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.