Cubic NLW/NLKG

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Revision as of 09:37, 19 November 2008 by Marco Frasca (talk | contribs) (Corrected an equation)
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The cubic nonlinear wave and Klein-Gordon equations have been studied on R, on R^2, and on R^3.

This kind of equation displays a class of solutions with a peculiar dispersion law. To show explicitly this, let us consider the massless equation

being . An exact solution of this equation is given by

being sn a Jacobi elliptic function and two integration constants, when

We see that we started with an equation without a mass term but the exact solution describes a wave with a dispersion relation proper to a massive solution. This can be seen as the superposition of an infinite number of massive linear waves through a Fourier series of the Jacobi function, that is

being an elliptic integral. We recognize the spectrum

Via the mapping theorem FraE2007 this is also an exact solution of Yang-Mills equations with the substitution for a SU(N) Lie group.