Nonlinear Transmission Line: Shock Waves and the Simple Wave Approximation
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors
Comments on “Nonlinear transmission line: shock waves and the simple wave approximation” submitted to Mathematics
This paper studies nonlinear electrical transmission lines, with a specific focus on shock waves in lossy half-nonlinear transmission lines, and the simple wave approximation in lossless nonlinear transmission lines, including cases with both nonlinear capacitors and inductors. The paper's key contributions include the derivation of exact analytical solutions for shock wave profiles (both voltage and current) using nonlinear approximations of the capacitor voltage-charge relationships, and the formulation of a generalized damped Helmholtz-Duffing equation to model these nonlinear dynamics. It also develops a simple wave decoupling method for nonlinear wave equations in lossless transmission lines, which extends earlier work on Josephson transmission lines. However, a significant aspect of the paper is the identification of connections to well-known equations, such as the KdV and mKdV equations, in the limit of small amplitudes.
The manuscript presents some developments relative to the existing literature. Compared to the references cited, which are mainly classical works on nonlinear wave dynamics, this study applies those ideas to a discrete electrical context by deriving explicit analytical solutions for shock wave profiles in nonlinear transmission lines. These types of solutions are not typically found in the cited sources. In relation to the author’s previous work, the main differences lie in the inclusion of current profiles alongside voltage or charge, and the generalization of the simple wave approximation to systems with both nonlinear capacitors and nonlinear inductors. Mathematically, the paper introduces an Abel-type reduction and reformulates the problem in terms of solvable ODEs, which extends the analytical tools used in earlier publications. Compared to the broader literature, which often relies on numerical methods or partial analytical approaches, the paper offers exact solutions for a specific nonlinear circuit model, and connects them to known PDEs such as the KdV and mKdV equations.
The manuscript can be accepted for publication in its present form.
Author Response
I am very grateful to the Reviewer for taking care.
Reviewer 2 Report
Comments and Suggestions for Authors- I certainly need more comments to this phrase:"Note that (36) doesn’t goes to (39) when γ → 0. More specifically, when γ → 0, the solution (36) goes to the weak shock solution." It's on the top of the left column on p. 4. I'm not happy because (35) must hold true when γ → 0, and this seems to entail that $\gamma/T^2\sim 1$, isn't it? Anyway, please explain this issue in the next version of the manuscript.
- I suggest removal of the passage at the bottom of right column on p. 2, which proclaims "changing gears" an so on. It gives nothing for understanding this paper. On the other hand, you can use it for future Fields Prize speech of yours.
- Your paper will benefit if you supply the graphical information that illustrate the waves profile, the shocks formation and so on.
Author Response
- "I certainly need more comments to this phrase:"Note that (36) doesn’t goes to (39) when γ → 0. More specifically, when γ → 0, the solution (36) goes to the weak shock solution." It's on the top of the left column on p. 4. I'm not happy because (35) must hold true when γ → 0, and this seems to entail that $\gamma/T^2\sim 1$, isn't it? Anyway, please explain this issue in the next version of the manuscript." After rereading the paper I understood that the paragraph in question in my paper is redundant so I have removed it altogether.
- "I suggest removal of the passage at the bottom of right column on p. 2, which proclaims "changing gears" an so on. It gives nothing for understanding this paper. On the other hand, you can use it for future Fields Prize speech of yours. " I have removed the passage. Now the Section begins with Consider
the generalized damped Helmholtz-Duffing equation. - "Your paper will benefit if you supply the graphical information that illustrate the waves profile, the shocks formation and so on. " The numerical calculations are being performed and the results hopefully will be presented in the next paper.
I am very grateful to the Reviewer for the constructing criticism.
Reviewer 3 Report
Comments and Suggestions for AuthorsIn the manuscript, a two-part of the theoretical study of shock wave behavior in nonlinear transmission lines was conveyed. In the first part, the authors obtain analytical expression describing the shock wave progression of a nonlinear lossy transmission line made out of inductors, capacitors, and resistors. This paper will contain the expression of capacitor charge and current of the inductor with respect to time and space. In the second part, a idealized approximation of the lossless transmission line is made in form of a simplified explanation of simple wave.
Strengths:
The subject matter has undeniable interest in applied physics and electrical engineering, especially signal propagation, high-frequency electronics and nonlinear circuits.
Lossy and lossless situations are dealt with so that a broader picture of nonlinear transmission line dynamics is presented in the paper.
Finding a solution to such a system that is analytic is no trifling matter, and of theoretical importance.
Weaknesses and Area of Improvements:
Clear and Grammar:
The paper has quite a number of grammatical problems. As another example, we should change the phrase: “both the charge of a capacitor and a current through the inductor” to: “both the charge on a capacitor and current through the inductor.”
The capitalization differs; the first word in the sentence is a small letter, and In the second part is a capital letter in the middle of the sentence.
Word arrangement and its use may be enhanced in terms of accuracy and comprehensibility.
Inability of the abstract to describe the tech nical detail:
There are no particular mathematical or physical details in the description. How does the analytic solution look, e.g.? Is it a Travelling wave solution or self-similar profile or what?
What is the form of dissipation: is it linear resistance or would it be some nonlinear damping term?
Is the nonlinearity in the line a nonlinear capacitance, nonlinear inductance or both?
Contribution, Novelty:
There is no clear articulation of novelty of the work in the abstract. Are there earlier literature dealing with similar problems? What makes this paper better or contributing to what has already been done by researchers about nonlinear transmission lines?
It would also be worth noting that the simple wave approximation was either newly derived in this paper or based on previously known approach.
Physical Relevance:
There is no mentioning of any physical implications of the solutions or practical applications thereof. Do the results apply to high-speed digital circuits, RF signal propagation, or to metamaterials?
A little bit of validation physically, either numerical simulation or at least, experimental comparison, would hold up the scientific merit.
Recommendation:
Minor Revision
Ways to improve it:
Improve your language use and grammar throughout the abstract and the main body so as to make them easier to read and accurate.
Add a more formal declaration on the novelty and possible areas of applications of the work.
Present a discussion of the physical interpretation of the findings, particularly the shock profile and whether the simple-wave approximation used is true or not.
Unless they already appear in the main text, it may be useful to add some comparison with numerical simulations or a note of the feasibility of experiments.
Author Response
We are grateful to the Reviewer both for the words of encouragement
and for the constructive criticism. Taking into account the latter we made the following changes in the text of the paper.
The Reviewer:
The paper has quite a number of grammatical problems. As another example, we should change
the phrase: “both the charge of a capacitor and a current through the inductor” to: “both the charge
on a capacitor and current through the inductor.”
The capitalization differs; the first word in the sentence is a small letter, and In the second part is a
capital letter in the middle of the sentence.
Word arrangement and its use may be enhanced in terms of accuracy and comprehensibility.
The grammatical mistakes were corrected.
The Reviewer.
"Inability of the abstract to describe the technical detail:
There are no particular mathematical or physical details in the description. How does the analytic
solution look, e.g.? Is it a Travelling wave solution or self-similar profile or what?
What is the form of dissipation: is it linear resistance or would it be some nonlinear damping term?
Is the nonlinearity in the line a nonlinear capacitance, nonlinear inductance or both?
There is no clear articulation of novelty of the work in the abstract. Are there earlier literature
dealing with similar problems? What makes this paper better or contributing to what has already
been done by researchers about nonlinear transmission lines?"
We have rewritten the abstract which is now nearly three times longer in comparison with the previous version. We emphasise that both the inductors and the capacitors are (in the general case)
nonlinear, while ohmic resistors considered in the first part of the paper are linear.
The Reviewer:
"It would also be worth noting that the simple wave approximation was either newly derived in this
paper or based on previously known approach."
We have added to the text the following paragraph
"The simple wave approximation, well known in the theory of nonlinear waves, refers to a wave where the relationship between the wave's properties (like displacement, pressure, or density) and its spatial and temporal coordinates can be described by a single, independent function. In essence, it's a wave where the relationship between its amplitude and other characteristics is straightforward, and not influenced by other waves or complex interactions. More specifically, it allows to decouple the waves moving in the opposite
directions (in a 1D case). "
We have also added the sentence:
"We emphasize that our approximation (see below) is based on previously known approach but the application of the approximation to the fully nonlinear transmission line was (to the best of our knowledge) was never done before."
The Reviewer:
"Physical Relevance:
There is no mentioning of any physical implications of the solutions or practical applications thereof.
Do the results apply to high-speed digital circuits, RF signal propagation, or to metamaterials?:
We have expanded the Introduction Section to explain the physical relevance of our studies.
The Reviewer.
"A little bit of validation physically, either numerical simulation or at least, experimental comparison,
would hold up the scientific merit."
The Reviewer.
"Improve your language use and grammar throughout the abstract and the main body so as to make
them easier to read and accurate."
We have improved the language of the paper as best as we could.
The Reviewer.
"Add a more formal declaration on the novelty and possible areas of applications of the work.
Present a discussion of the physical interpretation of the findings, particularly the shock profile and
whether the simple-wave approximation used is true or not."
The issue of the validity of the simple wave approximation is a delicate one.
Honestly speaking, the only argument we can give at the present moment that the approximation is widely used for different nonlinear waves and is believed to be a quite reasonable one.
The Reviewer.
"Unless they already appear in the main text, it may be useful to add some comparison with
numerical simulations or a note of the feasibility of experiments."
We are planning to present the comparison of our results with the numerical simulations in the next paper.
Round 2
Reviewer 2 Report
Comments and Suggestions for Authorsattached
Comments for author File: Comments.pdf
Author Response
"In more concrete words, the manuscript currently addresses three topics
(i) travelling waves in lossy transmission lines ,
(ii) the waves called simple in the lossless lines
(iii) some links to KdV equations.
Each one of these is interesting by itself but I have not seen any links between them explained in the manuscript. Moreover, only the considerations of (i) give a concrete result, which, nevertheless, has not been discussed properly. The author just put a couple of formulae practically with no comments. At the same time he has been saying a lot about the other two topics but with no concrete results."
I have to agree with the Reviewer - there is only weak connection between the two parts of the paper. It is certainly a drawback. I leave to the Editor to decide whether the drawback is serious enough to prevent the publication.
"One more serious flaw of the manuscript is the misuse of the common terminology that’s very misleading. The author talks about the shocks while the common name to what he has found out is kinks, and they are smooth. They can tend to form a shock sometimes, but there are no such results in the manuscript."
Here I would like to politely disagree with the Reviewer. I can recommend my relatively recent publication
E. Kogan,
The kinks, the solitons and the shocks in series connected discrete Josephson transmission lines,
Phys. Stat. Sol. (b) {\bf 259}, 2200160 (2022).
To additionally clarify the issue I have added to the text of the paper a new Section
Weak dissipation
There I show the connection between the waves considered in the paper and stationary dispersive shock waves.
"The `decoupling’ discussed as a new technique in connection with the lossless lines is the very classical and common procedure of setting to Riemann’s invariants, which, by the way, allows for applying powerful methods of integration."
I certainly don't claim to be a discoverer of the simple wave approximation. I just apply it to the problem considered.
"So I recommend the real and a very thorough revision. There should be one topic and one result, carefully elaborated and interestingly exposed"
In principle, I agree with the Reviewer that in a paper ideally should be one result. My (nonideal) paper is what it is. Here I repeat that I leave to the Editor to decide whether that nonideal paper can be published.
Reviewer 3 Report
Comments and Suggestions for AuthorsAccepted
Author Response
I am very grateful to the Reviewer for taking care.