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Comments on the Negative grade KdV Hierarchy
by Y. F. Adans, J.F. Gomes, G.V. Lobo and A.H. Zimerman
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Submission summary
Authors (as registered SciPost users):  Jose Gomes 
Submission information  

Preprint Link:  scipost_202212_00040v1 (pdf) 
Date submitted:  20221217 17:02 
Submitted by:  Gomes, Jose 
Submitted to:  SciPost Physics Proceedings 
Proceedings issue:  34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The construction of negative grade KdV hierarchy is proposed in terms of a Miuragauge transformation. Such gauge transformation is employed within the zero curvature representation and maps the Lax operator of the mKdV into its couterpart in the KdV setting. Each odd negative KdV flow is obtained from an odd and its subsequent even negative mKdV flows. The negative KdV flows are shown to inherit the two different vacua structure that characterises the associated mKdV flows.
Current status:
Reports on this Submission
Anonymous Report 1 on 2023112 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202212_00040v1, delivered 20230112, doi: 10.21468/SciPost.Report.6522
Strengths
1. The report provides a detailed account of a contribution to the 34th International Colloquium on Group Theoretical Methods in Physics.
2. The authors use previously developed algebraic techniques and extend their (and other authors') earlier work to demonstrate a new interesting relationship between the KdV and mKdV hierarchies.
3. The relationship between the hierarchies and some of the consequences of the discovered relationship are described in detail.
Weaknesses
1. The significance of the different 'vacuum structures' should be explained.
2. The article does need to be proofread carefully because there are many grammatical errors, including incorrectly spelled words, missing definite and indefinite articles, missing punctuation, etc. I have supplied examples below.
Report
The conference is devoted to the the many applications of group theory in physics, including mathematical physics, and the development of grouptheoretical methods. This article falls within the scope of the meeting.
The article contains new results that should be of use to mathematical physicists interested in the structure of integrable field theories and their organisation within 'hierarchies'.
There is a limit of eight pages maximum for parallel session submissions to the conference and this article meets that restriction.
Requested changes
1. Equations
Eq(34) is stated to be an equation of motion. However, it is not actually an equation but an expression (there is no '=' anywhere).
Eq(49) is too long and needs to be reformatted to fit properly.
2. Spellings/insertions (this is already extensive but probably not exhaustive)
p1 l4 'which allows its' > 'allowing their'
p2 l1 'properties as' > 'properties, such as' and a ',' after 'solutions'
p2 l7 'negative [3]' > 'negative flows [3]'
p2 l15 'there shall > 'there should'
p3 insert ',' at the end of eq(8)
p3 insert ',' at the end of eq(14)
p3 'well know' > 'well known'
p3 'be showed that' > 'be shown that'
p3 insert ',' at the end of eq(16)
p3 to p4 Suggest placing '.' at the end of eq(18) and then breaking the long sentence into several short ones to include eqs(19,20) in separate sentences.
p4 before eq(23) replace 'Propose' > 'We propose'
p4 after eq(23) insert 'the' before 'KdV' and also before 'subhierarchy'
p4 insert ',' at the ends of eqs(24,29)
p5 l1 and l4 'KdV' > 'the KdV'
p5 l6 insert 'operator' after 'Lax'
p5 after eq(31) insert 'the' before 'inverse', replace 'recurssion' > 'recursion', replace 'hamiltonian' > 'Hamiltonian'
p5 before eq(32) insert 'operator' after 'Lax' and insert ',' at the end of eq(32)
p5 eq(34) seems to be missing something since it is evidently not an equation of motion
p5 after eq(36) insert 'whether' after 'no matter', replace 'are' by 'is' before 'associated'
p5 why is part of the final sentence italicised? (It might be better to split the sentence containing eqs(35,36) into two sentences. In any case, eq(36) should end with '.' and on the following line 'this' > 'This')
p6 insert ',' at the end of eq(37)
p6 after eq(39) replace 'as local' > 'as a local'
p7 l12 is this a new paragraph? If not, there is no need to start a new line with 'For'; if it is then there should be a proper paragraph break
p7 l2 replace 'satisfy' > 'satisfies'
p7 insert ',' at the ends of eqs(45,46)
p7 replace 'Sinh' > 'sinh' and insert 'the' before 'Miura'
p7 final paragraph: replace 'inherits' > 'inherit', replace 'structure' > 'structures', insert 'and' before 'each KdV', replace 'pair' > 'pairs', remove space after 'multisolitons', remove 'its' before 'vacuum', change 'its' > 'their', change 'in the lines' > 'along the lines'
p8 ref[6] remove space and the extra ',' after the author's name
Author: Jose Gomes on 20230117 [id 3245]
(in reply to Report 1 on 20230112)We agree with the report and have made all requested changes regarding grammar errors, typos, punctuation, etc
We have rewritten text between eqns. (17) and (22) in order to clarify the the different vacuum structures employed in the paper.