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Analytical Issues in the Construction of Self-dual Chern–Simons Vortices
No full textIn this note we discuss the solvability of Liouville-type systems in presence of singular sources, which arise from the study of non-abelian Chern Simons vortices in Gauge Field Theory and their asymptotic behaviour (for limiting values of the physical parameters). This investigation has contributed towards the understanding of singular PDE ’s in Mean Field form, in connection to surfaces with conical singularities, sharp Moser–Trudinger and log(HLS)-inequalities, bubbling phenomena and point-wise profile estimates in terms of Harnack type inequalities. We shall emphasise mostly the physical impact of the rigorous mathematical results established and mention several of the remaining open problems
Selfdual gauge field vortices: an analytical approach
No full textIn modern theoretical physics, gauge field theories are of great importance since they keep internal symmetries and account for phenomena such as spontaneous symmetry breaking, the quantum Hall effect, charge fractionalization, superconductivity and supergravity. This monograph discusses specific examples of gauge field theories that exhibit a selfdual structure.The author builds a foundation for gauge theory and selfdual vortices by introducing the basic mathematical language of the subject and formulating examples ranging from the well-known abelian–Higgs and Yang–Mills models to the Chern–Simons–Higgs theories (in both the abelian and non-abelian settings). Thereafter, the electroweak theory and self-gravitating electroweak strings are also examined, followed by the study of the differential problems that have emerged from the analysis of selfdual vortex configurations; in this regard the author treats elliptic problems involving exponential non-linearities, also in relation to concentration-compactness principles and blow-up analysis.
Many open questions still remain in the field and are examined in this comprehensive work in connection with Liouville-type equations and systems. The goal of this text is to form an understanding of selfdual solutions arising in a variety of physical contexts. Selfdual Gauge Field Vortices: An Analytical Approach is ideal for graduate students and researchers interested in partial differential equations and mathematical physics
Blow-up analysis for a cosmic strings equation
No full textIn this paper we develop a blow-up analysis for solutions of an elliptic PDE of Liouville type over the plane. Such solutions describe the behavior of cosmic strings (parallel in a given direction) for a W-boson model coupled with Einstein's equation. We show how the blow-up behavior of the solutions is characterized, according to the physical parameters involved, by new and surprising phenomena. For example in some cases, after a suitable scaling, the blow-up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the “uniformization” of the Riemann sphere with conical singularities
Uniqueness of selfdual periodic Chern-Simons vortices of topological-type
No full textIn analogy with the abelian Maxwell-Higgs model (cf. Jaffe and Taubes in Vortices and monopoles, 1980) we prove that periodic topological-type selfdual vortex-solutions for the Chern-Simons model of Jackiw-Weinberg [Phys Rev Lett 64:2334-2337, 1990] and Hong et al. Phys Rev Lett 64:2230-2233, 1990 are uniquely determined by the location of their vortex points, when the Chern-Simons coupling parameter is sufficiently small. This result follows by a uniqueness and uniform invertibility property established for a related elliptic problem (see Theorem 3.6 and 3.7)
A harnack inequality for liouville-type equations with singular sources
No full textLet Omega subset of R-2 be a bounded domain, and V epsilon C-0,C-1 (Omega) satisfy: 0 < a <= V <= b, vertical bar del V vertical bar <= A in Omega. For given alpha > 0 and 0 G Q, we show that every solution of the equation: -Delta u = vertical bar z vertical bar(2 alpha)Ve(u), in Omega satisfies: u (0) + infu(Omega) <= C, with a suitable constant C depending only on a, b, A and dist (0, delta Omega). This furnishes a nontrivial extension of an analogous result established by Brezis-Li-Shafrir in [3], in case alpha > 0
On Non-Topological Solutions for Planar Liouville Systems of Toda-Type.
No full textMotivated by the study of non-abelian Chern Simons vortices of non-topological type in Gauge Field Theory, see e.g. Gudnason (Nucl Phys B 821:151–169, 2009), Gudnason (Nucl Phys B 840:160–185, 2010) and Dunne (Lecture Notes in Physics, New Series, vol 36. Springer, Heidelberg, 1995), we analyse the solvability of the following (normalised) Liouville-type system in the presence of singular sources:
(1)τ⎧⎩⎨⎪⎪−Δu1=eu1−τeu2−4Nπδ0,−Δu2=eu2−τeu1,β1=12π∫R2eu1andβ2=12π∫R2eu2,
with τ>0 and N>0 .
We identify necessary and sufficient conditions on the parameter τ and the “flux” pair: (β1,β2), which ensure the radial solvability of (1)τ.
Since for τ=12, problem (1)τ reduces to the (integrable) 2 × 2 Toda system, in particular we recover the existence result of Lin et al. (Invent Math 190(1):169–207, 2012) and Jost and Wang (Int Math Res Not 6:277–290, 2002), concerning this case.
Our method relies on a blow-up analysis for solutions of (1)τ , which (even in the radial setting) takes new turns compared to the single equation case.
We mention that our approach also permits handling the non-symmetric case, where in each of the two equations in (1)τ , the parameter τ is replaced by two different parameters τ1>0 and τ2>0 respectively, and also when the second equation in (1)τ includes a Dirac measure supported at the origin
Vortex condensates for the SU(3) Chern-Simons theory
No full textWe investigate SU(3)-periodic vortices in the self-dual Chern-Simons theory proposed by Dunne in [13, 15]. At the first admissible non-zero energy level E = 2 pi, and for each (broken and unbroken) vacuum state phi((0)) of the system, we find a family of periodic vortices asymptotically gauge equivalent to phi((0)), as the Chern-Simons coupling parameter k -> 0. At higher energy levels, we show the existence of multiple gauge distinct periodic vortices with at least one of them asymptotically gauge equivalent to the (broken) principal embedding vacuum, when k -> 0
Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations
No full textIn this paper we study the existence of multiple solutions for the non-Abelian Chem-Simons-Higgs (N x N)-system:Delta u(i) = lambda(Sigma(N)(j=1) Sigma(N)(k=1) K(kj)K(ji)e(uj)e(uk) - Sigma(N)(j=1) k(ji)e(uj)) + 4 pi Sigma(ni)(j=1) delta(pij), i=1, . . . , N;over a doubly periodic domain Omega, with coupling matrix K given by the Cartan matrix of SU(N + 1), (see (1.2) below). Here, lambda > 0 is the coupling parameter, delta(p )is the Dirac measure with pole at p and n(i) is an element of N, for i = 1,...,N. When N = 1, 2 many results are now available for the periodic solvability of such system and provide the existence of different classes of solutions known as: topological, non-topological, mixed and blow-up type. On the contrary for N >= 3, only recently in [27] the authors managed to obtain the existence of one doubly periodic solution via a minimization procedure, in the spirit of [46]. Our main contribution in this paper is to show (as in [46]) that actually the given system admits a second doubly periodic solutions of "Mountain-pass" type, provided that 3 <= N <= 5. Note that the existence of multiple solutions is relevant from the physical point of view. Indeed, it implies the co-existence of different non-Abelian Chem-Simons condensates sharing the same set (assigned component-wise) of vortex points, energy and fluxes. The main difficulty to overcome is to attain a "compactness" property encompassed by the so-called Palais-Smale condition for the corresponding "action" functional, whose validity remains still open for N >= 6. (C) 2019 Elsevier Masson SAS. All rights reserved
On a sharp Sobolev-type inequality on two-dimensional compact manifolds. Arch. Ration. Mech. Anal. 145 (1998), no. 2, 161–195.
No full textMotivated by the asymptotic analysis of double vortex condensates in the Chern-Simons-Higgs theory, we construct a suitable minimizing sequence for a sharp Sobolev inequality "a la MOSER" for two-dimensional compact manifolds. As a consequence, we first obtain a direct proof of the sharp character of such an inequality. Secondly, and more interestingly, we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum
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