Unstable kink and anti-kink profile for the sine-Gordon equation on a ${\mathcal {Y}}$-junction graph

Abstract

The sine-Gordon equation on a metric graph with a structure represented by a ${\mathcal {Y}}$-junction, is considered. The model is endowed with boundary conditions at the graph-vertex of $\delta '$-interaction type, expressing continuity of the derivatives of the wave functions plus a Kirchhoff-type rule for the self-induced magnetic flux. It is shown that particular stationary, kink and kink/anti-kink soliton profile solutions to the model are linearly (and nonlinearly) unstable. To that end, a recently developed linear instability criterion for evolution models on metric graphs by Angulo and Cavalcante (2020), which provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is applied. This leads to the spectral study to the linearized operator and of its Morse index. The analysis is based on analytic perturbation theory, Sturm-Liouville oscillation results and the extension theory of symmetric operators. The methods presented in this manuscript have prospect for the study of the dynamics of solutions for the sine-Gordon model on metric graphs with finite bounds or on metric tree graphs and/or loop graphs.

Appendix

1.1 Extension Theory

For the sake of completeness, in this part of the appendix 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 [35, 36]. The following classical result, known as the von-Neumann decomposition theorem, can be found in [41], p. 138.

Theorem A.1

Let A be a closed, symmetric operator, then

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

(A.1)

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

$$\begin{aligned} A^*u=Ax+(-i)y+iz. \end{aligned}$$

(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 and can be found in Naimark [36] (Theorem 16, p. 44) and Reed and Simon, vol. 2, [41] (see Theorem X.2, p. 140).

Proposition A.3

Let A be a densely defined lower semi-bounded symmetric operator (that is, $A\ge mI$) 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

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

with $A^*\phi _{\pm }=\pm i \phi _{\pm }$, and $\Vert \phi _+\Vert =\Vert \phi _-\Vert$.

Next Proposition can be found in Naimark [36] (see Theorem 9, p. 38).

Proposition A.5

All self-adjoint extensions of a closed, symmetric operator which has equal and finite deficiency indices have one and the same continuous spectrum.

The following result was used in the proof of Proposition  3.10.

Proposition A.6

Let ${\mathcal {Y}}$ be a ${\mathcal {Y}}$-junction of type I. Consider the following closed symmetric operator, $({\mathcal {H}}, D({\mathcal {H}}))$, densely defined on $L^2({\mathcal {Y}})$ by

$$\begin{aligned} \begin{aligned} {\mathcal {H}}&=\Big (\Big (-c_j^2\frac{d^2}{dx^2}\Big )\delta _{j,k} \Big ),\;\;1\leqq j, k\leqq 3,\\ D({\mathcal {H}})&= \Big \{(v_j)_{j=1}^3\in H^2({\mathcal {Y}}): c_1v_1'(0-)\\&=c_2v'_2(0+)=c_3v_3'(0+)=0,\;\; \sum _{j=2}^3 c_jv_j(0+)-c_1v_1(0-)=0 \Big \}, \end{aligned} \end{aligned}$$

(A.3)

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

$$\begin{aligned} D({\mathcal {H}}_\lambda )= & {} \Big \{(u_j)_{j=1}^3\in H^2({\mathcal {Y}}): c_1u'_1(0-)=c_2u'_2(0+)\nonumber \\= & {} c_3u'_3(0+),\;\; \sum _{j=2}^3 c_ju_j(0+)-c_1u_1(0-)=\lambda c_1u'_1(0-) \Big \}. \end{aligned}$$

(A.4)

Proof

We show initially that the adjoint operator $( {\mathcal {H}}^*, D( {\mathcal {H}}^*))$ of $( {\mathcal {H}}, D( {\mathcal {H}}))$ is given by

$$\begin{aligned} {\mathcal {H}}^*={\mathcal {H}}, \quad D( {\mathcal {H}}^*)=\{(u_j)_{j=1}^ 3\in H^2({\mathcal {Y}}) : c_1u'_1(0-)=c_2u'_2(0+)=c_3u'_3(0+)\}. \end{aligned}$$

(A.5)

Indeed, formally for $\mathbf{u} =(u_1, u_2, u_3), \mathbf{v} =(v_1, v_2, v_3)\in H^2({\mathcal {Y}})$ we have

$$\begin{aligned} \begin{aligned} \langle {\mathcal {H}}{} \mathbf{u} , \mathbf{v} \rangle&=c_1^2u_1(0-)v_1' (0-)- c_1^2u'_1(0-)v_1 (0-) \\&\quad + \sum _{j=2}^ 3c_j^2[u'_{j}(0+)v_j(0+) -u_{j}(0+)v'_j(0+)]+\langle \mathbf{u} ,{\mathcal {H}} \mathbf{v} \rangle \\&\equiv R+ \langle \mathbf{u} ,{\mathcal {H}} \mathbf{v} \rangle . \end{aligned} \end{aligned}$$

(A.6)

Hence, for $\mathbf{u} , \mathbf{v} \in D({\mathcal {H}})$ we obtain $R=0$ in (A.6) and so the symmetric property of ${\mathcal {H}}$. Next, let us denote $D_0^*:=\{(u_j)_{j=1}^ 3\in H^2({\mathcal {G}}) : c_1u'_1(0-)=c_2u'_2(0+)=c_3u'_3(0+)\}$. We shall show that $D_0^*=D( {\mathcal {H}}^*)$. Indeed, from (A.6) we obtain for $\mathbf{v} \in D_0^*$ and $\mathbf{u} \in D( {\mathcal {H}})$ that $R=0$ and so $\mathbf{v} \in D ({\mathcal {H}}^*)$ with ${\mathcal {H}}^*\mathbf{v} = {\mathcal {H}}{} \mathbf{v}$. Let us show the inclusion $D_0^*\supseteq D({\mathcal {H}}^*)$. Take $\mathbf{u} =(u_1, u_2, u_3)\in D({\mathcal {H}})$, then for any $\mathbf{v} =(v_1, v_2, v_3)\in D({\mathcal {H}}^*)$ we have from (A.6)

$$\begin{aligned} \langle {\mathcal {H}} \mathbf{u} , \mathbf{v} \rangle = R+\langle \mathbf{u} ,{\mathcal {H}}^* \mathbf{v} \rangle =\langle \mathbf{u} ,{\mathcal {H}} \mathbf{v} \rangle . \end{aligned}$$

(A.7)

Thus, we arrive for any $\mathbf{u} \in D({\mathcal {H}})$ at the equality

$$\begin{aligned} \sum _{j=2}^ 3c_j^2v'_{j}(0+)u_j(0+) - c_1^2v_1' (0-)u_1(0-)=0. \end{aligned}$$

(A.8)

Next, let us consider $\mathbf{u} =(u_1, u_2, 0)\in D({\mathcal {H}})$. Then $c_1 u_1'(0-)=c_2 u_2'(0+)=0$ and from Eqs. (A.3) and (A.8), we obtain $[c_2v'_2(0+)-c_1v'_1(0-)]c_1 u_1(0-)=0$. Thus, by choosing $u_1(0-)\ne 0$ follows $c_1v'_1(0-)=c_2v'_2(0+)$. Similarly, we have $c_1v'_1(0-)=c_3v'_3(0+)$. Therefore, $\mathbf{v} \in D_0^*$. This shows the relations in (A.5).

Now, from (A.5) we obtain that the deficiency indices for $( {\mathcal {H}}, D( {\mathcal {H}}))$ are $n_{\pm }( {\mathcal {H}})=1$. Indeed, $\ker ({\mathcal {H}}^*\pm iI)=[\Phi _{\pm }]$ with $\Phi _{\pm }$ defined by

$$\begin{aligned} \Phi _{\pm }= \left( \underset{x<0}{e^{\frac{ ik_{\mp }}{c_1}x}}, \underset{x>0}{-e^{\frac{- ik_{\mp }}{c_2}x}}, \underset{x>0}{-e^{\frac{ -ik_{\mp }}{c_3}x}}\right) , \end{aligned}$$

(A.9)

$k^2_{\mp }={\mp } i$, $\mathrm {Im}\,(k_{-})<0$ and $\mathrm {Im}\,(k_{+})<0$. Moreover, $\Vert \Psi _{-}\Vert _{L^ 2({\mathcal {Y}})}=\Vert \Psi _{+}\Vert _{L^ 2({\mathcal {Y}})}$.

Next, from extension theory for symmetric operators we have that every self-adjoint extension $(\widehat{{\mathcal {H}}}, D(\widehat{{\mathcal {H}}}))$ of $({\mathcal {H}}, D({\mathcal {H}}))$ is characterized by $D(\widehat{{\mathcal {H}}})\subset D({\mathcal {H}}^*)$, and $\mathbf{u} =(u_1,u_2,u_3)\in D(\widehat{{\mathcal {H}}})$ if and only if $\sum _{j=2}^3 c_ju_j(0+)-c_1u_1(0-)=\lambda c_1u'_{1}(0-),$ where, for $\theta \in [0, 2\pi )-\{\pi /2\}$,

$$\begin{aligned} \lambda = -\Big (\sum _{j=1}^3 c_j \Big )\frac{1+e^{i\theta }}{e^{-i\frac{\pi }{4}}+e^{i(\theta +\frac{\pi }{4})}} \in {\mathbb {R}}. \end{aligned}$$

Thus, the set of self-adjoint extensions $(\widehat{{\mathcal {H}}}, D(\widehat{{\mathcal {H}}}))$ of the symmetric operator $({\mathcal {H}}, D({\mathcal {H}}))$ can be seen as one-parametrized family $({\mathcal {H}}_\lambda , D({\mathcal {H}}_\lambda ))$ defined by ${\mathcal {H}}_\lambda ={\mathcal {H}}$ and $D({\mathcal {H}}_\lambda )$ by (A.4). This finishes the proof. $\square$

1.2 Morse index calculations

The following result gives a precise value for the Morse-index of the operator ${\mathcal {L}}_{\lambda }$ in (3.31) associated to the kink/anti-kink profile $(\phi _j)$ obtained in Sect. 3.2.1, when we consider the domain ${\mathcal {C}}_1\cap D ({\mathcal {L}}_{\lambda })$, ${\mathcal {C}}_1=\{(u_j)_{j=1}^3\in L^2({\mathcal {Y}}): u_2=u_3\}$. We establish this result in order to possible future study related to the stability properties of kink/anti-kink profiles. We conjecture that these profiles are in fact unstable (see Remark 3.18 and [3]).

Proposition A.7

Let $\lambda \in (-\infty , -\frac{\pi }{2})$ and consider ${\mathcal {B}}_2={\mathcal {C}}_1 \cap D ({\mathcal {L}}_{\lambda })$. Then ${\mathcal {L}}_{\lambda } : {\mathcal {B}}_2\rightarrow {\mathcal {C}}_1$ is well defined and $n( {\mathcal {L}}_{\lambda } |_{{\mathcal {B}}_2} )=2$.

Proof

The proof is based on analytic perturbation theory. By Proposition 3.21 we have for $\Phi '_{\lambda _0}=(2\phi '_1, \phi '_2, \phi '_2)\in {\mathcal {C}}_1$, $\phi _i\equiv \phi _{i, \lambda _0}$, that ${{\,\mathrm{span}\,}}\{\Phi '_{\lambda _0}\}=\ker ({\mathcal {L}}_{\lambda _0} |_{{\mathcal {B}}_2} )$ ($\lambda _0\equiv -\frac{\pi }{2}$). Moreover, by the proof of Proposition 3.22 we have for $\Lambda _1=(\phi '_1, \phi '_2, \phi '_2)\in {\mathcal {C}}_1\cap H^1({\mathcal {Y}})$ that the quadratic form ${\mathcal {Q}}$ in (3.36) satisfies ${\mathcal {Q}}(\Lambda _1)<0$ thus $n({\mathcal {L}}_{\lambda _0} |_{{\mathcal {B}}_2} )=1$. Let us denote by $\chi _{_{\lambda _0}}$ the unique negative eigenvalue for ${\mathcal {L}}_{\lambda _0}|_{{\mathcal {B}}_2}$. We note in this point of the analysis that by using the classical perturbation theory and the convergence ${\mathcal {L}}_{\lambda } \rightarrow {\mathcal {L}}_{\lambda _0}$ as $\lambda \rightarrow \lambda _0$ in the generalized sense (Kato [29]) we obtain that given a closed curve $\Gamma$ such that $\sigma _0=\{\chi _{_{\lambda _0}}, 0\}$ belongs to the inner domain of $\Gamma$, then we can only conclude that $n({\mathcal {L}}_{\lambda } |_{{\mathcal {B}}_2} )\geqq 1$ (by Proposition 3.21) for $\lambda \approx \lambda _0$. So we will need to determine exactly how the eigenvalue zero will move, either to right or to left . In this form, we divide our analysis into several steps:

• i) The mapping $\lambda \in (-\infty , \infty )\rightarrow \Psi _\lambda = (\phi _{i, a_i(\lambda )})$ is real-analytic and we have the convergence $\Psi _\lambda - \Psi _{\lambda _0}\rightarrow 0$, as $\lambda \rightarrow \lambda _0$, in $H^1({\mathcal {Y}})$. Then, it follows that ${\mathcal {L}}_{\lambda }$ converges to ${\mathcal {L}}_{\lambda _0}$ as $\lambda \rightarrow \lambda _0$ in the generalized sense (see proof of Proposition 3.24). Moreover, the family $\{{\mathcal {L}}_\lambda \}_{\lambda \in I}$ represents a real-analytic family of self-adjoint operators of type (B) in the sense of Kato (see [29]) on ${\mathcal {C}}_1$. Therefore, from Theorem IV-3.16 from Kato [29] we obtain $\Gamma \subset \rho ({\mathcal {L}}_{\lambda })$ for $|\lambda -\lambda _0|$ sufficiently small, and $\sigma ({\mathcal {L}}_{\lambda })$ is likewise separated by $\Gamma$ into two parts, such that the part of $\sigma ({\mathcal {L}}_{\lambda })$ inside $\Gamma$ consists of a finite number of eigenvalues with total multiplicity (algebraic) equal to two. Then, it follows from Kato-Rellich’s theorem (see [40]) the existence of two analytic functions, $\Omega$ and $\Pi$, defined in a neighborhood of $\lambda _0$ with $\Omega : (\lambda _0-\delta , \lambda _0+\delta )\rightarrow {\mathbb {R}}$ and $\Pi : (\lambda _0-\delta , \lambda _0+\delta )\rightarrow {\mathcal {C}}_1$ such that $\Omega (\lambda _0)=0$ and $\Pi (\lambda _0)=\Phi '_{\lambda _0}$. For all $\lambda \in (\lambda _0-\delta , \lambda _0+\delta )$, $\Omega (\lambda )$ is the simple isolated second eigenvalue of ${\mathcal {L}}_{\lambda }$, and $\Pi (\lambda )$ is the associated eigenvector for $\Omega (\lambda )$. Moreover, $\delta$ can be chosen small enough to ensure that, for $\lambda \in (\lambda _0-\delta , \lambda _0+\delta )$, the spectrum of ${\mathcal {L}}_{\lambda }$ in ${\mathcal {C}}_1$ is positive, except for, at most, the first two eigenvalues.

• ii) Using Taylor’s theorem, we obtain for sufficiently small $\delta$ the following expansions:

  $$\begin{aligned} \Omega (\lambda )=\beta (\lambda -\lambda _0)+ O(|\lambda -\lambda _0|^2), \;\; \Pi (\lambda )=\Phi '_{\lambda _0} + \Pi '(\lambda _0) (\lambda -\lambda _0) + \mathbf{O} (|\lambda -\lambda _0|^2),\nonumber \\ \end{aligned}$$

  (A.10)

  where $\beta = \Omega '(\lambda _0)$. Moreover, by analyticity we also obtain the expansion

  $$\begin{aligned} \Psi _\lambda -\Psi _{\lambda _0}=(\lambda -\lambda _0) \partial _\lambda \Psi _\lambda |_{\lambda =\lambda _0} + \mathbf{O} (|\lambda -\lambda _0|^2). \end{aligned}$$

  (A.11)

  Next, we determine the sign of $\beta$. Thus, we compute $\langle {\mathcal {L}}_{\lambda } \Pi (\lambda ), \Phi '_{\lambda _0} \rangle$ in two different ways. On one hand, using ${\mathcal {L}}_{\lambda } \Pi (\lambda )= \Omega (\lambda )\Pi (\lambda )$, (A.10) leads to

  $$\begin{aligned} \langle {\mathcal {L}}_{\lambda } \Pi (\lambda ), \Phi '_{\lambda _0} \rangle = \beta (\lambda -\lambda _0)\Vert \Phi '_{\lambda _0}\Vert ^2 + O(|\lambda -\lambda _0|^2). \end{aligned}$$

  (A.12)

  On the other hand, since $\Phi '_{\lambda _0}\in D({\mathcal {L}}_{\lambda } )\cap {\mathcal {C}}_1$ for all $\lambda \ne \lambda _0$ and ${\mathcal {L}}_{\lambda }$ is self-adjoint, we get $\langle {\mathcal {L}}_{\lambda } \Pi (\lambda ), \Phi '_{\lambda _0} \rangle =\langle \Pi (\lambda ), {\mathcal {L}}_{\lambda } \Phi '_{\lambda _0} \rangle$. Now, using the notation $\phi _{i,\lambda }=\phi _{i,a_i(\lambda )}$, we have from (3.13) the relation ${\mathcal {L}}_{\lambda } \Phi '_{\lambda _0} = {\mathcal {L}}_{\lambda _0} \Phi '_{\lambda _0} +{\mathcal {A}} \Phi '_{\lambda _0} ={\mathcal {A}} \Phi '_{\lambda _0}$ where ${\mathcal {A}}$ is the diagonal-matrix ${\mathcal {A}}=\Big ((\cos (\phi _{i,\lambda })-\cos (\phi _{i,\lambda _0}))\delta _{i,k}\Big )$, $1\leqq i,k\leqq 3$. Next, for $G_i(\lambda )=\cos (\phi _{i,\lambda })$ we have the expansion

  $$\begin{aligned} G_i(\lambda )-G_i(\lambda _0)=-\sin (\phi _{i,\lambda _0})\Big ( \partial _\lambda \phi _{i,\lambda }|_{_{\lambda =\lambda _0}}\Big )(\lambda -\lambda _0)+ O(|\lambda -\lambda _0|^2). \end{aligned}$$

  Then ${\mathcal {A}}=\Big ((-\sin (\phi _{i,\lambda _0})\partial _\lambda \phi _{i,\lambda }|_{_{\lambda =\lambda _0}}(\lambda -\lambda _0))\delta _{i,k}\Big ) + \mathbf{O} (|\lambda -\lambda _0|^2)$, $1\leqq i,k\leqq 3$. Thus,

  $$\begin{aligned} \begin{aligned} \langle {\mathcal {L}}_{\lambda } \Pi (\lambda ), \Phi '_{\lambda _0} \rangle&=\langle \Pi (\lambda ), {\mathcal {L}}_{\lambda } \Phi '_{\lambda _0} \rangle =\langle \Phi '_{\lambda _0} + \Pi '(\lambda _0) (\lambda -\lambda _0) + \mathbf{O} (|\lambda -\lambda _0|^2), {\mathcal {A}} \Phi '_{\lambda _0} \rangle \\&=\Big \langle \Phi '_{\lambda _0}, \Big ((-\sin (\phi _{i,\lambda _0})\partial _\lambda \phi _{i,\lambda }|_{_{\lambda =\lambda _0}}(\lambda -\lambda _0))\delta _{i,k}\Big )\Phi '_{\lambda _0} \Big \rangle + O(|\lambda -\lambda _0|^2)\\&\equiv (\lambda -\lambda _0) \eta + O(|\lambda -\lambda _0|^2), \end{aligned}\nonumber \\ \end{aligned}$$

  (A.13)

  with $\eta \equiv 4a_1'(\lambda _0) \int _{-\infty }^0 (\phi '_{1,\lambda _0})^3\sin (\phi _{1,\lambda _0})\, dx+2a_1'(\lambda _0) \int _0^{\infty } (\phi '_{2,\lambda _0})^3\sin (-\phi _{2,\lambda _0}) \, dx$. Next, we obtain the sign of $\eta$. Indeed, from the qualitative properties of the anti-kink profile $\phi _{i,\lambda _0}$ we get immediately that the two integrals above are positive. Next, from the relation in (3.28) we obtain that $a_1'(\lambda _0)=\frac{1}{3}$ (indeed we always have $a_1'(\lambda )>0$). Then $\eta >0$. Next, from (A.12) and (A.13) there follows $\beta =\frac{\eta }{\Vert \Phi '_{\lambda _0}\Vert ^2} + O(|\lambda -\lambda _0|)$. Therefore, from (A.10) there exists $\delta _0>0$ sufficiently small such that $\Omega (\lambda )>0$ for any $\lambda \in (\lambda _0, \lambda _0+\delta _0)$, and $\Omega (\lambda )<0$ for any $\lambda \in (\lambda _0-\delta _0, \lambda _0)$. Thus, in the space ${\mathcal {C}}_1$, the Morse index $n({\mathcal {L}}_\lambda |_{{\mathcal {B}}_2})=1$ for $\lambda >\lambda _0$ and $\lambda \approx \lambda _0$ (equality that is used in Proposition 3.22), and $n({\mathcal {L}}_\lambda |_{{\mathcal {B}}_2})=2$ for $\lambda <\lambda _0$ and $\lambda \approx \lambda _0$.

• iii) Next, since $\ker ({\mathcal {L}}_\lambda )=\{{0}\}$ for $\lambda \ne -\frac{\pi }{2}$, we obtain via a continuation argument based on the Riesz-projection operator (see proof of Proposition 3.24) that for any $\lambda \in (-\infty , -\frac{\pi }{2})$ follows $n({\mathcal {L}}_\lambda |_{{\mathcal {B}}_2})=2$ . This finishes the proof.

$\square$