On self-similar global textures in an expanding universe

Åminneborg, Stefan; Bergström, Lars

doi:10.1016/0370-2693(95)01156-k

Cited by 6 publications

(18 citation statements)

References 14 publications

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“…If we consider Gaussian Pulse data for various fixed values of R 0 and δ, we find that as A approaches its critical value A * , the corresponding solution approaches a particular self-similar solution. This critical solution is not the solution found by Turok and Spergel; rather it appears to be one of the sequence of self-similar solutions discovered by Aminneborg and Bergstrom [8], and subsequently Bizon [9]. These regular, self-similar solutions obey Eq.…”

Section: Self-similar Solutions At Criticalitymentioning

confidence: 64%

Critical phenomena in nonlinear sigma models

Liebling

Hirschmann

Isenberg

2000

Journal of Mathematical Physics

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We consider solutions to the nonlinear sigma model (wave maps) with target space S 3 and base space 3 + 1 Minkowski space, and we find critical behavior separating singular solutions from nonsingular solutions. For families of solutions with localized spatial support a self-similar solution is found at the boundary. For other families, we find that a static solution appears to sit at the boundary. This behavior is compared to the black hole critical phenomena found by 11.10.Lm, 11.27.+d

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Section: Self-similar Solutions At Criticalitymentioning

confidence: 64%

Critical phenomena in nonlinear sigma models

Liebling

Hirschmann

Isenberg

2000

Journal of Mathematical Physics

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“…To prepare the ground for the proof of Theorem 1 we first discuss some basic properties of solutions of equation (9). It is convenient to use new variables defined by…”

Section: Proof Of Theoremmentioning

confidence: 99%

Equivariant Self-Similar Wave Maps from Minkowski Spacetime into 3-Sphere

Bizoń

2000

Communications in Mathematical Physics

104

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We prove existence of a countable family of spherically symmetric self-similar wave maps from 3 + 1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the ground state wave map found previously by Shatah. The first excitation is particularly interesting in the context of the Cauchy problem since it plays the role of a critical solution sitting at the threshold of singularity formation. We analyze the linear stability of our wave maps and show that the number of unstable modes about a given map is equal to its excitation index. Finally, we formulate a condition under which these results can be generalized to higher dimensions.

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“…The previous results in spherical symmetry indicated that all initial data of compact support, when tuned, approached the same universal, self-similar solution [2]. This critical solution happens to be one of a family of solutions found by Aminneborg and Bergstrom by assuming self-similarity and solving the resulting ODE [12]. A linear perturbation analysis revealed the n = 1 solution to have the requisite single unstable mode necessary to be a proper critical solution (a so-called intermediate attractor).…”

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confidence: 58%

Singularity threshold of the nonlinear sigma model using 3D adaptive mesh refinement

Liebling

2002

Phys. Rev. D

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Numerical solutions to the nonlinear sigma model, a wave map from 3ϩ1 Minkowski space to S 3 , are computed in three spatial dimensions using adaptive mesh refinement. For initial data with compact support the model is known to have two regimes: one in which regular initial data forms a singularity and another in which the energy is dispersed to infinity. The transition between these regimes has been shown in spherical symmetry to demonstrate threshold behavior similar to that between black hole formation and dispersal in gravitating theories. Here, I generalize the result by removing the assumption of spherical symmetry. The evolutions suggest that the spherically symmetric critical solution remains an intermediate attractor separating the two end states. a ϭ ͩ sin ͑x,y,z,t ͒sin sin sin ͑x,y,z,t ͒sin cos sin ͑x,y,z,t ͒cos cos ͑x,y,z,t ͒ ͪ , ͑1͒where and are the usual spatial angles. The dynamics reduce to the scalar field (x,y,z,t) which satisfies the equation of motion ϭ ,xx ϩ ,yy ϩ ,zz Ϫ sin 2 r 2 , ͑2͒ RAPID COMMUNICATIONS

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On self-similar global textures in an expanding universe

Cited by 6 publications

References 14 publications

Critical phenomena in nonlinear sigma models

Critical phenomena in nonlinear sigma models

Equivariant Self-Similar Wave Maps from Minkowski Spacetime into 3-Sphere

Singularity threshold of the nonlinear sigma model using 3D adaptive mesh refinement

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