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In this chapter we study field equations arising from the classical Born—Infeld electromagnetic theory, a topic of current research activities in theoretical physics. In §12.1, we introduce the Born—Infeld theory and use the Bernstein type theorems for minimal surface equations to study its electrostatic solutions. In §12.2, we study electrostatic and magnetostatic solutions in view of finite energy, obtain a generalized Bernstein problem, and find a connection between the minimal surface equations in Euclidean spaces and the maximal surface equations in Minkowskian spaces. In §12.3, we study the Born—Infeld wave equations. In particular, we solve the one-dimensional equations explicitly. We shall also illustrate the connection between the Born—Infeld theory and the Nambu—Goto string theory. In §12.4, we study string-like solutions arising from an Abelian Higgs theory within the framework of the Born—Infeld electromagnetism.

In this chapter we present a study of an important planar Abelian gauge field theory arising in condensed matter physics in which electromagnetism is governed by a Chern—Simons dynamics. In §5.1 we consider the gauged Schrodinger equation, which is nonrelativistic, and we present its explicit solution. In §5.2 we introduce the relativistic Chern—Simons model and state our main results concerning its topological solutions. In §5.3 we make a systematic analysis of the problem and prove the existence theorem stated in §5.2. In §5.4 we obtain all possible symmetric non-topological solutions. In §5.5 and §5.6 we construct spatially periodic solutions modelling a condensed lattice structure.

The Hodge theorem states that, on a compact oriented manifold, each de Rham cohomology class can be represented by a harmonic form. Such a result has an important parallel in the Yang—Mills theory: each second Chern—Pontryagin class on S 4 can be represented by a family of self-dual instantons. The purpose of this chapter is to establish the general theorem that, for each m = 1, 2,..., a similarly defined 2m-th cohomology class on S 4m generalizing the Chern—Pontryagin class on S 4 can be represented by a family of self-dual instantons. In §3.1 we review some basic facts in 4 dimensions. In §3.2 we solve the Liouville equation. In §3.3 we present the explicit solutions of Witten in 4 dimensions based in the solution of the Liouville equation. In §3.4 we discuss the problem in all 4m dimensions and state our general representation theorem. In §3.5–§3.7 we prove the theorem.

In this chapter we present a thorough study of a most natural generalization of the Abelian Higgs theory in (2 + 1) dimensions containing m Higgs scalar fields. We are led to an m × m system of nonlinear elliptic equations which is not integrable. The main tool here is to use the Cholesky decomposition theorem to reveal a variational structure of the problem. In §4.1 we formulate our problem and state an existence and uniqueness theorem. In §4.2 we treat the problem as a pure differential equation problem and state a series of general results for the system defined on a compact surface and the full plane. In §4.3–§4.5, we establish the existence part of our results stated in §4.1 and §4.2. In §4.6 we establish the nonexistence results stated in §4.2. In §4.7 we extend our study to the situation when the coefficient matrix of the nonlinear elliptic system is arbitrary.

In this chapter we present static multisoliton solutions of the non-Abelian Chern—Simons equations. In §6.1 we review some basic facts about a complex semi-simple Lie algebra such as the Cartan—Weyl bases and Cartan matrices to be used in the development to follow. In §6.2 we consider the solution of the non-Abelian gauged Schrodinger equations coupled with a Chern—Simons dynamics via the Toda equations, which is a non-relativistic Chern—Simons theory. In §6.3 we introduce the relativistic Chern—Simons equations and state a general existence theorem. In §6.4 we reduce the governing equations into a nonlinear elliptic system and formulate a variational principle. In §6.5 we present an analysis of the elliptic system and prove all the results stated. In §6.6 we apply our existence theory to some concrete examples.

In this chapter we present a series of results concerning the existence of cosmic strings. Such solutions arise from the Einstein equations coupled with a suitable matter and gauge field system and lead to local concentration of energy and gravitational curvature, a structure relevant in the theory of galaxy formation in the early-time cosmology [321, 322]. In §10.1 we discuss some basic notions in the study of cosmic strings. In particular, we present the elegant solution of Comtet and Gibbons [86] of the harmonic map string equations. In §10.2–§10.6 we present a study of the existence of multiple strings arising from the self-dual Abelian Higgs theory. In §10.7 we consider non-Abelian strings arising from the Weinberg—Salam electroweak theory.

The purpose of this chapter is to present a very concise introduction to the basic concepts and terminology of field theory which will be encountered in the rest of the book. We shall begin in §1.1 with a discussion of classical mechanics in view of a variational formulation and then consider the Schrodinger equation in quantum mechanics. In §1.2 we present special relativity and relativistic wave equations. In particular, we derive the Maxwell equations for electromagnetism. In §1.3 we study the role of continuous symmetry in field theory and prove Noether’s theorem. In particular, we consolidate the notion of energy and momenta and introduce the concept of conserved charges and currents. In §1.4 we present the Abelian gauge field theory and related concepts such as symmetry-breaking and the Higgs mechanism. In §1.5 we discuss in general the non-Abelian gauge field theory. In §1.6 we establish Einstein’s equations of gravitation and discuss some simplest consequences in cosmology.

In this chapter we consider the coexistence of vortices (strings) and antivortices (antistrings) in an Abelian gauge theory. In §11.1, we introduce the gauge field model and state our main existence theorems. In §11.2, we calculate various components of the energy-momentum tensor and reduce the equations of motion into a self-dual system and boil the problem down to a nonlinear elliptic equation with sources through an integration of the Einstein equations. In §11.3, we prove the existence of a unique solution for the elliptic equation governing vortices, in absence of gravity, and, we prove the existence of a solution for the equation governing strings. In §11.4, we calculate the precise value of the quantized energy, and total curvature, of a solution possessing M vortices (strings) and N antivortices (antistrings) and observe the roles played by these two different types of vortices (strings), energetically. In §11.5, we consider the problem over a closed Riemann surface.

It is remarked that, if the gauge group is not compact, finite−time blowing−up Yang–Mills fields over the Minkowski spaces may exist. The example taken is G=SU(2,C). It is shown that blow−up can occur in an arbitrarily small time interval when the initial data of the fields are suitably chosen.

We present topological and nontopological self-dual soliton solutions in an O(2) gauged O(3) {sigma} model on R{sub 2} with Chern-Simons rather than Maxwell dynamics. These solutions are not vortices in the usual sense in that the {ital magnetic} {ital flux} is irrelevant to the stability of the topological solitons, which are stabilized by the degree {ital N}, but it plays a crucial role in the stabilization of the nontopological solitons. It turns out that {ital topological} and {ital nontopological} solitons of arbitrary vorticity {ital N} exist. We have studied both types of vortices with {ital N}=1 and {ital N}=2, and the nontopological soliton with {ital N}=0 numerically. We present analytic proofs for the existence of these topological and nontopological solitons. The qualitative features of the gauged O(3) solitons are contrasted with those of the gauged CP{sup 1} solitons. {copyright} {ital 1996 The American Physical Society.}

This paper presents a resolution of the gaugedO(3) sigma model proposed by B.J. Schroers in which the matter field o mapsR2 intoS2 while the vector gauge potential gives rise to a magnetic field. It is shown that for each natural numberN there are solutions to saturate the classical energy lower boundE≧4πN for the field configurations in the topological family deg(o)=N if and only ifN≠1. Furthermore the solutions obtained depend on at least 4N−3 continuous parameters, the associated magnetic flux can assume its value in an open interval, and the decay rates of the field strengths may be specified in a suitable range. These solutions are multisolitons represented byN prescribed lumps of the magnetic field, simulatingN identical particles in equilibrium, and are governed by a nonlinear elliptic equation with both vortex and anti-vortex source terms.

"This paper studies the existence and the minimization problem of the solutions of the Ginzburg-Landau equations in R┬▓ coupled with an external magnetic field or a source current. The lack of a suitable Sobolev inequality makes it necessary to consider a variational problem over a special admissible space so that the space norms of the gauge vector fields of a minimization sequence can be controlled by the corresponding energy upper bound and a solution may be obtained as a minimizer of a modified energy of the problem. Asymptotic properties and flux quantization are established for finite-energy solutions. Besides, it is shown that the solutions obtained also minimize the original Ginzburg-Landau energy when the admissible space is properly chosen."

"The paper studies the problem of numerical approximations of the critical transition temperature and the energy gap function in the Bardeen-Cooper-Schrieffer equation arising in superconductivity theory. The positive kernel function leads to a phonon dominant state at zero temperature. Much attention is given to the equation defined on a bounded region. Two discretized versions of the equation will be introduced. The first version approximates the desired solution from below, while the second, from above. Numerical examples are presented to illustrate the efficiency of the method. Besides, The [sic] approximations of a full space solution and the associated critical temperature by solution sequences constructed on bounded domains are also investigated."

The proof of Lemma 3.2 is corrected to take account of a missing term in a crucial Sobolev inequality used in the original work. As a result, Theorem 3.3 is much improved.

Recent work on magnetic phase transition in nanoscale systems indicates that new physical phenomena, in particular, the Bloch wall width narrowing, arise as a consequence of geometrical confinement of magnetization and leads to the introduction of geometrically constrained domain wall models. In this paper, we present a systematic mathematical analysis on the existence of the solutions of the basic governing equations in such domain wall models. We show that, when the cross section of the geometric constriction is a simple step function, the solutions may be obtained by minimizing the domain wall energy over the constriction and solving the Bogomol’nyi equation outside the constriction. When the cross section and potential density are both even, we establish the existence of an odd domain wall solution realizing the phase transition process between two adjacent domain phases. When the cross section satisfies a certain integrability condition, we prove that a domain wall solution always exists which links ...

"This paper establishes the existence of string-like static solutions of the coupled Einstein-matter-gage equations. Such solutions have important implications in cosmology and quantum physics. It is shown that for a prescribed string location, the system possesses a continuous family of distinct finite energy solution configurations. The proof relies on a two-side shooting method. Power-type decay estimates at spatial infinity are also obtained."

"This paper considers the Einstein equations coupled with the nonabelian gauge and Higgs fields. It is shown that, when cosmic string solutions are sought in the Einstein-Georgi-Glashow system and the Einstein-Weinberg-Salam system governing the gravitational-electromagnetic weak interaction forces, the self duality conditions lead to positive values of the cosmological constant which can be expressed by some fundamental parameters in particle physics."

In this paper, we use the method of calculus of variations to develop an existence theory for the steady state solutions of a nonlinear Schrodinger equation modeling light waves propagating in a photorefractive crystal. We show via direct minimization and mountain-pass argument that there exist steady state solutions realizing a continuous spectrum of energy points or wavenumbers.

The problem of existence of knot-like solitons as the energy-minimizing configurations in the Faddeev model, topologically characterized by an Hopf invariant, Q, is considered. It is proved that, in the full space situation, there exists an infinite set S of integers so that for any m ∈ S, the Faddeev energy, E, has a minimizer among the class Q = m; in the bounded domain situation, the same existence theorem holds when S is the set of all integers. One of the important technical results is that E and Q satisfy the sublinear inequality E ≤ C|Q|3/4, where C > 0 is a universal constant, which explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.

SynopsisIn this paper we study the global existence and nonexistence of spatially periodic solutions to the initial value problem of the Ginzburg–Landau equation.

This paper studies the solutions of the Ginzburg-Landau equations on ℝ3 in the presence of an arbitrarily distributed external magnetic field. The existence and regularity of the solutions at the lowest energy level are established. The solutions found are in the Coulomb gauge. If the external field is sufficiently regular, the solutions are shown to have nice asymptotic decay properties at infinity.

The Skyrme model is generalized to higher‐dimensional Riemannian manifolds and the geometric conditions under which homothetic maps are absolute or local minimizers of the model are obtained.

We establish under nonuniform nonresonance conditions an existence and uniqueness theorem for a linear, and the solvability for a nonlinear, fourth-order boundary value problem which occurs frequently in plate deflection theory.