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

Autor: Ramón G. Plaza, Jaime Angulo Pava
Rok vydání: 2021
Předmět:
Zdroj: Repositório Institucional da USP (Biblioteca Digital da Produção Intelectual)
Universidade de São Paulo (USP)
instacron:USP
ISSN: 1432-1823
0025-5874
DOI: 10.1007/s00209-021-02899-0
Popis: 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.
Databáze: OpenAIRE
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