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Department of Chemistry, University of New South Wales, Syndey, Australia
ABSTRACT
The Klein–Gordon equation (KGE), formulated in 1926 as the first relativistic wave equation, remains one of the most broadly consequential equations in theoretical physics. This review surveys the KGE across four interconnected dimensions: its historical genesis from de Broglie’s matter-wave hypothesis and five-dimensional relativity; the interpretational crisis of negative probability densities that motivated second quantization and the modern quantum-field-theoretic viewpoint; the rich mathematical theory of linear and nonlinear variants, including well-posedness, Strichartz estimates, solitary waves, stability theory, and breather nonexistence; and the expanding landscape of numerical and computational methods, from exponential integrators and filtered finite differences to physics-informed neural networks and quantum simulation algorithms. Particular attention is paid to recent developments (2020–2026), including codimension-one stability results, dispersive estimates on cones and inverse-square potentials, long-lived quantum oscillons, and machine-learning solvers for hyperbolic systems. Application domains are reviewed in depth: coherent scalar-field dynamics in cosmology, axion dark matter and oscillon formation, sine–Gordon phenomenology in Josephson junctions and quasi-one-dimensional conductors, Klein tunneling in graphene, and superradiant instabilities of massive scalar fields around rotating black holes. Open problems are identified at the interface of rigorous analysis, high-performance simulation, and data-driven modeling. This review is intended both as a self-contained introduction for graduate students and as a reference map of the current research frontier.
Keywords: Klein–Gordon Equation; Relativistic Wave Equation; Nonlinear Dispersive Equations; Solitons and Oscillons; Scalar Field Cosmology; Physics-Informed Neural Networks; Exponential Integrators; Quantum Simulation