Vortex structures for some geometric flows from pseudo-Euclidean spaces

Abstract

For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a threedimensional 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. ∂ t m = m × K,N m + |∇ K,N m| 2 m − m 3 e 3 + m 2 3 m , (2) and their stationary cases for the unknown maps m(τ, s

On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system

“…More details about the subject for the single complex component Ginzburg-Landau equation are shown in the literature [11], [9], [19], [21], [23] and [25]. For more results, the reader can refer to [19], [24], [13] and the references therein.…”

The helical vortex filaments of Ginzburg-Landau system in ${\mathbb R}^3$

“…In classical fluid and Bose-Einstein condensates, experiments and numerical methods show that the possible existence of arbitrary number of interacting helical vortex filaments, see [5,22]. The reader can also refer to [13,14,24,32,37] for the constructions of solutions with helical vortex structure, involving Schrödinger map equation, Gross-Pitaevskii equation and Euler equation.…”