2018/05/25 by Xiaosen Han, Han, Xiaosen, Gabriella Tarantello +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Topological Materials and Phenomena
paper · pdf · doi:10.48550/arxiv.1805.09970
openalex publication_date 2018/05/25 · openalex created_date 2018/06/01 · openalex updated_date 2026/07/28
In this paper we study the existence of multiple solutions for the non-Abelian Chern--Simons--Higgs (N× N)-system: Δui=λ(∑j=1N∑k=1N KkjKji\reuj\reuk-∑j=1N Kji\reuj)+4π∑j=1niδ_pij, i=1,…, N; over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of SU(N+1), (see \eqrefk1 below). Here, λ>0 is the coupling parameter, δp is the Dirac measure with pole at p and ni∈ 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 \citehaya1 the authors managed to obtain the existence of one doubly periodic solution via a minimisation procedure, in the spirit of \citenota . Our main contribution in this paper is to show (as in \citenota) 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 Chern--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.