Traveling vortex helices for Schrödinger map equations

Wei, Juncheng; Yang, Jun

doi:10.1090/tran/6379

Cited by 16 publications

(17 citation statements)

References 40 publications

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“…We have constructed vortex structures of 0-dimension(vortex pairs of degrees ±1) for K = 1, N = 2 and 1-dimension(elliptic vortex circles/helices) for K = 1, N = 3. Moreover, the existence of vortex structures of elliptic vortex circles/helices can be extended to higher dimensions, see results in [11] and [16]. For the case of K = 2, N = 2, we only find solutions with hyperbolic vortex circles, other than hyperbolic vortex helices due to technical reasons, see Remark 3.…”

Section: Ruiqi Jiang Youde Wang and Jun Yangmentioning

confidence: 87%

“…Solutions constructed of this form are called traveling vortex rings for the case of the dimension N ≥ 3. Later on, J. Wei and J. Yang [16] concerned the existence of traveling wave solutions with invariance under skew motions and possessing vortex helices for (8). Under the stereographic projection, setting…”

Section: Ruiqi Jiang Youde Wang and Jun Yangmentioning

confidence: 99%

“…For simplicity, in the rest part of the present paper, we only sketch the resolution arguments for (32), which basically follow the methods in [16] for the existence of elliptic vortex helices. From the viewpoint of analysis, we only have one more term than equation (3.12) in [16].…”

Section: Remarkmentioning

confidence: 99%

“…M. del Pino, M. Kowalczyk and M. Musso [2] were the first to use this procedure to study the Ginzburg-Landau equation for the existence of vortices in a bounded domain. The methods in the present paper basically follow those in dealing with vortex phenomena for Schrödinger map equations in [11] and [16]. Here are the main steps of the approach.…”

Section: 2mentioning

confidence: 99%

“…One of the main objectives of this section is to compute the error S[v 0 ], as those done in [16]. For the convenience of reader, we here sketch the computations and only show the method for dealing with the singular terms.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Vortex structures for some geometric flows from pseudo-Euclidean spaces

Jiang¹,

Wang²,

Yang³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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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

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Section: Ruiqi Jiang Youde Wang and Jun Yangmentioning

confidence: 87%

Section: Ruiqi Jiang Youde Wang and Jun Yangmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

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Vortex structures for some geometric flows from pseudo-Euclidean spaces

Jiang¹,

Wang²,

Yang³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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Existence and Uniqueness of Local Regular Solution to the Schrödinger Flow from a Bounded Domain in $${\mathbb {R}}^3$$ into $${\mathbb {S}}^2$$

Chen

Wang

2023

Commun. Math. Phys.

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Travelling helices and the vortex filament conjecture in the incompressible Euler equations

et al. 2022

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We consider the Euler equations in $$\mathbb R^3$$ R 3 expressed in vorticity form $$\begin{aligned} \left\{ \begin{array}{l} \vec \omega _t + (\mathbf{u}\cdot {\nabla } ){\vec \omega } =( \vec \omega \cdot {\nabla } ) \mathbf{u} \\ \mathbf{u} = \mathrm{curl}\vec \psi ,\ -\Delta \vec \psi = \vec \omega . \end{array}\right. \end{aligned}$$ ω → t + ( u · ∇ ) ω → = ( ω → · ∇ ) u u = curl ψ → , - Δ ψ → = ω → . A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of helical filaments, associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple polygonal helical filaments travelling and rotating together.

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Traveling vortex helices for Schrödinger map equations

Cited by 16 publications

References 40 publications

Vortex structures for some geometric flows from pseudo-Euclidean spaces

Vortex structures for some geometric flows from pseudo-Euclidean spaces

Existence and Uniqueness of Local Regular Solution to the Schrödinger Flow from a Bounded Domain in $${\mathbb {R}}^3$$ into $${\mathbb {S}}^2$$

Travelling helices and the vortex filament conjecture in the incompressible Euler equations

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