred to as
topological solitons) [21, 47, 48].

Initially, we look at the particular family of solutions with the same kink-
type structure with $c_{j}>0$,

$\varphi_{j}(x)=4\arctan\big{(}e^{-(x-b_{j})/c_{j}}\big{)},\qquad
x\in(0,\infty),\,\;j=1,2,3,\\\ $ (1.8)

where the constants $b_{j}$ are determined by the boundary conditions (1.7).
The family satisfies as well

$\varphi_{j}(+\infty)=0,\qquad j=1,2,3,$ (1.9)

ensuring that $\varPhi=(\varphi_{j})_{j=1}^{3}\in H^{2}(\mathcal{Y})$. Our
second class of solutions are the anti-kink/kink type soliton, namely,
profiles having the form

$\begin{cases}\varphi_{1}(x)=4\arctan\big{(}e^{(x-a_{1})/c_{1}}\big{)},&x\in(0,\infty),\;\;\lim_{x\to+\infty}\varphi_{1}(x)=2\pi,\\\
\varphi_{j}(x)=4\arctan\big{(}e^{(x-a_{j})/c_{j}}\big{)}-2\pi,&x\in(0,\infty),\;\;\lim_{x\to+\infty}\varphi_{j}(x)=0\;\;j=2,3,\\\
\end{cases}$ (1.10)

where each $a_{j}$ is a constant determined by the boundary conditions (1.7).
Expression in (1.10) is so-called $2\pi$-kink because the total Josephson
phase is $2\pi$ when one circles the branch point at large distances, i.e.,
$\sum_{j=1}^{3}\varphi_{j}(+\infty)=2\pi$. We have left open the stability
study of other Josephson configurations systems, such as a tricrystal junction
with a $\pi$-kink,
$\phi_{1}(x)=4\arctan\big{(}e^{(x-a_{1})/c_{1}}\big{)}-\pi$,
$\phi_{j}(x)=4\arctan\big{(}e^{(x-a_{j})/c_{j}}\big{)}-2\pi$, $j=2,3$. This
stability analysis is of important also for experimentalists since these
network systems open a large opportunity for applications in high-performance
computers (see [52] and references therein).

In the forthcoming analysis we establish the existence of two smooth mapping
of stationary profiles, the first one
$\lambda\in(-\infty,-\sum_{j=1}^{3}c_{j})\to\Pi_{\lambda,\delta^{\prime}}=(\varphi_{1},\varphi_{2},\varphi_{3},0,0,0)$,
with $\varphi_{j}=\varphi_{j,b_{j}(\lambda)}$ defined in (1.8), and the second
one
$\lambda\in(-\infty,+\infty)\to\Phi_{\lambda,\delta^{\prime}}=(\varphi_{1},\varphi_{2},\varphi_{3},0,0,0)$,
with $\varphi_{j}=\varphi_{j,a_{j}(\lambda)}$ defined in (1.10).

The main linear instability results of this manuscript are the following,

######  Theorem 1.1.

Let $\lambda\in(-\infty,-\sum_{j=1}^{3}c_{j})$ and consider the smooth family
of stationary profiles $\lambda\mapsto\Pi_{\lambda,\delta^{\prime}}$. Then
$\Pi_{\lambda,\delta^{\prime}}$ is linear and nonlinearly unstable for the
sine-Gordon model (2.1) on a tricrystal junction in the following cases:

  1. 1)

for $\lambda\in(-\frac{\pi}{2}\sum_{j=1}^{3}c_{j},-\sum_{j=1}^{3}c_{j})$ and
$c_{i}>0$,

  2. 2)

for $\lambda\in(-\infty,-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}]$ and
$c_{1}=c_{2}=c_{3}$.

######  Theorem 1.2.

Let $c_{1}=c_{2}=c_{3}$, $\lambda\in(-\infty,+\infty)$, and the smooth family
of stationary anti-kink/kink profiles
$\lambda\to\Phi_{\lambda,\delta^{\prime}}$ determined above. Then
$\Phi_{\lambda,\delta^{\prime}}$ is spectrally unstable for the sine-Gordon
model (1.5) on a tricrystal junction.

In our stability analysis below, the family of linearized operators around the
stationary profiles plays a fundamental role. These operators are
characterized by the following formal self-adjoint diagonal matrix operators,

$\mathcal{W}\mathbb{v}=\Big{(}\Big{(}-c_{j}^{2}\frac{d^{2}}{dx^{2}}v_{j}+\cos(\varphi_{j})v_{j}\Big{)}\delta_{j,k}\Big{)},\quad\l
1\leqq j,k\leqq 3,\;\;\mathbb{v}=(v_{j})_{j=1}^{3},$ (1.11)

where $\delta_{j,k}$ denotes the Kronecker symbol, and defined on domains with
$\delta^{\prime}$-type interaction at the vertex $\nu=0$,

$D(\mathcal{W})=\Big{\\{}\mathbb{v}=(v_{j})_{j=1}^{3}\in
H^{2}(\mathcal{Y}):c_{1}v^{\prime}_{1}(0+)=c_{2}v^{\prime}_{2}(0+)=c_{3}v_{3}(0+),\;\;\sum\limits_{j=1}^{3}c_{j}v_{j}(0+)=\lambda
c_{1}v^{\prime}_{1}(0+)\Big{\\}},$ (1.12)

with $\lambda\in\mathbb{R}$. It is to be observed that the particular family
(1.8) of kink-profile stationary solutions under consideration is such that
$(\varphi_{j})_{j=1}^{3}\in D(\mathcal{W})$ in view that they satisfy the
boundary conditions (1.7).

In section §2 we establish a general instability criterion for static
solutions for the sine-Gordon model (1.5) on a $\mathcal{Y}$-junction. The
reader can find this result in Theorem 2.4 below. It essentially provides
sufficient conditions on the flow of the semigroup generated by the
linearization around the stationary solutions, for the existence of a pair of
positive/negative real eigenvalues of this linearization, which determine the
Morse index for the associated self-adjoint operator
$(\mathcal{W},D(\mathcal{W})$. The proof of Theorems 1.1 and 1.2 follow as an
application of Theorem 2.4.

The structure of the paper is the following: in section §2, we review the
general instability criterion for stationary solutions for the sine-Gordon
model (1.5) on a $\mathcal{Y}$-junction developed in [10] (see Theorem 2.4
below; see also [4]). It is to be observed that this instability criterion is
very versatile, as it applies to any type of stationary solutions (such as
anti-kinks or breathers, for example) and for different interactions at the
vertex, such as both the $\delta$\- and $\delta^{\prime}$-types. The
subsection 3.1 is devoted to develop the instability theory of kink-profiles
and we show Theorem 1.1. A special space is defined in order to analyze the
Cauchy problem in $H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$. Subsection
3.2 is devoted to the proof of Theorem 1.2. By convenience of the reader and
by the sake of completeness we establish in the Appendix some results of the
extension theory of symmetric operators used in the body of the manuscript.

### On notation

For any $-\infty\leq a<b\leq\infty$, we denote by $L^{2}(a,b)$ the Hilbert
space equipped with the inner product
$(u,v)=\int_{a}^{b}u(x)\overline{v(x)}dx$. By $H^{n}(a,b)$ we denote the
classical Sobolev spaces on $(a,b)\subseteq\mathbb{R}$ with the usual norm. We
denote by $\mathcal{Y}$ the junction of type II parametrized by the edges
$E_{j}=(0,\infty)$, $j=1,2,3$, attached to a common vertex $\nu=0$. On the
graph $\mathcal{Y}$ we define the classical $L^{p}$-spaces

$L^{p}(\mathcal{Y})=L^{p}(0,+\infty)\oplus L^{p}(0,+\infty)\oplus
L^{p}(0,+\infty),\quad\,p>1,$

and Sobolev-spaces $H^{m}(\mathcal{Y})=H^{m}(0,+\infty)\oplus
H^{m}(0,+\infty)\oplus H^{m}(0,+\infty)$ with the natural norms. Also, for
$\mathbf{u}=(u_{j})_{j=1}^{3}$, $\mathbf{v}=(v_{j})_{j=1}^{3}\in
L^{2}(\mathcal{Y})$, the inner product is defined by

$\langle\mathbf{u},\mathbf{v}\rangle=\sum_{j=1}^{3}\int_{0}^{\infty}u_{j}(x)\overline{v_{j}(x)}\,dx$

Depending on the context we will use the following notations for different
objects. By $\|\cdot\|$ we denote the norm in $L^{2}(\mathbb{R})$ or in
$L^{2}(\mathcal{Y})$. By $\|\cdot\|_{p}$ we denote the norm in
$L^{p}(\mathbb{R})$ or in $L^{p}(\mathcal{Y})$. Finally, if $A$ is a closed,
densely defined symmetric operator in a Hilbert space $H$ then its domain is
denoted by $D(A)$, the deficiency indices of $A$ are denoted by
$n_{\pm}(A):=\dim\ker(A^{*}\mp iI)$, where $A^{*}$ is the adjoint operator of
$A$, and the number of negative eigenvalues counting multiplicities (or Morse
index) of $A$ is denoted by $n(A)$.

##  2 Preliminaries: Linear instability criterion for sine-Gordon model on a
tricrystal junction

We start our study by recasting the equations in (1.5) in the vectorial form

$\mathbb{w}_{t}=JE\mathbb{w}+F(\mathbb{w})$ (2.1)

where $\mathbb{w}=(u,v)^{\top}$, with $u=(u_{1},u_{2},u_{3})^{\top}$,
$v=(v_{1},v_{2},v_{3})^{\top}$, $u_{j},v_{j}:E_{j}\to\mathbb{R}$, $1\leqq
j\leqq 3$,

$J=\left(\begin{array}[]{cc}0&I_{3}\\\ -I_{3}&0\end{array}\right),\quad
E=\left(\begin{array}[]{cc}\mathcal{T}&0\\\ 0&I_{3}\end{array}\right),\quad
F(\mathbb{w})=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\ -\sin(u_{1})\\\
-\sin(u_{2})\\\ -\sin(u_{3})\end{array}\right),$ (2.2)

and where $I_{3}$ denotes the identity matrix of order $3$ and $\mathcal{T}$
is the diagonal-matrix linear operator

$\mathcal{T}=\Big{(}\Big{(}-c_{j}^{2}\frac{d^{2}}{dx^{2}}\Big{)}\delta_{j,k}\Big{)},\quad\l
1\leqq j,k\leqq 3.$ (2.3)

For the $\mathcal{Y}$-junction being a tricrystal junction, we will use the
$\delta^{\prime}$-interaction domain for
$\mathcal{T}_{\lambda}\equiv\mathcal{T}$ given by (1.12), namely,
$D(\mathcal{T}_{\lambda})=D(\mathcal{W})$:

$D(\mathcal{T}_{\lambda}):=\\{(v_{j})_{j=1}^{3}\in
H^{2}(\mathcal{Y}):c_{1}v^{\prime}_{1}(0+)=c_{2}v^{\prime}_{2}(0+)=c_{3}v^{\prime}_{3}(0+),\;\;\sum\limits_{j=1}^{3}c_{j}v_{j}(0+)=\lambda
c_{1}v^{\prime}_{1}(0+)\\},$ (2.4)

with $\lambda\in\mathbb{R}$ (see Proposition 3.1).

In the sequel, we review the linear instability criterion of stationary
solutions for the sine-Gordon model (1.5) on a $\mathcal{Y}$-junction
developed in [10]. Although the stability analysis in [10] pertains to
interactions of $\delta$-type at the vertex, it is important to note that the
criterion proved in that reference also applies to any type of stationary
solutions independently of the boundary conditions under consideration and,
therefore, it can be used to study the present configurations with boundary
rules at the vertex of $\delta^{\prime}$-interaction type, or even to other
types of stationary solutions to the sine-Gordon equation such as anti-kinks
and breathers, for instance. In addition, the criterion applies to both the
$\mathcal{Y}$-junction of type I (see Figure 1(a)) and of type II (see Figure
1(b)).

Let $\mathcal{Y}$ be a tricrystal junction. Let us suppose that $JE$ on a
domain $D(JE)\subset H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ is the
infinitesimal generator of a $C_{0}$-semigroup on $H^{1}(\mathcal{Y})\times
L^{2}(\mathcal{Y})$ and that there exists an stationary solution
$\Upsilon=(\zeta_{1},\zeta_{2},\zeta_{3},0,0,0)\in D(JE)$. Thus, every
component $\zeta_{j}$ satisfies the equation

$-c^{2}_{j}\zeta^{\prime\prime}_{j}+\sin(\zeta_{j})=0,\quad j=1,2,3.$ (2.5)

Now, we suppose that $\mathbb{w}$ satisfies formally equality in (2.1) and we
define

$\mathbb{v}\equiv\mathbb{w}-\Upsilon,$ (2.6)

then, from (2.5) we obtain the following linearized system for (2.1) around
$\Phi$,

$\mathbb{v}_{t}=J\mathcal{E}\mathbb{v},$ (2.7)

with $\mathcal{E}$ being the $6\times 6$ diagonal-matrix
$\mathcal{E}=\left(\begin{array}[]{cc}\mathcal{L}&0\\\
0&I_{3}\end{array}\right)$, and

$\mathcal{L}=\Big{(}\Big{(}-c_{j}^{2}\frac{d^{2}}{dx^{2}}+\cos(\zeta_{j})\Big{)}\delta_{j,k}\Big{)},\qquad
1\leqq j,k\leqq 3.$ (2.8)

We point out the equality $J\mathcal{E}=JE+\mathcal{Z}$, with

$\mathcal{Z}=\left(\begin{array}[]{cc}0&0\\\
\big{(}-\cos(\zeta_{j})\,\delta_{j,k}\big{)}&0\end{array}\right)$ (2.9)

being a _bounded_ operator on $H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$.
This implies that $J\mathcal{E}$ also generates a $C_{0}$-semigroup on
$H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ (see Pazy [42]).

The linear instability criterion provides sufficient conditions for the
trivial solution $\mathbb{v}\equiv 0$ to be unstable by the linear flow of
(2.7). More precisely, it underlies the existence of a growing mode solution
to (2.7) of the form $\mathbb{v}=e^{\lambda t}\Psi$ and
$\mathrm{Re}\,\lambda>0$. To find it, one needs to solve the formal system

$J\mathcal{E}\Psi=\lambda\Psi,$ (2.10)

with $\Psi\in D(J\mathcal{E})$. If we denote by
$\sigma(J\mathcal{E})=\sigma_{\mathrm{pt}}(J\mathcal{E})\cup\sigma_{\mathrm{ess}}(J\mathcal{E})$
the spectrum of $J\mathcal{E}$ (namely,
$\lambda\in\sigma_{\mathrm{pt}}(J\mathcal{E})$ if $\lambda$ is isolated and
with finite multiplicity) then we have the following

######  Definition 2.1.

The stationary vector solution $\Upsilon\in D(\mathcal{E})$ is said to be
spectrally stable for model sine-Gordon (2.1) if the spectrum of
$J\mathcal{E}$, $\sigma(J\mathcal{\mathcal{missing}}E)$, satisfies
$\sigma(J\mathcal{E})\subset i\mathbb{R}.$ Otherwise, the stationary solution
$\Upsilon\in D(\mathcal{E})$ is said to be spectrally unstable.

######  Remark 2.2.

It is well-known that $\sigma_{\mathrm{pt}}(J\mathcal{E})$ is symmetric with
respect to both the real and imaginary axes and
$\sigma_{\mathrm{ess}}(J\mathcal{E})\subset i\mathbb{R}$ under the assumption
that $J$ is skew-symmetric and that $\mathcal{E}$ is self-adjoint (by
supposing, for instance, Assumption $(S_{3})$ below for $\mathcal{L}$; see
[27, Lemma 5.6 and Theorem 5.8]). These cases on $J$ and $\mathcal{E}$ are
considered in the theory. Hence, it is equivalent to say that $\Upsilon\in
D(J\mathcal{E})$ is spectrally stable if
$\sigma_{\mathrm{pt}}(J\mathcal{E})\subset i\mathbb{R}$, and it is spectrally
unstable if $\sigma_{\mathrm{pt}}(J\mathcal{E})$ contains point $\lambda$ with
$\mathrm{Re}\,\lambda>0.$

It is widely known that the spectral instability of a specific traveling wave
solution of an evolution type model is a key prerequisite to show their
nonlinear instability property (see [27, 35, 50] and the references therein).
Thus we have the following definition.

######  Definition 2.3.

The stationary vector solution $\Upsilon\in D(\mathcal{E})$ is said to be
nonlinearly unstable in $X\equiv H^{1}(\mathcal{Y})\times
L^{2}(\mathcal{Y})$-norm for model sine-Gordon (2.1) if there is $\epsilon>0$
such that for every $\delta>0$ there exist an initial data $\mathbb{w}_{0}$
with $\|\Upsilon-\mathbb{w}_{0}\|_{X}<\delta$ and an instant
$t_{0}=t_{0}(\mathbb{w}_{0})$, such that
$\|\mathbb{w}(t_{0})-\Upsilon\|_{X}>\epsilon$, where
$\mathbb{w}=\mathbb{w}(t)$ is the solution of the sine-Gordon model with
initial data $\mathbb{w}(0)=\mathbb{w}_{0}$.

From (2.10), the eigenvalue problem to solve is now reduced to

$J\mathcal{E}\Psi=\lambda\Psi,\quad\mathrm{Re}\,\lambda>0,\;\;\Psi\in
D(\mathcal{E}).$ (2.11)

Next, we establish our theoretical framework and assumptions for obtaining a
nontrivial solution to problem in (2.11):

  1. ($S_{1}$)

$J\mathcal{E}$ is the generator of a $C_{0}$-semigroup $\\{S(t)\\}_{t\geqq
0}$.

  2. ($S_{2}$)

Let $\mathcal{L}$ be the matrix-operator in (2.8) defined on a domain
$D(\mathcal{L})\subset L^{2}(\mathcal{Y})$ on which $\mathcal{L}$ is self-
adjoint.

  3. ($S_{3}$)

Suppose $\mathcal{L}:D(\mathcal{L})\to L^{2}(\mathcal{Y})$ is invertible with
Morse index $n(\mathcal{L})=1$ and such that
$\sigma(\mathcal{L})=\\{\lambda_{0}\\}\cup J_{0}$ with
$J_{0}\subset[r_{0},+\infty)$, for $r_{0}>0$, and $\lambda_{0}<0$,

The criterion for linear instability reads precisely as follows (cf. [10]).

######  Theorem 2.4 (linear instability criterion).

Suppose the assumptions $(S_{1})$ \- $(S_{3})$ hold. Then the operator
$J\mathcal{E}$ has a real positive and a real negative eigenvalue.

###### Proof.

See [4, 10]. ∎

##  3 Instability of stationary solutions for the sine-Gordon equation with
$\delta^{\prime}$-interaction on a tricrystal junction

In this section we study the stability of stationary solutions determined by a
$\delta^{\prime}$-interaction type at the vertex $\nu=0$ of a tricrystal
junction. First we study the kink-profile type in (1.8)-(1.9), thus the local
well-posedness problem associated to (1.5) with a
$\delta^{\prime}$-interaction. Next, we apply the linear instability criterion
(Theorem 2.4) to prove that the family of stationary solutions (1.8) are
linearly (and nonlinearly) unstable (Theorem 1.1 above). Our second focus goes
to the study of the anti-kink-profile type in (1.10) and similarly as in the
former profile case we establish the necessary ingredients for obtaining
Theorem 1.2 above.

###  3.1 Kink-profile’s instability on a tricrystal junction

We start our stability study for the kink-profile type in (1.8)-(1.9) and so
our first focus is dedicated to the Cauchy problem associated to the sine-
Gordon model in (2.1). As this study is not completely standard in the case of
metric graphs we provide the new ingredients that arise.

####  3.1.1 Cauchy Problem in $H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$

In this subsection we establish the local well-posedness in
$H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ of the sine-Gordon equation on a
tricrystal junction (section §2 in [10]). We start with the following result
from the extension theory. The proof follows the same strategy as in
Proposition A.6 and Theorem 3.1 in Angulo and Plaza [11] (see Proposition A.4
in the Appendix) and we omit it.

######  Proposition 3.1.

Consider the closed symmetric operator $(\mathcal{T},D(\mathcal{T}))$ densely
defined on $L^{2}(\mathcal{Y})$, with $\mathcal{Y}$ being a tricrystal
junction, by

$\begin{split}&\mathcal{T}=\Big{(}\Big{(}-c_{j}^{2}\frac{d^{2}}{dx^{2}}\Big{)}\delta_{j,k}\Big{)},\;\;1\leqq
j,k\leqq 3,\\\ &D(\mathcal{T})=\Big{\\{}(v_{j})_{j=1}^{3}\in
H^{2}(\mathcal{Y}):c_{1}v_{1}^{\prime}(0+)=c_{2}v^{\prime}_{2}(0+)=c_{3}v_{3}^{\prime}(0+)=0,\;\;\sum_{j=1}^{3}c_{j}v_{j}(0+)=0\Big{\\}}.\end{split}$
(3.1)

Here $c_{j}\neq 0$, $1\leqq j\leqq 3$, and $\delta_{j,k}$ is the Kronecker
symbol. Then, the deficiency indices are $n_{\pm}(\mathcal{T})=1$. Therefore,
we have that all the self-adjoint extensions of $(\mathcal{T},D(\mathcal{T}))$
are given by the one-parameter family
$(\mathcal{T}_{\lambda},D(\mathcal{T}_{\lambda}))$, $\lambda\in\mathbb{R}$,
with $\mathcal{T}_{\lambda}\equiv\mathcal{T}$ and $D(\mathcal{T}_{\lambda})$
defined by

$D(\mathcal{T}_{\lambda})=\\{(u_{j})_{j=1}^{3}\in
H^{2}(\mathcal{Y}):c_{1}u^{\prime}_{1}(0+)=c_{2}u^{\prime}_{2}(0+)=c_{3}u^{\prime}_{3}(0+),\;\;\sum_{j=1}^{3}c_{j}u_{j}(0+)=\lambda
c_{1}u^{\prime}_{1}(0+)\\}.$ (3.2)

Moreover, the spectrum of the family of self-adjoint operators
$(\mathcal{T}_{\lambda},D(\mathcal{T}_{\lambda}))$ satisfies
$\sigma_{\mathrm{ess}}(\mathcal{T}_{\lambda})=[0,+\infty)$ for every
$\lambda\neq 0$. For $\lambda<0$, $\mathcal{T}_{\lambda}$ has precisely one
negative simple eigenvalue. If $\lambda>0$ then $\mathcal{T}_{\lambda}$ has no
eigenvalues (see [6]).

######  Theorem 3.2.

Let $\mathcal{Y}=(0,+\infty)\cup(0,+\infty)\cup(0,+\infty)$. For any $\Psi\in
H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ there exists $T>0$ such that the
sine-Gordon equation (2.1) has a unique solution $\mathbf{w}\in
C([0,T];H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y}))$ satisfying
$\mathbf{w}(0)=\Psi$. For each $T_{0}\in(0,T)$ the mapping data-solution

$\Psi\in H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})\to\mathbf{w}\in
C([0,T_{0}];H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})),$ (3.3)

is at least of class $C^{2}$.

###### Proof.

By applying the same strategy as in [11] (Theorems 3.2 and 3.3), we have the
following observations:

  1. 1)

Consider the linear operators $J$ and $E$ defined in (2.2) with
$(\mathcal{T}_{\lambda},D(\mathcal{T}_{\lambda}))$ defined in Proposition 3.1.
Then, $\mathcal{A}\equiv JE$ with
$D(\mathcal{A})=D(\mathcal{T}_{\lambda})\times H^{1}(\mathcal{Y})$ is the
infinitesimal generator of a $C_{0}$-semigroup on $H^{1}(\mathcal{Y})\times
L^{2}(\mathcal{Y})$.

  2. 2)

By using the contraction mapping principle, we obtain the local well-posedness
result for the sine-Gordon equation (2.1) on $H^{1}(\mathcal{Y})\times
L^{2}(\mathcal{Y})$. The $C^{2}$-property of the mapping data-solution follows
from the implicit function theorem.

∎

We are ready to draw conclusions about the linearized operator around the
family of stationary solutions of the form (1.8) on a tricrystal junction
required by the assumption in section §2.

######  Proposition 3.3.

Consider the operator $(\mathcal{W}_{\lambda},D(\mathcal{W}_{\lambda}))$
determined by $\mathcal{W}_{\lambda}\equiv\mathcal{W}$ defined in (1.11), on
the domain $D(\mathcal{W}_{\lambda})=D(\mathcal{T}_{\lambda})$ defined in
(3.2). Let $\mathcal{E}$ be the following diagonal-matrix

$\mathcal{E}=\left(\begin{array}[]{cc}\mathcal{W}_{\lambda}&0\\\
0&I_{3}\end{array}\right).$

Then $J\mathcal{E}$ is the generator of a $C_{0}$-semigroup on
$H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ with
$D(J\mathcal{E})=D(\mathcal{W}_{\lambda})\times H^{1}(\mathcal{Y})$. This
implies, in turn, that assumption $(S_{1})$ (see section §2) is satisfied.

###### Proof.

From the relation $J\mathcal{E}=JE+\mathcal{Z}$ (see (2.9) for
$\zeta_{j}=\varphi_{j}$), standard semigroup theory and item 1) in the proof
of Theorem 3.2 imply the result. ∎

####  3.1.2 Kink-profile for the sine-Gordon equation on a tricrystal-junction

We will consider stationary solutions $(\varphi_{j})_{j=1}^{3}$ for the sine-
Gordon equation on a tricrystal-junction of the form (1.8) satisfying the
$\delta^{\prime}$-interactions at the vertex given by (1.7), this means that
$(\varphi_{j})_{j=1}^{3}\in D(\mathcal{T}_{\lambda})$. Here we shall consider
the full continuity case at the vertex, under which

$\varphi_{1}(0+)=\varphi_{2}(0+)=\varphi_{3}(0+).$

Hence $b_{1}/c_{1}=b_{2}/c_{2}=b_{3}/c_{3}$ and, moreover, the conditions
$c_{1}\varphi^{\prime}_{1}(0+)=c_{2}\varphi^{\prime}_{2}(0+)=c_{3}\varphi^{\prime}_{3}(0+)$
hold. The Kirchhoff type condition in (1.7) implies, for $y=e^{b_{1}/c_{1}}$,
the relation

$\frac{1+y^{2}}{y}\arctan(y)\sum_{j=1}^{3}c_{j}=-\lambda.$ (3.4)

Thus from the strictly-increasing property of the positive mapping
$y\mapsto\frac{1+y^{2}}{y}\arctan(y)$, $y>0$, we obtain from (3.4) that
$\lambda\in(-\infty,-\sum_{j=1}^{3}c_{j})$ and the existence of a smooth
mapping $\lambda\mapsto b_{1}(\lambda)$ satisfying (3.4). Moreover,
$\lambda\in(-\infty,-\sum_{j=1}^{3}c_{j})\mapsto\Pi_{\lambda,\delta^{\prime}}=(\varphi_{1,b_{1}(\lambda)},\varphi_{2,b_{2}(\lambda)},\varphi_{3,b_{3}(\lambda)},0,0,0)$
represents a real-analytic family of static profiles for the sine-Gordon
equation on a tricrystal-junction. Thus, we have:

  1. 1)

for $\lambda\in(-\infty,-\frac{\pi}{2}\sum_{j=1}^{3}c_{j})$ we obtain that
$b_{i}>0$ and $\varphi_{i}^{\prime\prime}(b_{i})=0$, for every $i$. Moreover,
$\varphi_{i}(0+)\in(\eta,2\pi)$, $i=1,2,3$, with
$\eta=4\arctan\big{(}e^{b_{1}/c_{2}}\big{)}>\pi$. Thus, the profile
$(\varphi_{1},\varphi_{2},\varphi_{3})$ looks like that presented in Figure
2(a) below (bump-type profile);

  2. 2)

for $\lambda\in(-\frac{\pi}{2}\sum_{j=1}^{3}c_{j},-\sum_{j=1}^{3}c_{j})$ we
obtain $b_{i}<0$, $\varphi^{\prime\prime}_{i}>0$, $\varphi_{i}\in(0,\pi)$ for
every $i$. Thus, the profile $(\varphi_{1},\varphi_{2},\varphi_{3})$ is of
tail-type as that of Figure 2(b) below;

  3. 3)

for $\lambda=-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}$ we obtain $b_{i}=0$,
$\varphi_{i}(0)=\pi$ and $\varphi_{i}^{\prime\prime}(0)=0$, for every $i$.
Moreover, $\varphi_{i}^{\prime\prime}(x)>0$ for $x>0$. Thus, the profile
$(\varphi_{1},\varphi_{2},\varphi_{3})$ is similar to that of Figure 2(c)
below.

(a) $\lambda\in(-\infty,-\tfrac{3\pi}{2})$

(b) $\lambda\in(-\tfrac{3\pi}{2},-3)$

(c) $\lambda=-\tfrac{3\pi}{2}$

Figure 2: Plots of stationary solutions (1.8) in the case where $c_{j}=1$ for
all $j=1,2,3$, for different values of
$\lambda\in(\infty,-\sum_{j}c_{j})=(-\infty,-3)$. Panel (a) shows the
stationary profile solutions (“bump” configuration) for the case
$\lambda\in(-\infty,-\tfrac{3\pi}{2})$. Panel (b) shows the profiles of “tail”
type for the case $\lambda\in(-\tfrac{3\pi}{2},-3)$. Panel (c) shows the
“smooth” profile solutions when $\lambda=-\tfrac{3\pi}{2}$ (color online).

The stability result for the stationary profiles,
$\Pi_{\lambda,\delta^{\prime}}=(\varphi_{1},\varphi_{2},\varphi_{3},0,0,0)$,
with $\varphi_{j}=\varphi_{j,b_{j}(\lambda)}$ defined in (1.8) and (3.4) in
the continuous case is that established in Theorem 1.1. We leave for a
possible future study the stability analysis of other kink-type profiles, such
as those non-continuous at the vertex.

The proof of Theorem 1.1 is a consequence of Theorem 2.4. Thus we only need to
verify assumption $(S_{3})$ associated to the family of self-adjoint operators
$\mathcal{W}_{\lambda}$ in (1.11) (see Proposition 3.1), with the domain
$D(\mathcal{W}_{\lambda})=D(\mathcal{T}_{\lambda})$ in (3.2) and the kink-
profiles $\varphi_{j}$ defined in (1.8).

####  3.1.3 Spectral study for
$(\mathcal{W}_{\lambda},D(\mathcal{W}_{\lambda}))$ on a tricrystal-junction

In this subsection, the spectral behavior for $\mathcal{W}_{\lambda}$ on
$D(\mathcal{W}_{\lambda})$ will be study with the focus of verifying
assumption $(S_{3})$ in section 2.

######  Proposition 3.4.

Let $\lambda\in(-\infty,-\sum_{j=1}^{3}c_{j})$, $c_{j}>0$. Then for
$\lambda\neq-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}$ we have
$\ker(\mathcal{W}_{\lambda})=\\{0\\}$. For
$\lambda_{0}=-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}$,
$\ker(\mathcal{W}_{\lambda_{0}})=2$. Moreover,
$\sigma_{\mathrm{ess}}(\mathcal{W}_{\lambda})=[1,+\infty)$.

###### Proof.

We consider $\mathbb{u}=(u_{1},u_{2},u_{3})\in D(\mathcal{W}_{\lambda})$ and
$\mathcal{W}_{\lambda}\mathbb{u}=\mathbb{0}$. Then, from Sturm-Liouville
theory on half-lines (see [14], Chapter 2, Theorem 3.3),
$u_{j}(x)=\alpha_{j}\varphi^{\prime}_{j}(x)$, $x>0$, $j=1,2,3$. Thus, for
$\lambda\neq-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}$ we obtain
$\alpha_{1}/c_{1}=\alpha_{2}/c_{2}=\alpha_{3}/c_{3}$. Next, from the jump
conditions for $\mathbb{u}$ and
$\Psi_{\lambda,\delta^{\prime}}=(\varphi_{j})$, we obtain

$\alpha_{1}\varphi^{\prime}_{1}(0)\sum_{j=1}^{3}c_{j}=\alpha_{1}\lambda
c_{1}\varphi^{\prime\prime}_{1}(0).$ (3.5)

Next, suppose $\alpha_{1}\neq 0$. Since $\varphi^{\prime}_{1}(0)<0$ and for
$\lambda\in(-\infty,-\frac{\pi}{2}\sum_{j=1}^{3}c_{j})$ we have
$\varphi^{\prime\prime}_{1}(0)<0$, relation in (3.5) implies a contradiction
because of $c_{j}>0$. Now, considering
$\lambda\in(-\frac{\pi}{2}\sum_{j=1}^{3}c_{j},-\sum_{j=1}^{3}c_{j})$ and from
the specific values of $\varphi^{\prime}_{1}(0),\varphi^{\prime\prime}_{1}(0)$
we obtain from (3.5) again a contradiction. Thus, from the two cases above we
need to have $\alpha_{1}=\alpha_{2}=\alpha_{3}=0$. For
$\lambda_{0}=-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}$ we recall that
$\varphi^{\prime\prime}_{j}(0)=0$ for every $j$. Hence, from the jump-
condition for $\mathbb{u}$ follows $\sum_{j=1}^{3}\alpha_{j}=0$. Then
$\Psi_{1}=(\varphi^{\prime}_{1},-\varphi^{\prime}_{2},0)$ and
$\Psi_{2}=(0,\varphi^{\prime}_{2},-\varphi^{\prime}_{3})$ belong to
$D(\mathcal{W}_{\lambda_{0}})$ and
$\operatorname{span}\\{\Psi_{1},\Psi_{2}\\}=\ker(\mathcal{W}_{\lambda_{0}})$.

The statement $\sigma_{\mathrm{ess}}(\mathcal{W}_{\lambda})=[1,+\infty)$ is an
immediate consequence of Weyl’s Theorem because of
$\lim_{x\to+\infty}\cos(\varphi_{1}(x))=1=\lim_{x\to+\infty}\cos(\varphi_{j}(x))$
(see [45]). This finishes the proof. ∎

######  Proposition 3.5.

Let $\lambda\in[-\frac{\pi}{2}\sum_{j=1}^{3}c_{j},-\sum_{j=1}^{3}c_{j})$. Then
$n(\mathcal{W}_{\lambda})=1$.

###### Proof.

We will use the extension theory approach, which is based on the fact that the
family $(\mathcal{W}_{\lambda},D(\mathcal{W}_{\lambda}))$ represents all the
self-adjoint extensions of the closed symmetric operator
$(\widetilde{T},D(\widetilde{T}))$ with $D(\widetilde{T})\equiv
D(\mathcal{T})$ defined in (3.1), and

$\widetilde{T}=\Big{(}\Big{(}-c_{j}^{2}\frac{d^{2}}{dx^{2}}+\cos(\varphi_{j})\Big{)}\delta_{j,k}\Big{)},\;\l
1\leqq j,k\leqq 3,\\\ $ (3.6)

with $n_{\pm}(\widetilde{T})=1$. Next, we show that $\widetilde{T}\geqq 0$. If
we denote $L_{j}=-c_{j}^{2}\frac{d^{2}}{dx^{2}}+\cos(\varphi_{j})$ then we
obtain

$L_{j}\psi=-\frac{1}{\varphi^{\prime}_{j}}\frac{d}{dx}\Big{[}c_{j}^{2}(\varphi^{\prime}_{j})^{2}\frac{d}{dx}\Big{(}\frac{\psi}{\varphi^{\prime}_{j}}\Big{)}\Big{]},$
(3.7)

for any $\psi$. It is to be observed that $\varphi^{\prime}_{j}\neq 0$. Then
we have for any $\Lambda=(\psi_{j})\in D(\widetilde{T})$,

$\begin{split}\langle\widetilde{T}\Lambda,\Lambda\rangle=A-\sum_{j=1}^{3}c_{j}^{2}\psi^{2}_{j}(0)\frac{\varphi^{\prime\prime}_{j}(0)}{\varphi^{\prime}_{j}(0)}\equiv
A+P,\end{split}$ (3.8)

where $A\geqq 0$ represents the integral terms. Next we show that $P\geqq 0$.
Indeed, since $\varphi_{j}^{\prime\prime}(0)\geqq 0$ and
$\varphi_{j}^{\prime}(0)<0$, for $j=1,2,3$, we obtain immediately $P\geqq 0$.
Then, $\widetilde{T}\geqq 0$.

Due to Proposition A.3 (see Appendix), $n(\mathcal{W}_{\lambda})\leqq 1$.
Next, for
$\Psi_{\lambda,\delta^{\prime}}=(\varphi_{1},\varphi_{2},\varphi_{3})\in
D(\mathcal{W}_{\lambda})$ we obtain

$\langle\mathcal{W}_{\lambda}\Psi_{\lambda,\delta^{\prime}},\Psi_{\lambda,\delta^{\prime}}\rangle=\sum_{j=1}^{3}\int_{0}^{+\infty}[-\sin(\varphi_{j})+\cos(\varphi_{j})\phi_{j}]\varphi_{j}dx<0,$
(3.9)

because of $0<\varphi_{j}(x)\leqq\pi$ and $x\cos x\leqq\sin x$ for all
$x\in[0,\pi]$. Then from the minimax principle we arrive at
$n(\mathcal{W}_{\lambda})=1$. This finishes the proof. ∎

###### Proof of Theorem 1.1.

Let $\lambda\in(-\frac{\pi}{2}\sum_{j=1}^{3}c_{j},-\sum_{j=1}^{3}c_{j})$.
Then, from Propositions 3.4 and 3.5 we have
$\ker(\mathcal{W}_{\lambda})=\\{0\\}$ and $n(\mathcal{W}_{\lambda})=1$. Thus,
from Theorem 3.2, Corollary 3.3 and Theorem 2.4 follows the linear instability
property of the stationary profile
$\Pi_{\lambda,\delta^{\prime}}=(\varphi_{1},\varphi_{2},\varphi_{3},0,0,0)$.

Next, we consider the closed subspace in $L^{2}(\mathcal{Y})$,
$\mathcal{C}_{1}=\\{(u_{j})_{j=1}^{3}\in
L^{2}(\mathcal{Y}):u_{1}=u_{2}=u_{3}\\}$, and the case $c_{1}=c_{2}=c_{3}$
(hence $b_{1}=b_{2}=b_{3}$). Then, we can show that on
$\mathcal{B}_{1}=\mathcal{C}_{1}\cap D(\mathcal{W}_{\lambda})$ we obtain
$\mathcal{W}_{\lambda}:\mathcal{B}_{1}\to\mathcal{C}_{1}$ is well-defined.
Moreover, we note that for $\lambda_{0}=-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}$
follows $\ker(\mathcal{W}_{\lambda_{0}}|_{\mathcal{B}_{1}})=\\{0\\}$ and
$n(\mathcal{W}_{\lambda_{0}}|_{\mathcal{B}_{1}})=1$ (because
$\Psi_{\lambda_{0},\delta^{\prime}}=(\varphi_{1},\varphi_{1},\varphi_{1})\in\mathcal{B}_{1}$
and
$\langle\mathcal{W}_{\lambda_{0}}\Psi_{\lambda_{0},\delta^{\prime}},\Psi_{\lambda_{0},\delta^{\prime}}\rangle<0$).
Therefore, from Kato-Rellich Theorem, analytic perturbation and a continuation
argument (see Angulo and Plaza [10] and/or Propositions 3.10-3.11 below), we
can see that for all $\lambda\in(-\infty,-\frac{\pi}{2}\sum_{j=1}^{3}c_{j}]$
the following relations hold:
$\ker(\mathcal{W}_{\lambda}|_{\mathcal{B}_{1}})=\\{0\\}$ and
$n(\mathcal{W}_{\lambda}|_{\mathcal{B}_{1}})=1$. Thus, the static profiles
$\Pi_{\lambda,\delta^{\prime}}$ are also linearly unstable in this case.

Now, since the mapping data-solution for the sine-Gordon model on
$H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ is at least of class $C^{2}$
(indeed, it is smooth) by Theorem 3.2, it follows that the linear instability
property of $\Psi_{\lambda,\delta^{\prime}}$ is in fact of nonlinear type in
the $H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$-norm (see Henry et al. [29],
Angulo and Natali [9], and Angulo et al. [8]). This finishes the proof. ∎

######  Remark 3.6.

For the case $\lambda\in(-\infty,-\frac{\pi}{2}\sum_{j=1}^{3}c_{j})$ in
Proposition 3.5, the formula for $P$ in (3.8) satisfies $P<0$. Therefore, it
is not clear whether the extension theory approach provides an estimate of the
Morse-index of $\mathcal{W}_{\lambda}$; see also the related Remark 4.4 in
[10].

###  3.2 Kink/anti-kink instability theory on a tricrystal junction

In this subsection we study the existence and stability of kink/anti-kink
profile in (1.10). Since these stationary profiles do not belong to the
classical $H^{2}(\mathcal{Y})$-energy space, we need to work with specific
functional spaces suitable for our needs.

####  3.2.1 The anti-kink/kink solutions on a tricrystal junction

In the sequel, we describe the profiles $(\varphi_{j})_{j=1}^{3}$ in (1.10)
satisfying the $\delta^{\prime}$-condition in (1.7). Thus, we obtain the
following relations:

$\operatorname{sech}\big{(}\frac{a_{1}}{c_{1}}\big{)}=\operatorname{sech}\big{(}\frac{a_{2}}{c_{2}}\big{)}=\operatorname{sech}\big{(}\frac{a_{3}}{c_{3}}\big{)},$
(3.10)

and

$c_{1}\arctan(e^{-a_{1}/c_{1}})+c_{2}\arctan(e^{-a_{2}/c_{2}})+c_{3}\arctan(e^{-a_{3}/c_{3}})-\frac{(c_{2}+c_{3})\pi}{2}=\frac{\lambda}{2}\operatorname{sech}\big{(}\frac{a_{1}}{c_{1}}\big{)}.$
(3.11)

Here, we consider the specific class of anti-kink/kink profiles with the
condition $\frac{a_{1}}{c_{1}}=\frac{a_{2}}{c_{2}}=\frac{a_{3}}{c_{3}}$. Thus,
from (3.11) and $y=e^{-a_{1}/c_{1}}$ we get the following equality

$F(y)\equiv\frac{1+y^{2}}{y}\Big{[}\Big{(}\sum_{i=1}^{3}c_{i}\Big{)}\arctan(y)-\frac{(c_{2}+c_{3})\pi}{2}\Big{]}=\lambda.$
(3.12)

Next, we show that the mapping $F$ is strictly increasing for $y>0$. Indeed,
since for $y\in(0,1)$ we have that
$\frac{1}{y}+\frac{y^{2}-1}{y^{2}}\arctan(y)>0$, then $F^{\prime}(y)>0$. Now,
for $\theta=\sum_{i=1}^{3}c_{i}$, the relation

$F^{\prime}(y)=\theta\Big{[}\frac{1}{y}+\frac{y^{2}-1}{y^{2}}\Big{(}\arctan(y)-\frac{\pi}{2}\Big{)}\Big{]}+\theta\Big{[}\frac{\pi}{2}-\frac{(c_{2}+c_{3})\pi}{2\sum_{i=1}^{3}c_{i}}\Big{]}\frac{y^{2}-1}{y^{2}}$
(3.13)

shows that $F^{\prime}(y)>0$ for $y\in[1,+\infty)$. Moreover, it is no
difficult to see that $\lim_{y\to 0^{+}}F(y)=-\infty$, and
$\lim_{y\to+\infty}F(y)=+\infty$.

Then, from (3.12) we have the following specific behavior of the
$\lambda$-parameter:

  1. a)

for $a_{1}=0$, $\lambda=-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$,

  2. b)

for $a_{1}>0$, $\lambda\in(-\infty,-\frac{\pi}{2}(c_{2}+c_{3}-c_{1}))$,

  3. c)

for $a_{1}<0$, $\lambda\in(-\frac{\pi}{2}(c_{2}+c_{3}-c_{1}),+\infty)$.

Moreover, from (3.12) and the properties for $F$ we obtain the existence of a
smooth shift-map (also real analytic) $\lambda\in(-\infty,+\infty)\mapsto
a_{1}(\lambda)$ satisfying $F(e^{-a_{1}(\lambda)/c_{1}})=\lambda$. Thus, the
mapping

$\lambda\in(-\infty,+\infty)\,\mapsto\Phi_{\lambda,\delta^{\prime}}=(\varphi_{1,a_{1}(\lambda)},\varphi_{2,a_{2}(\lambda)},\varphi_{3,a_{3}(\lambda)},0,0,0),$

represents a real-analytic family of static profiles for the sine-Gordon
equation (1.5) on a tricrystal junction, with the profiles
$\varphi_{j,a_{j}(\lambda)}$ satisfying the boundary condition in (1.10).
Hence we obtain, for $a_{i}=a_{i}(\lambda)$ and
$\varphi_{i}=\varphi_{i,a_{i}(\lambda)}$, the following behavior:

  1. 1)

for $\lambda=-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$ we obtain
$a_{1}=a_{2}=a_{3}=0$, $\varphi_{2}(0)=\varphi_{3}(0)=-\pi$,
$\varphi_{1}(0)=\pi$, $\varphi_{i}^{\prime\prime}(0)=0$, $i=1,2,3$. Thus, the
profile of $(\varphi_{1},\varphi_{2},\varphi_{3})$ represents one-half
positive anti-kink $\varphi_{1}$ and two-half negative kink solitons profiles
$\varphi_{2}$, $\varphi_{3}$ (connected in the vertex of the graph) such as
Figure 3 shows below;

(a) Kink/anti-kink profiles, $\varphi_{j}$.

(b) “Fluxons”, $\varphi_{j}^{\prime}$.

Figure 3: Plots of the kink/anti-kink profiles (1.10) (panel (a), in blue) and
their corresponding derivatives or “fluxons” (panel (b), in red) in the case
where $c_{j}=1$, $a_{j}=0$, $j=1,2,3$, and
$\lambda=-\tfrac{\pi}{2}(c_{2}+c_{3}-c_{1})=-\tfrac{\pi}{2}$. Both plots are
depicted in the same scale for comparison purposes (color online).

  2. 2)

for $\lambda\in(-\infty,-\frac{\pi}{2}(c_{2}+c_{3}-c_{1}))$ we obtain
$a_{1}>0$, $\varphi_{i}^{\prime\prime}(a_{1})=0$, $i=1,2,3$. Therefore,
$\varphi_{1}(0)\in(0,\pi)$ and $\varphi_{2}(0)=\varphi_{3}(0)\in(-\pi,-2\pi)$.
Thus, the profile of $(\varphi_{1},\varphi_{2},\varphi_{2})$, looks similar to
the one shown in Figure 4 below (bump-profile type);

(a) Kink/anti-kink profiles, $\varphi_{j}$.

(b) “Fluxons”, $\varphi_{j}^{\prime}$.

Figure 4: Plots of the kink/anti-kink profiles (1.10) (panel (a), in blue) and
their corresponding derivatives or “fluxons” (panel (b), in red) in the case
where $c_{j}=1$, $a_{j}=1.8>0$, $j=1,2,3$, and
$\lambda\in(-\infty,-\tfrac{\pi}{2})$. Both plots are depicted in the same
scale for comparison purposes (color online).

  3. 3)

for $\lambda\in(-\frac{\pi}{2}(c_{2}+c_{3}-c_{1}),+\infty)$ we obtain
$a_{1}<0$ and therefore $\varphi^{\prime\prime}_{i}<0$ for $i=1,2,3$, Thus,
the profile of $(\varphi_{1},\varphi_{2},\varphi_{3})$ looks similar to that
in Figure 5 below (typical tail-profile).

(a) Kink/anti-kink profiles, $\varphi_{j}$.

(b) “Fluxons”, $\varphi_{j}^{\prime}$.

Figure 5: Plots of the kink/anti-kink profiles (1.10) (panel (a), in blue) and
their corresponding derivatives or “fluxons” (panel (b), in red) in the case
where $c_{j}=1$, $a_{j}=-0.5<0$, $j=1,2,3$, and
$\lambda\in(-\tfrac{\pi}{2},\infty)$. Both plots are depicted in the same
scale for comparison purposes (color online).

Here, our instability study of anti-kink/kink profiles will be in the case
where the “magnetic field” or “fluxon” is continuous at the vertex of the
tricrystal, namely, whenever
$\varphi^{\prime}_{1}(0)=\varphi^{\prime}_{2}(0)=\varphi^{\prime}_{3}($0)
($c_{1}=c_{2}=c_{3}$). It is to be observed that we have left open a possible
instability study of other anti-kink/kink soliton configurations, for
instance, profiles not satisfying the continuity property at zero for the
components $\varphi_{2},\varphi_{3}$
($\frac{a_{2}}{c_{2}}\neq\frac{a_{3}}{c_{3}}$) and/or the non-continuity of
the magnetic field.

The stability result for the stationary profiles,
$\Phi_{\lambda,\delta^{\prime}}=(\varphi_{1},\varphi_{2},\varphi_{3},0,0,0)$,
with $\varphi_{j}=\varphi_{j,a_{j}(\lambda)}$ defined in (1.10) is that
established in Theorem 1.2.

####  3.2.2 Functional space for stability properties of the anti-kink/kink
profile

The natural framework space for studying stability properties associated to
the anti-kink/kink soliton profile $\Phi=(\varphi_{j})_{j=1}^{3}$ described in
the former subsection for the sine-Gordon model is
$\mathcal{X}(\mathcal{Y})=H^{1}_{\mathrm{loc}}(0,\infty)\bigoplus
H^{1}(0,\infty)\bigoplus H^{1}(0,\infty)$. Thus we say that a flow
$t\to(u(t),v(t))\in\mathcal{X}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ is
called a _perturbed solution_ for the anti-kink/kink profile
$\Phi\in\mathcal{X}(\mathcal{Y})$ if for $(P(t),Q(t))\equiv(u(t)-\Phi,v(t))$
we have that $(P(t),Q(t))\in H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ and
$\mathbb{z}=(P,Q)^{\top}$ satisfies the following vectorial perturbed sine-
Gordon model

$\begin{cases}\mathbb{z}_{t}=JE\mathbb{z}+F_{1}(\mathbb{z})\\\
P(0)=u(0)-\Phi\in H^{1}(\mathcal{Y}),\\\ Q(0)=v(0)\in
L^{2}(\mathcal{Y}),\end{cases}$ (3.14)

where for $P=(p_{1},p_{2},p_{3})$ we have

$F_{1}(\mathbb{z})=\left(\begin{array}[]{cc}0\\\ 0\\\ 0\\\
\sin(\varphi_{1})-\sin(p_{1}+\varphi_{1})\\\
\sin(\varphi_{2})-\sin(p_{2}+\varphi_{2})\\\
\sin(\varphi_{3})-\sin(p_{3}+\varphi_{3})\end{array}\right).$ (3.15)

Then, the stability analysis of the stationary anti-kink/kink
$\Phi_{\lambda,\delta^{\prime}}=(\varphi_{1},\varphi_{2},\varphi_{3},0,0,0)$
by the sine-Gordon model on $\mathcal{X}(\mathcal{Y})\times
L^{2}(\mathcal{Y})$ reduces to studying the stability properties of the
trivial solution $(P,Q)=(0,0)$ for the linearized model associated to (3.14)
around $(P,Q)=(0,0)$. Thus, via Taylor’s Theorem we obtain the linearized
system in (2.7) but with the Schrödinger diagonal operator $\mathcal{L}$ in
(2.8) now determined by the anti-kink/kink profile $\Phi=(\varphi_{j})$. We
denote this operator by $\mathcal{L}_{\lambda}$ in (3.16) below, with the
$\delta^{\prime}$-interaction domain $D(\mathcal{L}_{\lambda})$ in (2.4). In
this form, we can apply _ipsis litteris_ the semi-group theory results in
subsection 3.1.1 to the operator $JE$ and to the local well-posedness problem
in $H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ for the vectorial perturbed
sine-Gordon model (3.14). Lastly, we note that the anti-kink/kink profile
$\Phi\in\mathcal{X}(\mathcal{Y})$, but $\Phi^{\prime}\in H^{2}(\mathcal{Y})$.

####  3.2.3 The spectral study in the anti-kink/kink case

In this subsection we provide the necessary spectral information for the
family of self-adjoint operators
$(\mathcal{L}_{\lambda},D(\mathcal{L}_{\lambda}))$, where

$\mathcal{L}_{\lambda}=\Big{(}\Big{(}-c_{j}^{2}\frac{d^{2}}{dx^{2}}+\cos(\varphi_{j})\Big{)}\delta_{j,k}\Big{)},\quad\l
1\leqq j,k\leqq 3,$ (3.16)

associated to the anti-kink/kink solutions
$(\varphi_{1},\varphi_{2},\varphi_{3})$ determined in the previous subsection.
Here $D(\mathcal{L}_{\lambda})$ is the $\delta^{\prime}$-interaction domain
defined in (1.12).

By completeness and convenience of the reader, we prove the following results
associated to the kernel of the operators $\mathcal{L}_{\lambda}$ for all
values of $c_{1},c_{2},c_{3}$ and the admissible $\lambda$-values. We note
that these results are expected, due to the break in the translation symmetry
of solutions for the sine-Gordon model on tricrystal configurations.

######  Proposition 3.7.

Let $c_{1},c_{2},c_{3}>0$ and $\lambda\in(-\infty,+\infty)$. Then,
$\ker(\mathcal{L}_{\lambda})=\\{\mathbf{0}\\}$ for all
$\lambda\neq-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$. For
$\lambda=-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$ we have
$\dim(\ker(\mathcal{L}_{\lambda}))=2$. Moreover, for all $\lambda$ we obtain
$\sigma_{\mathrm{ess}}(\mathcal{L}_{\lambda})=[1,+\infty)$.

###### Proof.

Let $\mathbb{u}=(u_{1},u_{2},u_{3})\in D(\mathcal{L}_{\lambda})$ and
$\mathcal{L}_{\lambda}\mathbb{u}=\mathbb{0}$. Then, since
$(\varphi^{\prime}_{1},\varphi^{\prime}_{3},\varphi^{\prime}_{2})\in
H^{2}(\mathcal{Y})$, it follows from Sturm-Liouville theory on half-lines that

$u_{j}(x)=\alpha_{j}\phi^{\prime}_{j}(x),\;\;x>0,\;\;j=1,2,3,$ (3.17)

for some $\alpha_{j}$, $j=1,2,3$, real constant. In what follows, we assume
$\lambda\neq-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$. Thus, from (2.4) and by
supposing $\alpha_{1}\neq 0$ we have

$\frac{\alpha_{2}}{\alpha_{1}}=\frac{c_{1}}{c_{2}}\frac{\varphi^{\prime\prime}_{1}(0)}{\varphi^{\prime\prime}_{2}(0)}=\frac{c_{2}}{c_{1}},\;\;\;\;\frac{\alpha_{3}}{\alpha_{1}}=\frac{c_{1}}{c_{3}}\frac{\varphi^{\prime\prime}_{1}(0)}{\varphi^{\prime\prime}_{3}(0)}=\frac{c_{3}}{c_{1}}$
(3.18)

and

$\lambda=\frac{1}{\alpha_{1}}\Big{(}\sum_{j=1}^{3}\alpha_{j}\Big{)}\frac{\varphi^{\prime}_{1}(0)}{\varphi^{\prime\prime}_{1}(0)}=(c_{1}+c_{2}+c_{3})\frac{\cosh^{2}(a_{1}/c_{1})}{\sinh(a_{1}/c_{1})}.$
(3.19)

Now, we consider the following cases:

  1. a)

Suppose $a_{1}<0$ ($\lambda\in(-\frac{\pi}{2}(c_{2}+c_{3}-c_{1}),+\infty)$):
then, since

$\frac{\cosh^{2}(a_{1}/c_{1})}{\sinh(a_{1}/c_{1})}\leqq-2,\;\;\text{for
all}\;\;a_{1}<0,$ (3.20)

we have, from (3.19),
$-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})<-2(c_{1}+c_{2}+c_{3})$, which is a
contradiction. So, we need to have $0=\alpha_{1}=\alpha_{2}=\alpha_{3}$ and
therefore $\mathbb{u}=\mathbb{0}$.

  2. b)

Suppose $a_{1}>0$ ($\lambda\in(-\infty,-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$):
then, from (3.19) we have $\lambda>2(c_{1}+c_{2}+c_{3})$. Thus for the case
$c_{2}+c_{3}-c_{1}\geqq 0$ we obtain immediately a contradiction. Now, for
$c_{2}+c_{3}-c_{1}<0$ is sufficient to study the case
$0<\lambda<-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$. Indeed, since
$2(c_{1}+c_{2}+c_{3})>-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$ we have again a
contradiction. So, we need to have $0=\alpha_{1}=\alpha_{2}=\alpha_{3}$ and
therefore $\mathbb{u}=\mathbb{0}$.

Now, suppose that $\lambda=-\frac{\pi}{2}(c_{2}+c_{3}-c_{1})$. In this case
the Kirchhoff’s condition for $\mathbb{u}$, $\varphi^{\prime}_{1}(0)\neq 0$
and $\varphi^{\prime\prime}_{1}(0)=0$ we get the relation
$\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. Therefore

$(u_{1},u_{2},u_{3})=\alpha_{2}(-\varphi^{\prime}_{1},\varphi^{\prime}_{2},0)+\alpha_{3}(-\varphi^{\prime}_{1},0,\varphi^{\prime}_{3}).$

Since
$(-\varphi^{\prime}_{1},\varphi^{\prime}_{2},0),(-\varphi^{\prime}_{1},0,\varphi^{\prime}_{3})\in
D(\mathcal{L}_{\lambda})$ we obtain that
$\dim(\ker(\mathcal{L}_{\lambda}))=2$.

The statement $\sigma_{\mathrm{ess}}(\mathcal{L}_{\lambda})=[1,+\infty)$ is an
immediate consequence of Weyl’s Theorem because of
$\lim_{x\to+\infty}\cos(\varphi_{1}(x))=1=\lim_{x\to+\infty}\cos(\varphi_{j}(x))$.
This finishes the proof. ∎

######  Proposition 3.8.

Let $c_{1}=c_{2}=c_{3}$ and $\lambda\in[-\frac{\pi}{2}c_{1},0)$. Then
$n(\mathcal{L}_{\lambda})=1$.

###### Proof.

Similarly as in the proof of Proposition 3.5 (see formula (3.8)), we obtain
via the extension theory that $n(\mathcal{L}_{\lambda})\leqq 1$ for all
$\lambda\in[-\frac{\pi}{2}c_{1},+\infty)$ because of
$\varphi_{j}^{\prime\prime}(0)\leqq 0$ for $j=1,2,3$.

Now we show that $n(\mathcal{L}_{\lambda})\geqq 1$ for
$\lambda\in[-\frac{\pi}{2}c_{1},0)$. Indeed, we consider the following
quadratic form $\mathcal{Q}_{\lambda}$ associated to
$(\mathcal{L}_{\lambda},D(\mathcal{L}_{\lambda}))$ for $\Lambda=(\psi_{i})\in
H^{1}(\mathcal{Y})$,

$\mathcal{Q}(\Lambda)=\frac{1}{\lambda}\Big{(}\sum_{j=1}^{3}c_{j}\psi_{j}(0)\Big{)}^{2}+\sum_{j=1}^{3}\int_{0}^{\infty}c^{2}_{j}(\psi^{\prime}_{j})^{2}+\cos(\varphi_{j})\psi_{j}^{2}dx.$
(3.21)

Next, for
$\Lambda_{1}=(\varphi^{\prime}_{1},\varphi^{\prime}_{2},\varphi^{\prime}_{3})\in
H^{1}(\mathcal{Y})$ we obtain from the equalities
$-\varphi^{\prime\prime\prime}_{j}+\cos(\varphi_{j})\varphi^{\prime}_{j}=0$,
$c_{1}\varphi^{\prime}_{1}(0)=c_{2}\varphi^{\prime}_{2}(0)=c_{3}\varphi^{\prime}_{3}(0)$,
and from integration by parts, the relation

$\mathcal{Q}_{\lambda}(\Lambda_{1})=\frac{9c_{1}^{2}}{\lambda}[\varphi^{\prime}_{1}(0)]^{2}-c_{1}\varphi^{\prime}_{1}(0)\sum_{j=1}^{3}c_{j}\varphi^{\prime\prime}_{j}(0).$
(3.22)

Thus, for $\lambda=-\frac{\pi}{2}c_{1}$ we have
$\varphi^{\prime\prime}_{j}(0)=0$ and therefore
$\mathcal{Q}_{\lambda}(\Lambda_{1})<0$. Thus
$n(\mathcal{L}_{-\frac{\pi}{2}c_{1}})=1$. Now, for
$\lambda\in(-\frac{\pi}{2}c_{1},0)$ we have $\varphi^{\prime\prime}_{1}(0)<0$
($a_{1}<0$) and so since
$\varphi^{\prime\prime}_{1}(0)=\varphi^{\prime\prime}_{2}(0)=\varphi^{\prime\prime}_{3}(0)$
we get $\mathcal{Q}_{\lambda}(\Lambda_{1})<0$ if and only if

$3\frac{\varphi^{\prime}_{1}(0)}{\varphi^{\prime\prime}_{1}(0)}=3c_{1}\frac{\cosh^{2}(a_{1})}{\sinh(a_{1})}<\lambda,\;\;\text{for}\;\;a_{1}<0,$
(3.23)

which is true by (3.20). Therefore, $n(\mathcal{L}_{\lambda})=1$ for
$\lambda\in(-\frac{\pi}{2}c_{1},0)$. ∎

######  Remark 3.9.

For the case $\lambda\in[0,+\infty)$ in Proposition 3.8, it was not possible
to show (in an easy way) that the quadratic form $\mathcal{Q}_{\lambda}$ in
(3.21) has a negative direction. But we will see in the following, via
analytic perturbation approach, that we still have
$n(\mathcal{L}_{\lambda})=1$.

######  Proposition 3.10.

Let $c_{1}=c_{2}=c_{3}$ and $\lambda\in[0,+\infty)$. Then
$n(\mathcal{L}_{\lambda})=1$.

###### Proof.

We will use analytic perturbation theory. Initially, from subsection 3.2.1 we
have that $\lambda\in(-\infty,+\infty)\to a_{1}(\lambda)$ represents a real-
analytic mapping function and so from the relations
$\varphi_{1,a_{1}(\lambda)}-\varphi_{1,a_{1}(0)}\in L^{2}(0,+\infty)$ for
every $\lambda$ and
$\|\varphi_{1,a_{1}(\lambda)}-\varphi_{1,a_{1}(0)}\|_{H^{1}(0,+\infty)}\to 0$
as $\lambda\to 0$, we obtain for
$\Phi_{a_{1}(\lambda)}=(\varphi_{1,a_{1}(\lambda)},\varphi_{3,a_{1}(\lambda)},\varphi_{2,a_{1}(\lambda)})$
(where we have used that $a_{1}(\lambda)=a_{2}(\lambda)=a_{3}(\lambda)$) the
convergence

$\|\Phi_{a_{1}(\lambda)}-\Phi_{a_{1}(0)}\|_{H^{1}(\mathcal{Y})}\to
0\;\;\text{as}\;\;\lambda\to 0.$

Thus, we obtain that $\mathcal{L}_{\lambda}$ converges to $\mathcal{L}_{0}$ as
$\lambda\to 0$ in the generalized sense. Indeed, denoting
$W_{\lambda}=\Big{(}\cos(\varphi_{j,a_{1}(\lambda)})\delta_{j,k}\Big{)}$ we
obtain

$\begin{split}\widehat{\delta}(\mathcal{L}_{\lambda},\mathcal{L}_{0})&=\widehat{\delta}(\mathcal{L}_{0}+(W_{\lambda}-W_{0}),\mathcal{L}_{0})\leqq\|W_{\lambda}-W_{0}\|_{L^{2}(\mathcal{Y})}\to
0,\qquad\text{as}\;\;\lambda\to 0,\end{split}$

where $\widehat{\delta}$ is the gap metric (see [32, Chapter IV]).

Now, we denote by $N=n(\mathcal{L}_{0})$ the Morse-index for $\mathcal{L}_{0}$
(from the proof of Proposition 3.8, $N\leqq 1$). Thus, from Proposition 3.7 we
can separate the spectrum $\sigma(\mathcal{L}_{0})$ of $\mathcal{L}_{0}$ into
two parts, $\sigma_{0}=\\{\gamma:\gamma<0\\}\cap\sigma(\mathcal{L}_{0})$ and
$\sigma_{1}$ by a closed curve $\Gamma$ belongs to the resolvent set of
$\mathcal{L}_{0}$ with $0\in\Gamma$ and such that $\sigma_{0}$ belongs to the
inner domain of $\Gamma$ and $\sigma_{1}$ to the outer domain of $\Gamma$.
Moreover, $\sigma_{1}\subset[\theta_{0},+\infty)$ with
$\theta_{0}=\inf\\{\theta:\theta\in\sigma(\mathcal{L}_{0}),\;\theta>0\\}>0$
(we recall that $\sigma_{\mathrm{ess}}(\mathcal{L}_{0})=[1,+\infty)$). Then,
by [32, Theorem 3.16, Chapter IV], we have
$\Gamma\subset\rho(\mathcal{L}_{\lambda})$ for
$\lambda\in[-\delta_{1},\delta_{1}]$ and $\delta_{1}>0$ small enough.
Moreover, $\sigma(\mathcal{L}_{\lambda})$ is likewise separated by $\Gamma$
into two parts so that the part of $\sigma(\mathcal{L}_{\lambda})$ inside
$\Gamma$ will consist of a negative eigenvalue with exactly total (algebraic)
multiplicity equal to $N$. Therefore, by Proposition 3.8 we need to have
$n(\mathcal{L}_{\lambda})=N=1$ for $\lambda\in[-\delta_{1},\delta_{1}]$.

Next, using a classical continuation argument based on the Riesz-projection,
we can see that $n(\mathcal{L}_{\lambda})=1$ for all $\lambda\in[0,+\infty)$
(see [10]). This finishes the proof. ∎

The following result gives a precise value for the Morse-index of the operator
$\mathcal{L}_{\lambda}$, with $\lambda\in(-\infty,-\frac{\pi}{2}c_{1})$, when
we consider the domain $\mathcal{C}\cap D(\mathcal{L}_{\lambda})$,
$\mathcal{C}=\\{(u_{j})_{j=1}^{3}\in L^{2}(\mathcal{Y}):u_{1}=u_{2}=u_{3}\\}$.
This strategy will allow the use of the linear instability framework
established in section 2 above.

######  Proposition 3.11.

Let $c_{1}=c_{2}=c_{3}$ and $\lambda\in(-\infty,-\frac{\pi}{2}c_{1}]$.
Consider $\mathcal{B}=\mathcal{C}\cap D(\mathcal{L}_{\lambda})$. Then
$\mathcal{L}_{\lambda}:\mathcal{B}\to\mathcal{C}$ is well defined. Moreover,
$\ker(\mathcal{L}_{\lambda}|_{\mathcal{B}})=\\{0\\}$ and
$n(\mathcal{L}_{\lambda}|_{\mathcal{B}})=1$.

###### Proof.

Initially, since $\varphi_{1}-2\pi=\varphi_{2}=\varphi_{3}$ and
$\cos(\varphi_{1})=\cos(\varphi_{1}-2\pi)=\cos(\varphi_{2})=\cos(\varphi_{3})$
follows immediately that, for $\mathbb{u}=(u_{j})_{j=1}^{3}\in\mathcal{B}$, we
have $\mathcal{L}_{\lambda}\mathbb{u}\in\mathcal{C}$.

Next, from the proof of Proposition 3.7 for $\lambda_{0}=-\frac{\pi}{2}c_{1}$
we have $\ker(\mathcal{L}_{\lambda_{0}}|_{\mathcal{B}})=\\{\mathbb{0}\\}$.
Moreover, the anti-kink/kink profile for $\lambda_{0}$,
$\Phi_{0}=(\varphi_{1,0},\varphi_{2,0},\varphi_{2,0})$, satisfies that
$\Phi^{\prime}_{0}=(\varphi^{\prime}_{1,0},\varphi^{\prime}_{2,0},\varphi^{\prime}_{2,0})\in\mathcal{C}\cap
H^{1}(\mathcal{Y})$. Then, from (3.22) we know that the quadratic form
$Q_{\lambda_{0}}$ associated to $(\mathcal{L}_{\lambda_{0}},\mathcal{B})$
satisfies $Q_{\lambda_{0}}(\Phi^{\prime}_{0})<0$. Therefore,
$n(\mathcal{L}_{\lambda_{0}}|_{\mathcal{B}})=1$. Hence, by using a similar
strategy as in the proof of Proposition 3.10 we obtain that
$n(\mathcal{L}_{\lambda}|_{\mathcal{B}})=1$ for
$\lambda\in(-\infty,-\frac{\pi}{2}c_{1})$. This finishes the proof. ∎

###### Proof of Theorem 1.2.

We consider initially the case $\lambda\in(-\frac{\pi}{2}c_{1},+\infty)$. From
Propositions 3.7, 3.8 and 3.10 we have $\ker(\mathcal{L}_{\lambda})=\\{0\\}$
and $n(\mathcal{L}_{\lambda})=1$. Moreover, from subsection 3.2.2 and based in
Proposition 3.3 we verify Assumption $(S_{1})$ for $J\mathcal{E}$ on
$H^{1}(\mathcal{Y})\times L^{2}(\mathcal{Y})$ in the linear instability
criterion in subsection 3.1. Thus, from Theorem 2.4 follows the linear
instability property of the stationary anti-kink/kink soliton profile
$\Pi_{\lambda,\delta^{\prime}}$.

For the case $\lambda\in(-\infty,-\frac{\pi}{2}c_{1}]$ we need to adjust our
instability criterion in section 2 to the space $\mathcal{R}=(\mathcal{C}\cap
H^{1}(\mathcal{Y}))\times\mathcal{C}$ determined by (3.14). Thus, initially we
need to see that $J\mathcal{E}$ is the generator of a $C_{0}$-semigroup on the
restricted subspace $\mathcal{R}$ (assumption $(S_{1})$). Indeed, this is a
consequence of the mapping $J\mathcal{E}:(\mathcal{C}\cap
D(\mathcal{L}_{\lambda}))\times(\mathcal{C}\cap
H^{1}(\mathcal{Y}))\to(\mathcal{C}\cap H^{1}(\mathcal{Y}))\times\mathcal{C}$
is well defined and by using the same strategy as in Angulo and Plaza [11]
(subsection 3.1.1 and Proposition 3.11). Now, from Proposition 3.11 and
Theorem 2.4 we finish the proof.

∎

## Acknowledgements

J. Angulo was supported in part by CNPq/Brazil Grant and by FAPERJ/Brazil
program PRONEX-E - 26/010.001258/2016.

##  Appendix A Appendix

For the sake of completeness, in this section we develop the extension theory
of symmetric operators suitable for our needs. For further information on the
subject the reader is referred to the monographs by Naimark [38, 39]. The
following classical result, known as _the von-Neumann decomposition theorem_ ,
can be found in [43, 44].

######  Theorem A.1.

Let $A$ be a closed, symmetric operator, then

$D(A^{*})=D(A)\oplus\mathcal{N}_{-i}\oplus\mathcal{N}_{+i}.$ (A.1)

with $\mathcal{N}_{\pm i}=\ker(A^{*}\mp iI)$. Therefore, for $u\in D(A^{*})$
and $u=x+y+z\in D(A)\oplus\mathcal{N}_{-i}\oplus\mathcal{N}_{+i}$,

$A^{*}u=Ax+(-i)y+iz.$ (A.2)

######  Remark A.2.

The direct sum in (A.1) is not necessarily orthogonal.

The following propositions provide a strategy for estimating the Morse-index
of the self-adjoint extensions (see Naimark [39]).

######  Proposition A.3.

Let $A$ be a densely defined lower semi-bounded symmetric operator (that is,
$A\geq mI$) with finite deficiency indices, $n_{\pm}(A)=k<\infty$, in the
Hilbert space $\mathcal{H}$, and let $\widehat{A}$ be a self-adjoint extension
of $A$. Then the spectrum of $\widehat{A}$ in $(-\infty,m)$ is discrete and
consists of, at most, $k$ eigenvalues counting multiplicities.

######  Proposition A.4.

Let $A$ be a densely defined, closed, symmetric operator in some Hilbert space
$H$ with deficiency indices equal $n_{\pm}(A)=1$. All self-adjoint extensions
$A_{\theta}$ of $A$ may be parametrized by a real parameter
$\theta\in[0,2\pi)$ where

$\begin{split}D(A_{\theta})&=\\{x+c\phi_{+}+\zeta e^{i\theta}\phi_{-}:x\in
D(A),\zeta\in\mathbb{C}\\},\\\ A_{\theta}(x+\zeta\phi_{+}+\zeta
e^{i\theta}\phi_{-})&=Ax+i\zeta\phi_{+}-i\zeta
e^{i\theta}\phi_{-},\end{split}$

with $A^{*}\phi_{\pm}=\pm i\phi_{\pm}$, and $\|\phi_{+}\|=\|\phi_{-}\|$.

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