Beyond-mean-field analysis of the Townes soliton and its breathing mode

I warmly thank the referee.

I warmly thank Referee #1 for this report and suggestions. Here is my response.

Referee:
-the abstract is still a bit technical. E.g., I don't like having formulas in there. The author may consider reformulating it in a more accessible way. Also the last paragraph of the intro may be made more accessible and slightly more detailed.

Response:
This formula is there on purpose to attract attention. In any case, given its level of complexity, I think that this is really a question of personal style. I take this opportunity to make the following change in the abstract, hopefully compatible with the Referee's request as it increases the text-to-formula ratio and makes the abstract slightly more informative. I replace the sentence

"The oscillatory breathing motion results from the quantum-mechanical breaking of the mean-field scaling symmetry."

by

"The breathing oscillations, absent on the mean-field level, result from the quantum-mechanical breaking of the mean-field scale symmetry."

In the last paragraph of the introduction I added a couple of clarifying sentences about the preexponential term and about conjectures made at the end of the article.


Referee:
-on pag.5, I am puzzled by the formula |g-gc|~g^2<<|gc|. I don't see what it means to formally set g=gc. The author should spend a few words to explain what he has in mind.

Response:
I am interested in the systematic expansion of the energy in powers of g. The Bogoliubov level corresponds to terms ~g^2. The beyond-mean-field correction is of this order of magnitude. Thus, when calculating it with the Bogoliubov accuracy I can replace g by g_c since the difference g-g_c is of higher order. I think this should be understandable from the context (see the paragraph preceding the discussion and the last paragraph of Sec.2). Nevertheless, I added the sentence "Distinguishing between $g$ and $g_c$ in this case would exceed the Bogoliubov accuracy." To avoid confusion I also replaced "to formally set g=g_c" by "to use the approximation g=g_c". I hope this is more clear now.

All this discussion would be much more straightforward, if I could just set g=g_c from the very beginning. However, as I explain in the text, I cannot do this because of the constraints on the potential range. So, I have to keep g different from g_c, but the difference can be chosen to be of higher order. This choice is made possible by the fact that the dependence of g on h is very slow and that effective-range corrections are algebraic in h, i.e., exponentially small as a function of g.

Referee:
-when discussing after eq.(27) the condition for adiabaticity, the author mentions the risk of non-adiabatic features at large R, but says nothing on what may happen at smnall R. Even though the gap is here the largest, I see from Fig.1 that the effective potential grows very fast. I am therefore wondering if the strong acceleration felt around this left turning point may induce any additional adiabatic effect.

Response:
As far as I know, the adiabatic theorem does not impose constraints on the acceleration.

I guess that account of the acceleration may lead to quantitative corrections, but I do not know how important they are. In the paper I argue that breaking of the adiabatic condition at large R may lead to a qualitatively different description of the higher part of the spectrum just below the continuum. It is not obvious to me that nonadiabatic effects at the left turning point can lead to qualitatively important consequences. At least, being close or far from the threshold E=0 does not seem to play a big role there. We can as well discuss the left and right turning points of small-amplitude oscillations.