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

### Submission summary

 As Contributors: Jasper van Wezel Arxiv Link: https://arxiv.org/abs/2009.00355v1 (pdf) Date submitted: 2020-09-02 07:53 Submitted by: van Wezel, Jasper Submitted to: SciPost Physics Lecture Notes Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

As a low-energy effective model emerging in disparate fields throughout all of physics, the ubiquitous $\varphi^4$-theory is one of the central models of modern theoretical physics. Its topological defects, or kinks, describe stable, particle-like excitations that play a central role in processes ranging from cosmology to particle physics and condensed matter theory. In these lecture notes, we introduce the description of kinks in $\varphi^4$-theory and the various physical processes that govern their dynamics. The notes are aimed at advanced undergraduate students, and emphasis is placed on stimulating qualitative insight into the rich phenomenology encountered in kink dynamics. The appendices contain more detailed derivations, and allow enquiring students to also obtain a quantitative understanding. Topics covered include the topological classification of stable solutions, kink collisions, the formation of bions, resonant scattering of kinks, and kink-impurity interactions.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2009.00355v2 on 11 December 2020

Submission 2009.00355v1 on 2 September 2020

## Reports on this Submission

### Strengths

Detailed and transparent

### Weaknesses

Lack of Physical examples

### Report

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
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.

### Requested changes

• validity: high
• significance: -
• originality: good
• clarity: good
• formatting: perfect
• grammar: acceptable

### Strengths

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.

### 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.

### Requested changes

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}
}

• validity: top
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: perfect

Anonymous on 2020-11-10

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 behaviour 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.