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.

Exact solutions

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

Similarly, when there is a mass term as in

the exact solution is given by

being now the dispersion relation

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

that is given by

being and the following dispersion relation holds

So, these 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 has no real zeros and so the field is never zero. This effect is known as spontaneous breaking of symmetry in physics.