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arXiv · 2606.30204

Bifurcation and global continuation of travelling-rotating Schr\"odinger maps on the sphere

Juan Carlos Sampedro·Luis Vega
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Abstract

We study travelling-rotating solutions of the Schr\"odinger map equation into the sphere, viewed as tangent profiles of rigid vortex filaments. Two first integrals reduce the profile equation to a scalar cubic equation for the vertical component, giving an elliptic-function description and explicit closure conditions. We prove bifurcation from the equatorial branch at $\lambda_k=R\sqrt{k^2-1}$, $k\ge2$, and establish a global continuation alternative inside the regular non-polar class. The possible boundary mechanisms are pole contact, vertical collapse, and double-root degeneration. Numerical continuation of the equatorial branches suggests convergence to the north-pole boundary. Up to gauge, the reconstructed vortex filaments are of Kida type.

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Juan Carlos Sampedro, Luis Vega. 2026-06-29. Bifurcation and global continuation of travelling-rotating Schr\"odinger maps on the sphere. https://arxiv.org/abs/2606.30204

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