Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations

Round 1

Reviewer 1 Report

The authors construct a (2+1)-d coupled Korteweg De-Vries equation with variable-coefficients by an expansion method. The topic is of current interest and its exact solutions could amend the existing literature on studies on nonlinear equations. The adopted expansion is also a particular application of the transformed rational function method (see Chaos Solitons Fractals, 42(2009), 1356-1363), since all the involved expansions in (3) in the second section are polynomials and thus rational functions of a solution to an integrable ODE. Many language errors should be removed. For example,  in the abstract, "which its coefficients are variables" should be "whose coefficients are variables"; after (2) and before (3) in section 2, "we can be expressed of" should be "we can express"; and in section 5, "can be describe" should be "can describe". Very recently, N-soliton solutions to integrable equations have pretty systematically been studied by the Hirota direct method for both (1+1)-dimensional and (2+1)-dimensional integrable equations (Partial Differ Equ Appl Math, 5(2022), 100220), and also some important class of novel equations in (2+1)-dimensions (see, e.g., Math Comput Simul, 190(2021), 270-279), and for nonlocal integrable equations (see, e.g., Partial Differ Equ Appl Math, 4(2021), 1000190, Acta Mathematica Scientia, 42(2022), 127-140 and Commun Theoret Phys, 74(2022), 065002). The manuscript successfully presents some interesting and potentially applicable special function traveling wave solutions to the selected nonlinear model partial differential equation. Therefore, I recommend publishing this manuscript provided that the authors could amend and/or revise their manuscript to discuss those related studies on exact and explicit solutions to nonlinear wave equations in the literature.  In this way, the interested readers could also benefit more from reading such an article on exact solutions to nonlinear equations.

Author Response

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Reviewer 2 Report

Basically, this is a paper that works with KdV equations and uses F-expansions to deduce NG-Functional solutions.

This paper has some serious flaws in its structure. We can find wrong and duplicate labeling of equations and math expressions.

Introduction seems to be only a cut and paste of several references without the proper care of merging the text between them and give a fluid and good purpose to the collected information. Introduction must be reorganized, since as it is written is a little bit hard to read and to understand the line of reasoning.

Section 2 has a few mistakes in math expressions, and it is to see where the mistakes come from since the algebraic systems are presented in final form.

Some of the expressions in case I, case II and case III supposed emerging from appendix 1, can be simplified to simpler forms. Where is the novelty of this?

Section 3 has some interesting results, but after a briefly and nice introduction, very well constructed, by the way, we can see only the final expressions of NG-Stochastic Solutions. So, it is a very dry section, for sure, not compatible with the intense and hard work that authors made to obtain these expressions.

Section 4 is only one example? Nothing more? What is the real purpose of showing this example, its importance and effect in the world of KdV equations?

Author Response

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Round 2

Reviewer 2 Report

The authors made the proper improvements to the paper and enhanced all its qualities. Thanks for the modifications made. The outcome is, for sure, a better and suitable paper to share with the community. 

 

Also grateful to the authors in properly answering all my questions and doubts properly, and I am satisfied with the answers.