Analytical solutions to scalar field equation with Higgs-like potential

Question

I'm considering the classical field theory of a real scalar field $\phi$ with the Higgs "Mexican hat" potential: $$\tag{1} \mathcal{V}(\phi) = \frac{\lambda}{4} (\phi^2 - \phi_0^2)^2, $$ where $\lambda > 0$ and $\phi_0$ are constants. $\phi_0$ is the value of the "true vacuum" field. The equation of motion of the self-interacting field is this (I'm using the metric signature $\eta = (1, -1, -1, -1)$ and units such that $c \equiv 1$): $$\tag{2} \eta^{ab} \, \partial_a \, \partial_b \, \phi + \lambda (\phi^2 - \phi_0^2) \, \phi = 0. $$ I already know the "kink" solution to this equation (and its Lorentz boosted variation): $$\tag{3} \phi(x) = \phi_0 \tanh \Bigl( \sqrt{\frac{\lambda}{2}} \, \phi_0 \, x \Bigr).$$ It's usefull to scale the variables to simplify things a bit: \begin{align} \Phi &= \frac{\phi}{\phi_0}, \tag{4} \\[1ex] T&= \sqrt{\frac{\lambda}{2}} \, \phi_0 \, t, \tag{5} \\[1ex] X &= \sqrt{\frac{\lambda}{2}} \, \phi_0 \, x, &Y &= \sqrt{\frac{\lambda}{2}} \, \phi_0 \, y, &Z &= \sqrt{\frac{\lambda}{2}} \, \phi_0 \, z. \tag{6} \end{align} This way, equation (2) becomes $$\tag{7} \frac{\partial^2 \Phi}{\partial T^2} - \frac{\partial^2 \Phi}{\partial X^2}- \frac{\partial^2 \Phi}{\partial Y^2}- \frac{\partial^2 \Phi}{\partial Z^2} + 2(\Phi^2 - 1) \, \Phi = 0. $$ So my question is this: What are the other known analytical solutions to this equation, apart the "kink" solution (3) ? (i.e. $\Phi(X) = \tanh X$ and its trivial Lorentz boosted version). Are there other non-trivial solutions known of this equation?

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EDIT1:. The first part of this paper (may 2020 !) is interesting and appears to answer my question: https://arxiv.org/abs/1911.02064. Apparently, there is no other known analytical solution! :-(

EDIT2: The author of the previous paper is wrong (about non-existence of other analytical solutions). I've found a nice solution, that I exposed in an answer below. It comes from this book (see section 13.2.4, page 376):

Cosmic Strings and Other Topological Defects. Vilenkin and Shellard.

https://books.google.ca/books?id=eW4bB_LAthEC

Answer 1

There's definitely other non-trivial solutions, although there is no situation I can think of where they would be useful. But anyway, you can for example try solutions of the form $$ \Phi(x,y,z,t)=g(x+\alpha t) $$ for some constant $\alpha$. Plugging this ansatz into your equation we get $$ (\alpha ^2-1) g''(x)+2g(x)^3-2 g(x)=0 $$ with solution $$ g(x)=\epsilon\, \mathrm{sn}\left(\frac{x-x_0}{\sqrt{\frac{\alpha ^2-1}{\epsilon ^2-2}}}\left|\frac{\epsilon ^2}{2-\epsilon ^2}\right.\right) $$ where $\epsilon,x_0$ are integration constants, and $\mathrm{sn}$ denotes the Jacobi elliptic function. In a certain limit this function becomes the hyperbolic tangent, unsurprisingly.

You can try other ansätze and get other analytical solutions, if you are lucky. Again, this feels like a somewhat pointless exercise but that's none of my busyness :-)

Answer 2

I just found an interesting analytical and highly non-trivial solution to equations (2) and (7) ! $$\tag{1} \Phi(T, X, Y, Z) = \tanh \bigl(Z + \mathcal{G}(a T - b X - c Y) \bigr), $$ where $a$, $b$ and $c$ are arbitrary constants satisfying the light like condition on the orthogonal plane of $Z$: $$\tag{2} a^2 - b^2 - c^2 = 0, $$ and $\mathcal{G}(u)$ is a completely arbitrary function (not necessarily linear)!

This solution describes an arbitrary plane wave traveling on the kink solution, at the speed of light.