Cubic NLW/NLKG

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 $\lambda >0$. 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 $\mu ,\theta$ 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

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

Similarly, when there is a mass term as in

$\Box \phi +\mu _{0}^{2}\phi +\lambda \phi ^{3}=0$

the exact solution is given by

$\phi (x)={\sqrt {\frac {2\mu ^{2}}{\mu _{0}^{2}+{\sqrt {\mu _{0}^{4}+2\lambda \mu ^{4}}}}}}{\rm {sn}}\left(p\cdot x+\theta ,{\sqrt {\frac {-\mu _{0}^{2}+{\sqrt {\mu _{0}^{4}+2\lambda \mu ^{4}}}}{-\mu _{0}^{2}-{\sqrt {\mu _{0}^{4}+2\lambda \mu ^{4}}}}}}\right)$

being now the dispersion relation

$p^{2}=\mu _{0}^{2}+{\frac {\lambda \mu ^{4}}{\mu _{0}^{2}+{\sqrt {\mu _{0}^{4}+2\lambda \mu ^{4}}}}}.$

Finally, we can write down the exact solution for the case

$\Box \phi -\mu _{0}^{2}\phi +\lambda \phi ^{3}=0$

that is given by

$\phi (x)=v\cdot {\rm {dn}}(p\cdot x+\theta ,i),$

being $v={\sqrt {\frac {2\mu _{0}^{2}}{3\lambda }}}$ and the following dispersion relation holds

$p^{2}={\frac {\lambda v^{2}}{2}}.$

So, this wave solutions are interesting as, notwithstanding we started with an equation with a wrong mass sign, the dispersion relation has the right one. Besides, Jacobi function dn has no real zeros and so the field is never zero. This effect is known as spontaneous breaking of symmetry in physics.