Does the following equation $$ \nabla^\mu \nabla_\mu \psi + a \psi^3 = b \psi $$ where $\psi$ is a real function, $a$ and $b$ are real constants, have other solutions that extend beyond a one dimensional line? From wiki: enter link description here

They have a solution $$ \psi(x) = \pm\sqrt{\frac{2\mu^4}{\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}}{\rm sn}\left(p\cdot x+\theta,\sqrt{\frac{-\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}{-\mu_0^2 - \sqrt{\mu_0^4 + 2\lambda\mu^4}}}\right) $$ which only varies along $\bar{x}=t\bar{p}$, where $t$ is a variation parameter, and constant along any other direction.

For example, are there solutions which depend on a radius relative to an origin? Are there solutions that exists beyond a line?


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