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arXiv · 2606.14104

Symmetry breaking for the complex sine-Gordon equation

Shibing Chen·Changfeng Gui·Yong Liu·Jianmin Yang·Wen Yang
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Abstract

We consider the existence of vortex solutions to the complex sine-Gordon II (CSG2) equation, which can be viewed as an analogy of the Ginzburg-Landau (GL) equation. Using the nontrivial kernels $\eta_{\pm}$ of the linearized CSG2 equation at the standard degree-2 vortex solution $\Psi_2$, we show that it bifurcating to a one-parameter family of symmetry-breaking solutions $\Psi_{2,\alpha}$. Explicit formulas of these solutions are also available, from which we propose a new bilinear system for this equation. Our method can be generalized to higher degree case. Nondegenracy and stability of degree-1 solution are also proved. Finally, we formally discuss the Lyapunov-Schmidt reduction procedure for the multivortex solutions of the CSG2 and GL equation.

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Shibing Chen, Changfeng Gui, Yong Liu, Jianmin Yang, Wen Yang. 2026-06-12. Symmetry breaking for the complex sine-Gordon equation. https://arxiv.org/abs/2606.14104

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