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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?


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

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  • $\begingroup$ Is the kink solution dependent on only one space-time variable? $\endgroup$ – ggcg Dec 5 '20 at 2:31
  • $\begingroup$ @ggcg, yes, this solution only depends on one spatial coordinate ($x$ here), but it may be made dependent on $x - v t$ from a Lorentz boost (change of referential). I'm looking for other nontrivial solutions. $\endgroup$ – Cham Dec 5 '20 at 18:23
  • $\begingroup$ The general solution is $\phi \left( T,X,Y,Z \right) =\tanh \left( {C_1}+{C_2}\,T+{ C_3}\,X+{C_4}\,Y-\sqrt {{{C_2}}^{2}-{{C_3}}^{2}-{{ C_4}}^{2}+1}\,Z \right)$ $\endgroup$ – Eli Dec 12 '20 at 17:38
  • $\begingroup$ @Eli, that's not the general solution. It's just the kink solution with some rotation of the axes, and a Lorentz boost, plus a simple translation. $\endgroup$ – Cham Dec 12 '20 at 17:52
  • $\begingroup$ this is more then your solution? $\endgroup$ – Eli Dec 12 '20 at 20:57
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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 :-)

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  • $\begingroup$ Is this solution gives some simple expressions, in some limits (excluding the hyperbolic $\tanh$)? I don't think this is a pointless exercice, since the solution could be used as a base for some numerical resolution with perturbations. And the solutions could teach us something about the field behavior. Do you think the kink solution is pointless? $\endgroup$ – Cham Dec 8 '20 at 17:22
  • $\begingroup$ @Cham Yes, I was commenting on my lack of imagination rather than the legitimacy of your query. I was trying to say "here's the solution, I don't know what to do with it". I'm sure people smarter than me will find plenty of applications to this type of solutions :-) $\endgroup$ – AccidentalFourierTransform Dec 9 '20 at 14:13
  • $\begingroup$ I just found a paper which describes your solution in more details: arxiv.org/abs/0907.4053. I'm now very sceptic that we could find some simple analytical solutions similar to the famous kink solution. $\endgroup$ – Cham Dec 9 '20 at 14:34
  • $\begingroup$ Look out the interesting solution that I gave as an answer! $\endgroup$ – Cham Dec 25 '20 at 21:50
  • $\begingroup$ @Cham huh, that is definitely a neat find! It has a cute physical interpretation, I would have never guessed that solution. $\endgroup$ – AccidentalFourierTransform Dec 26 '20 at 21:28
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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.

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