Abstract
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.
Highlights
We look for a solution to (14) in the form, W (t, τ, s) = U τ, s, sN − c4t eiω5τK+iω6sN with 1 + |ω5|2 − |ω6|2 > 0. (24)
For (15), if we look for a solution with a standard vortex of degree +1 in the form u = w+ := ρ( )eiθ, in polar coordinate (, θ) in R2, ρ satisfies ρ 2ρ(ρ
For given parameters d and small ε, instead of considering (32) we look for a ψ to the projected form (89)
Summary
There exist κ ∈ R and small constant μ > 0 such that (20) and (21) possess the following four types of vortex solutions on the domain ΩK,N which is defined by ΩK,N =. For the proof of Theorems 1.2, we shall construct solutions to these two-dimensional elliptic problems by the standard Lyapunov-Schmidt reduction methods in a way such that each possesses a vortex of degree +1 at (d, 0) and its antipair of degree −1 at (−d, 0). To prove Theorem 1.2 we will use the finite-dimensional Lyapunov-Schmidt reduction method to find solutions to problems such as (32) in Section 2, see Remark 4. For the existence of solutions of the Type C form in (19) with elliptic vortex helix structures to (14) (i.e., (2)) in Theorem 1.2, we shall consider (32) with constants, κ = c3 , μ = ε| log ε| =. Other detailed relations in Theorems 1.2 can be derived
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