Instability of Static Solutions of the sine-Gordon Equation on a ${\mathcal {Y}}$-Junction Graph with $\delta$-Interaction

Abstract

The aim of this work is to establish a linear instability result of static, kink and kink/anti-kink soliton profile solutions for the sine-Gordon equation on a metric graph with a structure represented by a ${\mathcal {Y}}$-junction. The model considers boundary conditions at the graph-vertex of $\delta$-interaction type. It is shown that kink and kink/anti-kink soliton type static profiles are linearly (and nonlinearly) unstable. For that purpose, a linear instability criterion that provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is established. As a result, the linear stability analysis depends upon the spectral study of this linear operator and of its Morse index. The extension theory of symmetric operators, Sturm–Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. A comprehensive study of the local well-posedness of the sine-Gordon model in ${\mathcal {E}}({\mathcal {Y}}) \times L^2({\mathcal {Y}})$ where ${\mathcal {E}}({\mathcal {Y}}) \subset H^1({\mathcal {Y}})$ is an appropriate energy space, is also established. The theory developed in this investigation has prospects for the study of the instability of static wave solutions of other nonlinear evolution equations on metric graphs.

Appendix A

For convenience of the reader, and because of the use of non-standard techniques along the manuscript, in this section we formulate the extension theory of symmetric operators suitable for our needs (see (Naimark 1967, 1968) for further information). The following result is classical and can be found in Reed and Simon (1975), p. 138.

Theorem A. 1

(von-Neumann decomposition) 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 see Reed and Simon, vol. 2, chapter X Reed and Simon (1975).

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 (1968) (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 proposition is the main result of this appendix and characterizes all self-adjoint extensions of the symmetric operator under consideration. It plays a key role in the proof of Proposition  4.4.

Proposition A. 6

Consider the closed symmetric operator densely defined on $L^2({\mathcal {Y}})$, $( {\mathcal {M}}, D( {\mathcal {M}}))$, by

$$\begin{aligned} {\mathcal {M}}= & {} \Big (\Big (-c_j^2\frac{d^2}{dx^2}\Big )\delta _{j,k} \Big ),\;1\leqq j, k\leqq 3,\nonumber \\ D({\mathcal {M}})= & {} \Big \{(v_j)_{j=1}^3\in H^2({\mathcal {Y}}): v_1(0-)=v_2(0+)=v_3(0+)=0,\;\;\nonumber \\&\sum \limits _{j=2}^3 c_j^2v_j'(0+)-c_1^2v_1'(0-)=0 \Big \}, \end{aligned}$$

(A.3)

with $\delta _{j,k}$ being the Kronecker symbol. Then, the deficiency indices are $n_{\pm }( {\mathcal {M}})=1$. Therefore, we have that all the self-adjoint extensions of $( {\mathcal {M}}, D( {\mathcal {M}}))$, namely $({\mathcal {J}}_Z, D({\mathcal {J}}_Z))$, $Z\in {\mathbb {R}}$, are defined by ${\mathcal {J}}_Z\equiv {\mathcal {M}}$ and $D({\mathcal {J}}_Z)$ by (1.11).

Proof

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

$$\begin{aligned} {\mathcal {M}}^*={\mathcal {M}}, \quad D( {\mathcal {M}}^*)=\{u\in H^2({\mathcal {Y}}) : u_1(0-)=u_2(0+)=u_3(0+)\}. \end{aligned}$$

(A.4)

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

$$\begin{aligned} \langle {\mathcal {M}}{\varvec{v}}, {\varvec{u}}\rangle&=- c_1^2v_1' (0-)u_1(0-)+ c_1^2v_1 (0-)u'_1(0-) \nonumber \\&\quad + \sum _{j=2}^ 3c_j^2v'_{j}(0+)u_j(0+) - \sum _{j=2}^ 3c_j^ 2v_{j}(0+)u'_j(0+)\nonumber \\&\quad +\langle {\varvec{v}},{\mathcal {M}} {\varvec{u}}\rangle . \end{aligned}$$

(A.5)

From (A.5), we obtain immediately for ${\varvec{u}}=(u_1, u_2, u_3), {\varvec{v}}=(v_1, v_2, v_3)\in D({\mathcal {M}})$ the symmetric property of ${\mathcal {M}}$. Next, we denote by $D^*=\{u\in H^2({\mathcal {Y}}) : u_1(0-)=u_2(0+)=u_3(0+)\}$ and we will show $D^*=D( {\mathcal {M}}^*)$. Indeed, from (A.5) we obtain for ${\varvec{u}}\in D^*$ and ${\varvec{v}}\in D( {\mathcal {M}})$ that $\langle {\mathcal {M}}{\varvec{v}}, {\varvec{u}}\rangle = \langle {\varvec{v}}, {\mathcal {M}}{\varvec{u}}\rangle$ and so ${\varvec{u}}\in D ({\mathcal {M}}^*)$ with ${\mathcal {M}}^*{\varvec{u}}= {\mathcal {M}}{\varvec{u}}$. Let us show the inverse inclusion $D^*\supseteq D({\mathcal {M}}^*)$. Take $\varvec{u}=(u_1, u_2, u_3)\in D({\mathcal {M}}^*)$, then for any $\varvec{v}=(v_1, v_2, v_3)\in D({\mathcal {M}})$ we have from (A.5)

$$\begin{aligned} \langle {\mathcal {M}} {\varvec{v}}, {\varvec{u}}\rangle&=- c_1^2v_1' (0-)u_1(0-)+ \sum _{j=2}^ 3c_j^2v'_{j}(0+)u_j(0+) +\langle \varvec{v},{\mathcal {M}} {\varvec{u}}\rangle \nonumber \\&=\langle {\varvec{v}},{\mathcal {M}}^* \varvec{u}\rangle =\langle {\varvec{v}},{\mathcal {M}} {\varvec{u}}\rangle . \end{aligned}$$

(A.6)

Thus, we arrive for any ${\varvec{v}}\in D({\mathcal {M}})$ 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.7)

Next, it consider ${\varvec{v}}=(v_1, 0, v_3)\in D({\mathcal {M}})$ then $c_1^2 v_1'(0-)=c_3^2 v_3'(0+)$. Then, from (A.7) we obtain

$$\begin{aligned}{}[u_3(0+)-u_1(0-)]c_1^ 2 v_1'(0-)=0. \end{aligned}$$

So, by choosing $v_1'(0-)\ne 0$ we obtain $u_1(0-)=u_3(0+)$. Then, (A.7) is reduced to $[u_2(0+)-u_1(0-)]c_2^ 2v_2'(0+)=0$. Therefore, by choosing ${\varvec{v}}=(v_1, v_2, v_3)\in D({\mathcal {M}})$ with $v_2'(0+)\ne 0$ we conclude that ${\varvec{u}}\in D^*$. This shows relations in (A.4).

From (A.4) we obtain that the deficiency indices for $( {\mathcal {M}}, D( {\mathcal {M}}))$ are $n_{\pm }( {\mathcal {M}})=1$. Indeed, $\ker ({\mathcal {M}}^*\pm iI)=[\Psi _{\pm }]$ with $\Psi _{\pm }$ defined by

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

(A.8)

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

Next, let us show that the domain of any self-adjoint extension $\widehat{{\mathcal {M}}}$ of the operator $({\mathcal {M}}, D({\mathcal {M}}))$ in (A.3) (and acting on complex-valued functions) is given by $D(\widehat{{\mathcal {M}}})=D({{\mathcal {J}}_Z})$ in (1.10). Indeed, we recall from extension theory for symmetric operator that $D(\widehat{{\mathcal {M}}})$ is a restriction of $D({{\mathcal {M}}}^*)$ (von-Neumann decomposition above), so $D(\widehat{{\mathcal {M}}})\subset D({{\mathcal {M}}}^*)$ (continuity at the vertex $\nu =0$). Next, due to Proposition A. 4 follows

$$\begin{aligned} D(\widehat{{\mathcal {M}}})=\left\{ \varvec{u}\in H^2({\mathcal {Y}}), \, {\varvec{u}}= u_0+ \zeta \Psi _{-} + \zeta e^{i\theta }\Psi _{+}:\, u_0\in D({\mathcal {M}}), \zeta \in {\mathbb {C}},\theta \in [0,2\pi )\right\} , \end{aligned}$$

Thus, it is easily seen that for ${\varvec{u}}=(u_1,u_2, u_3)\in D(\widehat{{\mathcal {M}}})$, we have

$$\begin{aligned} \sum \limits _{j=2}^3 c_j^2u_j'(0+)-c_1^2u_1'(0-)= - \zeta \sum \limits _{j=1}^3 c_j \Big (e^ {i\frac{\pi }{4}}+e^ {i(\theta -\frac{\pi }{4})}\Big ) ,\;\; u_{1}(0-)= \zeta (1+e^{i\theta }). \end{aligned}$$

(A.9)

From the last equalities, it follows that

$$\begin{aligned} \sum \limits _{j=2}^3 c_j^2u_j'(0+)-c_1^2u_1'(0-)&=Zu_{1}(0-) ,\,\, \text {where}\,\, Z=Z(\theta )\nonumber \\&= -\sum \limits _{j=1}^3 c_j \frac{e^{i\frac{\pi }{4}}+e^{i(\theta -\frac{\pi }{4})}}{1+e^{i\theta }}\in {\mathbb {R}}, \end{aligned}$$

(A.10)

with $\theta \in [0, 2\pi )-\{\pi \}$. Thus, the set of self-adjoint extensions $(\widehat{{\mathcal {M}}}, D(\widehat{{\mathcal {M}}}))$ of the symmetric operator $({\mathcal {M}}, D({\mathcal {M}}))$ can be seen as one-parametrized family $({\mathcal {J}}_Z, D({\mathcal {J}}_Z))$ defined by (1.11). This finishes the proof. $\square$