An introduction to kinks in $\varphi^4$-theory

Detailed and transparent

Lack of Physical examples

This is an interesting and useful review article which is very carefully drafted.

This review deals solitons arising in $\phi^4$ which is not integrable and hence
interactions between the kinks lead to interesting behavior in comparison to the
(integrable) sine-Gordon model. Major focus of the review is on the dynamics and
interactions of kinks in the $\phi^4$ theory.
The basic theory of solitons and and kink collisions are nicely discussed
in chapters 2 and 3. Collective coordinate approximation, gluing static
solutions and kink-impurity interactions are discussed in subsequent
sections. Overall, the review is self-sufficient and mathematical
steps and arguments are quite transparent. The discussion in the
main text is supplemented by three appendices. Indeed, authors have
discussed a wide range of using numerical calculations and effective
models.

I have just two suggestions:

1. It is very heartening to that authors have provides some relevant
exercise in every section? What about providing some hints and answers
at the end

2.Secondly, I find this article a bit dry. This is not a criticism
but, I do believe, review would look far more enriched if examples of physical
situations are added wherever possible. For example, as
given in Chaikin Lubensky chapter 9. The article would then attract
a wider audience.

With these minor comments, I would accept this review for publication.

Already in report

The authors have done a nice job of producing a readable, even
pedagogical, summary of $\phi^4$ kinks. They have managed to do so
without neglecting mathematical aspects, including some that were new
to me (the elliptic sine solution). The standard of english is excellent.

The authors have done a nice job of producing a readable, even
pedagogical, summary of $\phi^4$ kinks. They have managed to do so
without neglecting mathematical aspects, including some that were new
to me (the elliptic sine solution). The standard of english is excellent.

I would change the title of Section 3 to "Kink-antikink collisions".
On page 12, replace "loose" by "lose".

Although it is difficult to cite all literature, I do feel that
the tutorial would be enhanced by mentioning the following:

@article{kevrekidis2019dynamical,
title={A Dynamical Perspective on the $\varphi^4$ model},
author={Kevrekidis, Panayotis G and Cuevas-Maraver, Jes{\'u}s},
journal={Past, present and future. Nonlinear Systems and Complexity},
volume={26},
year={2019},
publisher={Springer}
}

@article{rice1983physical,
title={Physical dynamics of solitons},
author={Rice, MJ},
journal={Physical Review B},
volume={28},
number={6},
pages={3587},
year={1983},
publisher={APS}
}

@article{bishop1980solitons,
title={Solitons in condensed matter: a paradigm},
author={Bishop, AR and Krumhansl, JA and Trullinger, SE},
journal={Physica D: Nonlinear Phenomena},
volume={1},
number={1},
pages={1--44},
year={1980},
publisher={Elsevier}
}

@article{goodman2005kink,
title={Kink-Antikink Collisions in the $\phi^4$ Equation: The n-Bounce Resonance and the Separatrix Map},
author={Goodman, Roy H and Haberman, Richard},
journal={SIAM Journal on Applied Dynamical Systems},
volume={4},
number={4},
pages={1195--1228},
year={2005},
publisher={SIAM}
}