For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a three-dimensional Euclidean space, we construct solutions with various vortex structures (vortex pairs, elliptic/hyperbolic vortex circles, and also elliptic vortex helices). The approaches base on the transformations associated with the symmetries of the nonlinear problems, which will lead to two-dimensional elliptic problems with resolution theory given by the finite-dimensional Lyapunov-Schmidt reduction method in nonlinear analysis. © ...
