The nonlinear pde
$$
 \partial\_t^2\phi-\partial\_x^2\phi+\lambda\phi^3=0
$$
has the exact solution
$$
 \phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\_0t-p\cdot x+\varphi,i)
$$
with $\mu$ and $\varphi$ two integration constants and sn the snoidal Jacobi function, provided the dispersion relation holds
$$
 p^2\_0=p^2+\mu^2\sqrt{\frac{\lambda}{2}}.
$$
If I interpret $p\_0$ as the energy, it seems that is finite. Computing the integral
$$
 E=\int d^Dx\left[\frac{1}{2}(\partial\_t\phi)^2+\frac{1}{2}(\partial\_x\phi)^2+\frac{\lambda}{4}\phi^4\right]
$$
and extending the volume to infinity, this is divergent. Can one find a sound mathematical explanation for this? Is there a way to "regularize" this integral?


Thanks.



Exact solutions to nonlinear Klein-Gordon equation
regularization of the Klein-Gordon equation proceeds in the same way as for the Schrödinger equation; you restrict $x$ to the interval $(0,L)$ and impose periodic boundary conditions $\phi(0,t)=\phi(L,t)$; this quantizes the wave vector $p=p\_n(t)$, $n\in\mathbb{Z}$ --- for the linear Schrödinger equation the quantization would be time independent, now it depends on time; in the limit $L\rightarrow\infty$ you would ignore the quantization, calculate the energy $E\_L$ by integrating $x$ from $0$ to $L$, and then obtain a finite limit $L^{-1}E\_L$; this just means that the energy density per unit length is finite, even though the entire energy content of the infinite system is infinite (as it should be).


