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.

这项工作的目的是在度量图上用\（{\ mathcal {Y} \）-交界处。模型考虑\（\ delta \）的图顶点处的边界条件-互动类型。结果表明，扭结和扭结/反扭结孤子类型的静态轮廓是线性（和非线性）不稳定的。为此，建立了线性不稳定性标准，该标准为波周围的线性化算子提供了足够的条件，以使其具有一对真实的正/负本征值。结果，线性稳定性分析取决于该线性算子及其莫尔斯指数的谱研究。对称算子的扩展理论，Sturm-Liouville振荡结果以及算子的解析扰动理论是稳定性分析的基本要素。对\（{\ mathcal {E}}（{\ mathcal {Y}}）\ times L ^ 2（{\ mathcal {Y}}）\）中正弦Gordon模型的局部适定性的全面研究在哪里\（{\ mathcal {E}}（{\ mathcal {Y}}）\ subset H ^ 1（{\ mathcal {Y}}）\）是一个合适的能量空间，它也得以建立。本研究开发的理论具有研究度量图上其他非线性发展方程的静波解的不稳定性的前景。