Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity

Yang, Jun

doi:10.3934/dcds.2014.34.2359

“…A solvability theory for the linearized operator is of crucial importance in the use of singular perturbation methods for the construction of various solutions with much more complicated vortex structures of problems where the rescaled vortex w provide a canonical profile. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Duan¹,

Yang²

2021

DCDS

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the radially symmetric solution w(x) = (w + , w − ) : R 2 → C 2 of degree pair (1, 1) was given by A. Alama and Q. Gao in J. Differential Equations 255 (2013), 3564-3591. We will concern its linearized operator L around w and prove the non-degeneracy result under one more assumption B < 0: the kernel of L is spanned by the functions ∂w ∂x 1 and ∂w ∂x 2 in a natural Hilbert space. As an application of the non-degeneracy result, a solvability theory for the linearized operator L will be given.

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“…A solvability theory for the linearized operator is of crucial importance in the use of singular perturbation methods for the construction of various solutions with much more complicated vortex structures of problems where the rescaled vortex w provide a canonical profile. More details about the subject for the single complex component Ginzburg-Landau equation are shown in the literature [15], [17], [19] and [20].…”

Section: Now We Give the Main Resultsmentioning

confidence: 99%