Bubbling solutions of mixed type for a general non-Abelian Chern–Simons–Higgs system of rank 2 over a torus. (January 2022) - Huang, Hsin-Yuan; Lee, Youngae; Moon, Sang-Hyuck
Abstract: We consider a general non-Abelian Chern–Simons–Higgs system of rank 2 (0.1) Δ u i + 1 ɛ 2 ∑ j = 1 2 K j i e u j − ∑ j = 1 2 ∑ k = 1 2 K k j K j i e u j e u k = 4 π ∑ l = 1 N m i, l δ p l ( i = 1, 2 ) over a flat torus, where m 1, l ≥ 0, m 2, l ≥ 0, ( m 1, l, m 2, l ) ≠ ( 0, 0 ) for l = 1, …, N, δ p is the Dirac measure at p, K is a non-degenerate 2 × 2 matrix of the form K = 1 + a − a − b 1 + b . When a > − 1, b > − 1, and a + b > − 1, Eqs (0.1) are expected to have three types solutions: topological, non-topological and mixed type solutions. Concerning the existence of various type solutions, there are requirements that a > 0 and b > 0, or a and b are close to 0 in the literature. It is still open for generic a and b . We partially answer this question and show that (0.1) possesses bubbling mixed type solutions provided that ɛ is small and ( a, b ) satisfies (1.17).