In 1967 Gerald Feinberg hypothesized the existence of tachyons as particles represented by the imaginary solutions of relativistic quantum wave equations, such as the Klein-Gordon equation.
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Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schr�dinger equation with a covariant equation such as the Klein Gordon equation or the Dirac equation.
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The Schr�dinger equation can also be derived from a first order form similar to the manner in which the Klein-Gordon equation can be derived from the Dirac equation.
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Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein Gordon equation, and Poisson's equation.
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For example, the Klein Gordon equation is the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave-function.
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Continuing with the nonlinear Klein Gordon equation, using the average variational principle on equation, the wave action equation becomes by variation with respect to the wave phase \ theta:
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Many exactly solvable models have soliton solutions, including the Korteweg de Vries equation, the nonlinear Schr�dinger equation, the coupled nonlinear Schr�dinger equation, and the sine-Gordon equation.
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The Fourier transform ( in all four variables ) of such measures yields positive-energy, finite-energy solutions of the Klein-Gordon equation defined on Minkowski space, namely
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The sine-Gordon equation is a relativistic wave equation in 1 + 1 dimensions that also exhibits the soliton phenomenon and which became an important model of solvable relativistic field theory.
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Mathematically, a semifluxon can be constructed by joining two tails of conventional ( integer ) fluxon ( kink of the sine-Gordon equation ) at the 0-? boundary.
