Cubic NLW/NLKG

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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

$\Box\phi + \lambda\phi^3 = 0$

being $λ > 0$ . An exact solution of this equation is given by

$\phi(x) = \mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x+\theta,i),$

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

$p^2=\mu^2\left(\frac{\lambda}{2}\right)^{1\over 2}.$

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

$\phi(t,0)=\mu\left(\frac{2}{\lambda}\right)^{1\over 4}\sum_{n=0}^\infty(-1)^n\frac{2\pi}{K(i)}\frac{e^{\left(n+{1\over 2}\right)\pi}}{1+e^{-(2n+1)\pi}}\sin\left((2n+1)\frac{\pi}{2K(i)}\left(\frac{\lambda}{2}\right)^{1\over 4}\mu t\right)$

being $K (i)$ an elliptic integral. We recognize the spectrum

$m_n = (2n+1)\frac{\pi}{2K(i)}\left(\frac{\lambda}{2}\right)^{1\over 4}\mu.$

Via the mapping theorem FraE2007 this is also an exact solution of Yang-Mills equations with the substitution $\lambda\rightarrow Ng^2$ for a SU(N) Lie group.

Cubic NLW/NLKG

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