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In this paper, we prove the existence of finite-energy electrically and magnetically charged vortex solutions in the full Chern–Simons–Higgs theory, for which both the Maxwell term and the Chern–Simons term are present in the Lagrangian density. We consider both Abelian and non-Abelian cases. The solutions are smooth and satisfy natural boundary conditions. Existence is established via a constrained minimization procedure applied on indefinite action functionals. This work settles a long-standing open problem concerning the existence of dually charged vortices in the classical gauge field Higgs model minimally extended to contain a Chern–Simons term.

The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of ‘ideal’ knots. In this paper, we show that the energy infimum E N stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration ( n ≥1) in general dimensions obeys the sharp fractional-exponent growth law , where the exponent p is universally rendered as , which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.

The nonlinear non-local elliptic equation governing the deflection of charged plates in electrostatic actuators is studied under the pinned and the clamped boundary conditions. Results concerning the existence, construction and approximation, and behaviour of classical and singular solutions with respect to the variation of physical parameters of the equation in various situations are presented.

In this paper we develop in all dimensions an existence theory for the Efros energy–domain self–consistent equation governing the density of single–electron states. When the energy domain extends to infinity, we prove that solutions exist except in dimension one. When the energy domain is finite, we prove that solutions exist in all dimensions. The boundary condition near the Fermi level makes the integral diverge. Such a difficulty can be overcome by using a perturbed equation as an approximation and then passing to the zero perturbation limit.

An Abelian gauge field theory framed on a complex line bundle L over a compact Riemann surface M is developed which allows the coexistence, simultaneously in the same model, of magnetic vortices and antivortices represented by the N zeros and P poles of a section of L. The quantized minimum energy E is given in terms of the first Chern class c 1 (L) and by a certain intersection number obtained from the multivortices. We show that E = 2π(N + P). To realize such topological invariants as minimum energies, an existence and uniqueness theorem is established under the necessary and sufficient condition that |c 1 (L)| = |N − P

Locally concentrated solutions in the Born–Infeld theory are presented. In particular, existence and uniqueness theorems are established for multicentred magnetic string solutions induced from a Higgs field over a closed Riemann surface or a Euclidean plane. On any given compact surface, the Born–Infeld parameter may be adjusted under a necessary and sufficient condition to allow the existence of an arbitrarily large number of strings.

We show that there are soliton–like solutions representing arbitrarily located M vortices and N antivortices with opposite magnetic charges in an Abelian gauge field theory. We establish an existence and uniqueness theorem and prove that the total magnetic flux is proportional to M − N but the total energy is proportional to M + N . In the presence of Einstein's gravity, we establish an existence theorem under a necessary and sufficient condition that ensures the geodesic completeness of the corresponding gravitational metric. The coexisting vortices and antivortices are now cosmic strings and antistrings and the total magnetic flux and matter–gauge energy are given by similar expressions as those for vortices. Besides the energy, the total curvature also provides an exact, quantized, measurement of the number of the strings of the two types.

This work establishes the existence of electrically and magnetically charged static particle–like solutions known as dyons in the Weinberg–Salam theory for the unified electromagnetic and weak interactions. The solutions are obtained through a constrained variational study of the associated action functional which assumes an indefinite form due to the presence of the electric field.

In this paper we construct static cosmic string solutions in the Bogomol9nyi phase of the Abelian Higgs model in an S 2 setting. The strings are located at the north and south poles and have identical winding numbers. This construction, in particular, solves an open problem regarding the existence of a 2–string solution on S 2 . Furthermore, we prove that there are no symmetric solutions realizing N strings superimposed at a single point. Such a result implies, in particular, that there is no symmetric 1–string solution on S 2 and is the only known non–existence theorem for cosmic strings confined on a compact surface. The symmetry of the underlying system enables us to work on a nonlinear ordinary differential equation subject to some prescribed boundary data at t = ±∞ A shooting method is employed as an analytic tool in the study of the existence problem.

We study radially symmetric solutions of a nonlinear elliptic partial differential equation in R 2 with critical Sobolev growth, i. e. the nonlinearity is of exponential type. This problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology. Our results lead to many interesting implications for the physical problems considered. For example, for the self-dual Chern–Simons theory, we are able to conclude that the electric charge, magnetic flux, or energy of a non-topological N -vortex solution may assume any prescribed value above an explicit lower bound. For the Einstein-matter-gauge equations, we find a necessary and sufficient condition for the existence of a self-dual cosmic string solution. Such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant.