Lie Symmetry Analysis, Particular Solutions and Conservation Laws of Benjiamin Ono Equation
Round 1
Reviewer 1 Report
Authors prepared an interesting manuscript. However, some modifications are required before publication of this manuscript. I have some observations which are given below:
(i) Literature part need to be enhance with the help of latest references.
(ii) motivation of the manuscript is not clear.
(iii) What is the contribution of the manuscript. It is not clear to us.
(iv) Check mathematical derivation carefully.
(v) Rewrite the conclusion carefully.
(vi) Check the English carefully.
Based on the above mention comments, minor revision is required.
Author Response
Point 1: Literature part need to be enhance with the help of latest references.
Response 1: Added new references.
[1]Gangwei Wang; A new (3+1)-dimensional Schrodinger equation: derivation, soliton solution s and conservation laws， Nonlinear Dynamics, 2021, 104(2): 1595-1602.
[2]Gangwei Wang; Symmetry analysis, analytical solutions and conservation laws of a general ized KdV-Burgers-Kuramoto equation and its fractional version， Fractals-Complex Geometry Pattern s and Scaling in Nature and Society, 2021, 29(04): 2150101
[3]Gangwei Wang; A novel (3+1)-dimensional sine-Gorden and a sinh-Gorden equation: Derivati on, symmetries and conservation laws， Applied Mathematics Letters, 2021, 113: 106768.
[4]Gangwei Wang; Kaitong Yang; Haicheng Gu; Fei Guan; AH Kara; A (2+1)-dimensional sine-Gor don and sinh-Gordon equations with symmetries and kink wave solutions， Nuclear Physics B, 2020, 953(4): 114956 .
Point 2: motivation of the manuscript is not clear.
Response 2: Motivation
In this paper, by applying the Lie group method and the direct symmetry method, Lie algebras of the Benjiamin Ono equation are obtained, and we find that results of the two methods are same. Based on the Lie algebra, Lie symmetry groups, relationships between new solutions and old solutions, two kinds of ODEs as symmetry reductions are obtained. Making use of the power series method, the exact power series solution of the Benjiamin Ono equation has been derived. We also give the conservation laws of Benjiamin Ono equation by means of Ibragimovs new conservation Theorem.
Point 3: What is the contribution of the manuscript. It is not clear to us.
Response 3:Contribution of the manuscript
Making use of the power series method, the exact power series solution of the Benjiamin Ono equation has been derived. We also give the conservation laws of Benjiamin Ono equation by means of Ibragimovs new conservation Theorem.
Point 4:Check mathematical derivation carefully.
Response 4: I checked the derivation of the formula one by one, while correcting it.
Point 5:Rewrite the conclusion carefully.
Response 5: To summarize, by applying the Lie symmetry analysis to the BO equation, we get the classical Lie point symmetry of the equation. We also obtained power series solutions. To the best of our knowledge, power series solutions obtained in this paper are completely new . The BO equation can be solved by using the relationship between the new solutions and the old ones. Finally, we also get some new exact analytic solutions and give the conservation laws of BO equation. These conclusions may be useful for the explanation of some practical physical problems.
Point 6:Check the English carefully.
Response 6:I checked the English grammar one by one and corrected it at the same time.
Reviewer 2 Report
Comments for author File: 
Comments.pdf
Author Response
Point 1: I recommend adding 2-3 explicit examples of integrals to the section "Conservation Laws" with n = 0; 1; 2 allowing the readers to imagine the form of laws.
Response 1: We consider the conserved vectors for generators of BO equation.We have the following cases for classical generators. Four cases are discussed in the paper. No more explanation here.
Point 2: In section 3.2 term "particular solution" should be added.
Response 2: Added "Particular solutions" in section 3.2.
Point 3: There is no citation of papers by AP Chupakhin who is an expert in the Lie algebras of the nonlinear partial differential equations.
Response 3: Added AP Chupakhin's paper as a reference.
[1]Izmaylova, K. K. , and A. P. Chupakhin . "Group theoretical solutions of Schrodinger equation generated by three-dimensional symmetry algebras." nelin.dinam 3(2007).
[2]Golod, A. , and A. P. Chupakhin . "Invariant solution of dynamics of polytropic gas generated by three-dimensional algebras of symmetry." Sib.èlektron.mat.izv 5(2008):229–250.
