Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions
Round 1
Reviewer 1 Report
The paper presents, in principle, a nice work on the borderline of mathematics and physics. However, serious deficiencies need to be addressed. Specifically, four questions need to be answered:
[1.] The original generalized Benjamin-Bona-Mahony-Burgers equation (gBBMB) equation
[see https://arxiv.org/pdf/2103.03907.pdf] contains, explicitly, only terms dependent
on t and x. It is a partial differential equation for a function u=u(t,x). The
authors explicitly consider a form advocated in Ref.[14], Applied Mathematics and Computation
Volume 367, 15 February 2020, 124765, where u=u(t,x,y). This other form has y-derivatives, but the
physical interpretation of the y-dependent terms is unclear. The journal SYMMETRY
is a physics journal, and some motivational material must be added to justify,
and clarify the physical role, and the structure of the y-dependent terms.
[2.] The authors are required to give their paper another very careful reading
and correct a number of missing definitions, as well as notational
inconsistencies. For example, the X operators are sometimes put in boldface,
other times, just written as normal operators. This is not acceptable. Also,
just to add another example for inconsistencies, the "\cdot" operator (in LaTeX
notation) is missing in comparison to the third term in Eq.(1) of of Ref.[14],
Applied Mathematics and Computation Volume 367, 15 February 2020, 124765,
making the notation incomprehensible. I could be harsher in my criticism of
the inaccuracies; rather, I just point out that another very careful reading is
required and in the interest of the authors. This is serious.
[3.] The authors are required to add explanations of their considerations to
make the paper accessible to physicists. For example, the authors point out
"symmetries" and use them synonymously with "operators", while, of course, a
"symmetry" is a property of an equation or of the solution of an equation. An
example: Equation (7), at face value, is completely incorrect. The operator PHI
itself cannot be zero, universally. The operator, applied to a function, could
be zero, but this needs to be explained. Also, in which way, in general, do
the authors consider "symmetries" and use them synonymously with "operators"?
This needs to be explained to a physicist reader in an understandable way.
[4.] Last, but certainly not least, I would strongly suggest the addition
of a three-dimensional figure, where a newly found solution u=u(t=t_i, x, y)
is plotted for t_1=0, t_2=1, t_3=2 [in the units chosen by the authors],
and given as a physical illustration of the work of the authors. This would
tremendously enahce the article.
I am tentatively 51% inclined to reach final acceptance, but more, and
diligent, work is needed.
Author Response
Please find as attached.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Report on MS
SYMMETRY ANALYSIS, EXACT SOLUTIONS AND CONSERVATION
LAWS OF A BENJAMIN-BONA-MAHONY-BURGERS EQUATION IN
2 + 1-DIMENSIONS
by María S. Bruzón, Tamara María Garrido-Letrán and Rafael de la Rosa.
A. The results:
The authors considered a generalized family of Benjamin-Bona-Mahony-Burgers PDEs,
and they performed its conventional Lie symmetry analysis and symmetry reduction.
They listed the options and explored several exact solutions, with emphasis upon
the traveling waves. Using the multiplier method they derived the conservation
laws and deduced and discussed solutions of a specific ultimate second-order
nonlinear ordinary differential equation.
B. The presentation strategy:
A complete classification of the point symmetries admitted by gBBMB PDE equations
is followed by the use of the one-dimensional
point symmetry groups to reduce the PDE into ordinary differential equations (ODEs).
Kudryashov method is then recalled and shown to lead to exact solutions. A complete classification of the low-order conservation laws is finally based on the method
of multipliers. Trivial line-soliton solutions and the corresponding differential
invariants finally lead to a reduction and integration of the problem by quadratures.
C. A few marginal technical comments:
1. The Abstract is too long. It should not contain
the trivial explanatory sentences sampled by the first six lines:
"Partial differential equations are used to describe ...",
or by the statement that 'It is well known that the integrability
of a differential equation is strongly related to the existence
of conservation laws".
2. Concerning the style one can add that everywhere, the authors might
also reduce the use of some of the space-filling redundant words
like "Firstly, Moreover, In particular, Then, so, in addition," etc.
3. These are symptomatic observations: the scientific level of the
message is low. The text contains only too few, and not too impressive,
new results (not adding, typically, too much to the older results and/or
to the more recent results of the papers like [14]).
4. In places, the MS makes an impression of an introductory reading
for students rather than of a paper designed for an international scientific
Journal.
5. The style is only too casual, with
a.
misprints:
= "of a a general"
b.
omissions:
= "We study [what?] from the standpoint ..."
= "to approximate [the solutions of] the gBBMB equation (2)".
c.
repetitiveness:
= the pair of introductory sentences
"Partial differential equations are used to describe ..."
initiates a section twice.
D. Recommendation:
Reject.
The main single reason is that without a proper citation(!)
the MS is only too closely related to the older paper
"Conservation laws and exact solutions of a Generalized
Benjamin–Bona–Mahony–Burgers equation"
published, in 2016, by the same authors, in
Chaos, Solitons & Fractals 89, 578-583.
Author Response
Please find as attached.
Author Response File: Author Response.pdf
Reviewer 3 Report
The manuscript is devoted to investigation of a generalized family of Benjamin-Bona-Mahony-Burgers (gBBMB) equations by using the methods of classical and modern group analysis. Lie point group classification of gBBMB equation with respect to an arbitrary function in convective term is performed, and symmetry reduction of the considered equation on several subalgebras of the obtained symmetry algebras is done. It is shown that in some special cases the obtained reduced equations can be solved explicitly by Kudryashov’s method. As an example, the line soliton solution as a group invariant solution is constructed. Also, by employing the multiplier method, conservation laws for the equation in question are obtained.
In spite of the fact that all methods that are used in the manuscript are well-known, the obtained results for the gBBMB equation are new and lies within the scope of Symmetry. Therefore, the manuscript is suitable for publication
in this journal. However, some improvements are needed before publication.
- A nonclassical definition of operator \nabla is used in Eq. (2). It is not a good practice. Usually, \nabla is a vector differential operator, not scalar one. I recommend to change the symbol for this operator in Eq. (2) or rewrite this equation in the form similar to Eq. (5).
- In Introduction it is necessary to add references to books and papers on modern and classical Lie group analysis that are presented in the reference list.
- It follows from Eq. (5) that in fact the group classification was performed with respect to the function (F_u-\beta), not F. I recommend denote this function by a new character, say G. In this case the results of group classification presented in Theorem 2.1 can be written more simply because in this case \beta=0 and f2=0.
- Why the linear case of Eq. (2) is not considered during the group classification?
- It is not clear for me why in Section 3 the symmetry reduction is performed only for given four cases. Usually, an optimal system of subalgebras is constructed and symmetry reduction is performed for each subalgebra from this optimal system. It follows from Theorem 2.1 that symmetry algebras for Eq. (2) are only 3- and 4-dimensional. For such low-dimensional algebras optimal systems of subalgebras are well-known and can be found, e.g., in [Patera J., Winternitz P. Subalgebras of real three- and four-dimensional Lie algebras // J. Math. Phys. 1977. Vol. 18, №. 7. P. 1449–1455].
- In Theorem 5.1 parameter c is not arbitrary because the inequality с \gamma > \alpha should be fulfilled. Also, a special case c=\alpha/ \gamma arises. This case should be considered separately.
- There is an incorrect reference on Page 9: “Splitting equation (37)” should be printed instead of “Splitting equation (45)”.
Recommendation: revision.
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
Dear Editor,
this paper deals with the generalized Benjamin-Bona-Mahony-Burgers equations depending on three arbitrary constants and an arbitrary function. The Authors classify the symmetry reductions of the gBBMB, find several exact solutions, and obtain wave solutions of this equation. The paper is well written and sound. My only concern regards previous literature: somehow in contrast with the recent academic habits, the Authors do not cite some of their own works which could have been useful from the perspective of the reader. In particular, I would like to understand what is the relation between the results of the present paper and those found in Bruzón et al, Chaos, Solitons and Fractals 89 (2016).
Apart from that, I think that the paper deserves publication on Symmetry.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
I support acceptance provided a larger and better figure is
included with a revised version. The different t parameters
for the three jointly plotted curves would be better
discernible if three figures, with the solitons further apart
(t=0, 5, 10) were provided [figures (a), (b) and (c) stacked
on top of each other]. Does the solitions have infinite extent
for given t, in the x and y directions? This should be clarified.
Author Response
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Author Response File: Author Response.pdf
Reviewer 2 Report
The authors amended the text appropriately.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
The authors have made satisfactory revisions in accordance with my previous comments and suggestions. As a result, I believe that the current version of the manuscript is applicable for publication in Symmetry.
Author Response
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Author Response File: Author Response.pdf
