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how $\varphi^6$ topological soliton kink and anti-kink are calculated ?, what is an anti-kink?

$$L = -\frac{1}{4}F_{\mu\nu}^2− |\varphi|^6 − ieA_\mu(\varphi^\ast\partial_\mu\varphi−\varphi\partial_\mu\varphi^\ast)$$

This is an exercise for my university and i don't know how to solve

I have an idea of how to calculate vortices but not topological solitons, I have been reading a lot about homotopy and vortices but I do not understand well how the Lagrangian without having boundary conditions in the exercise, on the other hand I have read about topological defects and mechanisms a more difficult like Kibble – Zurek mechanism (https://arxiv.org/abs/1306.4523) which are primordial waves in the Higgs that use solitonic solutions but I think I am walking away from how to solve a topological soliton but according to what I have read all the monopoles and other vertices and topological defects were caused by the kibble-zurek mechanism.

"The kink-antikink interaction is generically attractive, obtein $\varphi^6$ potencial theory for $\mathscr{L}=\dfrac{1}{2}\partial^\mu\phi\partial_\mu\phi=\dfrac{1}{2}\left(\partial_\mu\phi\right)^2$. kinks, in ϕ and $L = -\frac{1}{4}F_{\mu\nu}^2− |\varphi|^6 − ieA_\mu(\varphi^\ast\partial_\mu\varphi−\varphi\partial_\mu\varphi^\ast)$ answer to the question how kink-kink (K-K) and kink-antikink (K-AK) long range (power-law) interactions occur in higher-order field theories that exhibit such topological defects."

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  • $\begingroup$ I think you are missing a large part of you Lagragian. Is there no kinetic and mass term for the scalar field? $\endgroup$
    – mike stone
    Commented Dec 11, 2021 at 20:15
  • $\begingroup$ @mikestone this is the lagragian that i have, i dont know if i need to add more terms, im not familiar with instantones, i will ask to my class. $\endgroup$
    – Juan Carlos Dominguez Solis.
    Commented Dec 11, 2021 at 21:04
  • $\begingroup$ Can you up load apicture of the whole problem? As it's a hw problem we can't solve it for you but gave give some hints. $\endgroup$
    – mike stone
    Commented Dec 11, 2021 at 21:19
  • $\begingroup$ 2n+4 field theories for n > 1. The kink-antikink interaction is generically attractive, obtein \varphi^6 potencial theory for \mathscr{L}=\dfrac{1}{2}\partial^\mu\phi\partial_\mu\phi=\dfrac{1}{2}\left(\partial_\mu\phi\right)^2. kinks, in ϕ and L = -\frac{1}{4}F_{\mu\nu}^2− |\varphi|^6 − ieA_\mu(\varphi^\ast\partial_\mu\varphi−\varphi\partial_\mu\varphi^\ast) answer to the question how kink-kink (K-K) and kink-antikink (K-AK) longrange (power-law) interactions occur in higher-order field theories that exhibit such topological defects. $\endgroup$
    – Juan Carlos Dominguez Solis.
    Commented Dec 11, 2021 at 23:26
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    $\begingroup$ Because youy have two theories One with $\varphi$ which is free, and one with a diferentfield $\phi$. Then there is weird potential that has no kink solutions. The usual problem has a $\phi^4$ interaction and an $m^2\phi^2$ term to give a $\phi$ condensate. It look like a half-baked attempt to describe Nielsen-Ollsen vortices (hencethe gauge field. Vortices are not kinks. Those are strickly one space one time dimensional. $\endgroup$
    – mike stone
    Commented Dec 11, 2021 at 23:49

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