Topologically Stratified Energy Minimizers in a Product Abelian Field Theory

We study a recently developed product Abelian gauge field theory by Tong and Wong hosting magnetic impurities. We first obtain a necessary and sufficient condition for the existence of a unique solution realizing such impurities in the form of multiple vortices. We next reformulate the theory into an extended model that allows the coexistence of vortices and anti-vortices. The two Abelian gauge fields in the model induce two species of magnetic vortex-lines resulting from $N_s$ vortices and $P_s$ anti-vortices ($s=1,2$) realized as the zeros and poles of two complex-valued Higgs fields, respectively. An existence theorem is established for the governing equations over a compact Riemann surface $S$ which states that a solution with prescribed $N_1, N_2$ vortices and $P_1,P_2$ anti-vortices of two designated species exists if and only if the inequalities $$\left|N_1+N_2-(P_1+P_2)\right|<\frac{|S|}{\pi},\quad \left|N_1+2N_2-(P_1+2P_2)\right|<\frac{|S|}{\pi},$$ hold simultaneously, which give bounds for the `differences' of the vortex and anti-vortex numbers in terms of the total surface area of $S$. The minimum energy of these solutions is shown to assume the explicit value $$E= 4\pi (N_1+N_2+P_1+P_2),$$ given in terms of several topological invariants, measuring the total tension of the vortex-lines.Comment: 22 page

Multiple Solutions for the Non-Abelian Chern--Simons--Higgs Vortex Equations

In this paper we study the existence of multiple solutions for the non-Abelian Chern--Simons--Higgs $(N\times N)$-system: \Delta u_i=\lambda\left(\sum_{j=1}^N\sum_{k=1}^N K_{kj}K_{ji}\re^{u_j}\re^{u_k}-\sum_{j=1}^N K_{ji}\re^{u_j}\right)+4\pi\sum_{j=1}^{n_i}\delta_{p_{ij}},\quad i=1,\dots, N; over a doubly periodic domain $\Omega$, with coupling matrix $K$ given by the Cartan matrix of $SU(N+1),$ (see \eqref{k1} below). Here, $\lambda>0$ is the coupling parameter, $\delta_p$ is the Dirac measure with pole at $p$ and $n_i\in \mathbb{N},$ for $i=1, \dots, N.$ 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 $N\ge 3,$ only recently in \cite{haya1} the authors managed to obtain the existence of one doubly periodic solution via a minimisation procedure, in the spirit of \cite{nota} . Our main contribution in this paper is to show (as in \cite{nota}) that actually the given system admits a second doubly periodic solutions of "Mountain-pass" type, provided that $3\le N\le 5$. 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 $N\ge 6$.Comment: 34 page

Spatial ceasing and decay of solutions to nonlinear hyperbolic equations with nonlinear boundary conditions

AbstractIn this work we study the spatial behavior of solutions to some nonlinear hyperbolic equations with nonlinear boundary conditions. Under suitable conditions, by using the weighted energy method, we prove that the solutions either cease to exist for a finite value of the spatial variable or decay algebraically in the spatial variable

Existence of U(1)U(1) Gauged Q-balls for A Field Model with Sixth-order Potential

Q-balls are non-topological solitons in a large family of field theories. We focus on the existence of $U(1)$ gauged Q-balls for a field theory with sixth-order potential. The problem can be reduced to proving the existence of critical points for some indefinite functional. For this, we use a constrained minimization approach to obtain the existence of critical points. Moreover, we establish some qualitative properties of the Q-ball solution, such as monotonicity, boundedness and asymptotic behavior

Chern--Simons Vortices in the Gudnason Model

We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the ${\cal N}=2$ supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern--Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an-inequality constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.Comment: 39 page