Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents a pedagogical review of the oscillatory and exponential behaviors of second-order differential equations, with a focus on the Airy equation, the nonlinear Lorenz equation, and the nonlinear Schrodinger equation. It highlights how the universal behavior of second-order differential equations governs both microscopic and macroscopic phenomena. Overall, the paper is clear and well-organized, and it provides a valuable reference for the public and also for researchers in the field. 

Minor comments or suggestions:

*Figure 1 is basically a screenshot from YouTube, with pause button and progress bar. The author may want to replace it with a proper figure.

*Line 71, maybe remove "In Calculus III" or explicitly mention which section in Calculus III the related topic is discussed.

*Line 114, even though m and hbar are just parameters from a mathematical perspective, it might be good to mention their physical meaning.

*Lines 189 and 492, the classical Lorenz model (1963) should be properly cited.

*Lines 200 and 203, it is hard to separate the headings and the text. The author may want to add a colon or start a new line after the heading.

*Lines 239 and 284, remove the period symbols after the headings.

*Line 326, "Appendix A Appendix A".

*Lines 607 and 608, I don't see the point of using double asterisk symbols.

The author could consider adding a few representative references for completeness and broader context:

* The author can add some references in the first paragraph for the application of tunneling to nuclear fusion in the stars (like the seminal work of Gamow on alpha decay, and the following works on stellar fusion), semiconductor devices (1973 Nobel prize), and scanning tunneling microscopy (1986 Nobel prize).

* The author can add the original WKB papers in addition to mathematical textbook references. Classical derivation from physics textbooks (like the Quantum Mechanics by Landau & Lifshitz, or the Modern Quantum Mechanics by Sakurai) can also be cited.

* The author can also add some references on the nonlinear Schrodinger equation and its application in Bose-Einstein condensation and water waves.

* References on the Lorenz model (like the original Lorenz 1963 paper I mentioned in the report, and some reviews on the follow-up development on the nature of chaos, and its application in the atmosphere) can also be added.

Author Response

Please refer to the attached file for detailed responses. 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors present a work that connects concepts from calculus, nonlinear dynamics and quantum physics. The manuscript is well structured and easy to follow. I particularly liked the boards and figures, which make the text accessible and appealing to a broad general audience. In my opinion, the manuscript deserves to be published after addressing a few main points. Below, I list them.

\subsection*{Comments}
\begin{enumerate}

\item Since the goal of the manuscript is to connect different fields, the main drawback lies in the lack of discussion of previous results available in the literature. The reference list is very limited. I therefore recommend that the authors significantly expand the bibliography to better situate their contribution within the existing body of work.

\item There is a interesting connect between chaos and the NLS equation, see for instance doi.org/10.1103/PhysRevLett.96.024104 ; doi.org/10.1016/0378-4371(95)00434-3 and also with external forces doi.org/10.1007/s40722-025-00413-w

\item The NLS equation plays a central role in the study of modulational instability (Benjamin–Feir instability), which depends critically on the combination of signs of its coefficients. The authors should mention this either in Appendix A or in the introduction, and include references to a few recent works on the topic, for instance, doi.org/10.1007/978-3-319-20690-5_4;

\item The authors mention nonlinear oscillators and solitons (or solitary waves). These two topics are closely related, particularly in the study of soliton interactions with external forces. For instance, the authors should consider to mention the review paper doi.org/10.1063/5.0210903, and the recent articles within different frameworks (doi.org/10.1063/1.5017559; doi.org/10.3390/sym15081478; 10.1016/j.chaos.2023.113799)

\item It would be helpful if the authors included a brief discussion of weakly nonlinear oscillators in Section 2, for example by referencing the Poincaré–Lindstedt expansion.

\item Figure 2 could be expanded to include the dissipative case, accompanied by corresponding graphs that illustrate each scenario more clearly.

\item In Section 4.2, the authors mention the double-well problem. They should include, or at least comment on, the effects of adding dissipation and an external forcing term, as this leads naturally to the possibility of chaotic behavior; 

\item Authors should mention the classical paper of Lorenz "deterministic nonperiodic flow " in the references.

Author Response

Please refer to the attached file for detailed responses. 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed most of the points I previously raised, and I am satisfied with the revisions. I am happy to recommend the manuscript for publication.