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.
KeywordsHamiltonian Density Abelian Higgs Model Real Scalar Field Bernstein Theorem Minimal Surface Equation
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