Solitary wave solutions of FKPP equation using Homogeneous balance method(HB method)

Wei Kou; Xiaopeng Wang; Xurong Chen; Yirui Yang

REVIEW

Solitary wave solutions of FKPP equation using Homogeneous balance method(HB method)

Yirui Yang, Wei Kou, Xiaopeng Wang, Xurong Chen

Pattern Formation and Solitons · Submitted 24 Sep 2020 · 5 references

Pith's verdict: The final two explicit solutions are correct up to a sign typo in Eq. (26), but the paper is a known-solution exercise with no new result.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 27, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 5 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2009.11378 (version 1)

Pith IDO555KOCN, Pith's permanent record of this paper and its review

CitationWei Kou, Xiaopeng Wang, Xurong Chen, Yirui Yang. "Solitary wave solutions of FKPP equation using Homogeneous balance method(HB method)." arXiv:2009.11378 (2020). https://pith.science/paper/O555KOCN

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Short version: the paper does not do what it claims. Eq. (26), presented as the general solitary wave solution, has the wrong sign on the tanh term. Eq. (24) gives the coefficient as (-amk)/2 = -3ak for m=6, while Eq. (26) writes +3ak. With b=0 and a=1/√6, the printed expression does not satisfy u_t - u_xx - u + u^2 = 0. Flip the sign to -3ak and you get Eq. (27), which does satisfy the PDE (I checked the polynomial residual in tanh(z/2) and all coefficients vanish; Eq. (28) works after the same correction).

What is actually new: nothing. The solutions are the standard Fisher-KPP traveling waves, expressible with tanh/sech in the usual way. They are not wrong after the sign fix, but they are known. The paper cites only Wang's HB-method papers, not any prior exact solution of the FKPP equation, and it does not position itself against that literature.

What the paper does well is modest: the homogeneous balance derivation is transparent up to Eq. (24), and the two explicit cases are genuinely checked. The plots are unnecessary but they don't hurt.

Soft spots, in order: (1) the load-bearing step is the line 'It's very easy to get that a can only be ±1/√6' — that is the algebraic condition that makes the candidate solve the PDE, and it is simply asserted; (2) Eq. (17) is called a general solution while a is still free; (3) the sign error in Eq. (26) contradicts both Eq. (24) and the later solutions; (4) there are typos and the abstract is hard to parse.

This is not a serious peer-review candidate. It is a worked exercise with a known result and a one-character slip. If the journal is very tolerant, a corrected short note could be published, but it would add little. I would not cite it.

Referee report

Major objections (2)

[Section III, Eq. (26)] There is a sign error in the general solution. From Eq. (24) with m=6, the hyperbolic-tangent coefficient is (-amk)/2 = -3ak, but Eq. (26) prints +3ak tanh[...]. As printed, Eq. (26) does not reduce to the b=0 case Eq. (27) and does not satisfy Eq. (1): for a=1/sqrt(6), b=0, k=-1/sqrt(6), c=5/6, substitution and expansion in T=tanh((kx+ct)/2) gives a residual of (5/12)(-1+T+T^2-T^3), not zero. Replacing +3ak by -3ak makes Eq. (26) consistent with Eqs. (24), (27), and (28), and the residual then vanishes identically. Since Eq. (26) is presented as the general solution and is the basis of the paper's central claim, this correction is essential.
[Section III, after Eq. (28)] The determination of the parameter a is asserted but never derived. The balance equations (15)-(16) determine k and c in terms of a and b, leaving a free; the sentence 'It's very easy to get that a can only be +/-1/sqrt(6)' is the only step that fixes a. This is load-bearing because the final solutions depend on it. Please include the substitution: for the b=0 case u = -1/(24a^2) sech^2(z/2) + (1/2) tanh(z/2) + 1/2, the FKPP residual reduces to (1/24 - 2/(3(24a^2)^2))(T^2 - 1), which vanishes exactly when a^2 = 1/6. This is a one-line calculation and should be shown in the paper.

Minor objections (5)

[Eq. (7)] The last term in the expression for u_xx, written as f'(phi_xxxx + a phi_xxxx), appears to contain a typo; the second term should presumably be a phi_xxx or an analogous first-order derivative term. Please correct the displayed equation.
[Eq. (2)] The upper limit of the sum in Eq. (2) is missing; it should presumably be N. Please make the summation range explicit.
[Abstract and Introduction] The text contains repeated words and misspellings, for example 'we use the homogeneous balance(HB) method is used to construct' and 'Fisher-Kolomogror-Pertrovskii-Piskmov'. The manuscript needs a careful language edit.
[Section III and Fig. 1] The solutions in Eqs. (27) and (28) are traveling fronts connecting u=0 to u=1, not localized solitary waves. The title and discussion should use 'traveling front' or explicitly justify the authors' terminology.
[Section III, verification statement] The sentence 'the rationality of the solution in this paper can be verified by using the pictures' is not a mathematical verification. The plots illustrate the profiles, but the authors should state the explicit residual check instead of relying on the figures.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

This paper claims that the homogeneous balance method yields two exact solitary-wave solutions of the FKPP equation, with the amplitude parameter forced to ±1/√6.

The reading

The authors aim to show that the homogeneous balance method produces exact solitary-wave solutions of the FKPP equation $u_t - u_{xx} - u + u^2 = 0$. Starting from the ansatz $u = f''(\phi)\phi_x^2 + f'(\phi)\phi_{xx} + a f'(\phi)\phi_x + b$, with $f(\phi) = -6\ln\phi$ and $\phi = 1 + e^{kx+ct}$, they reduce the PDE to a system of algebraic equations for the wave parameters. The resulting solution is a combination of $\operatorname{sech}^2$ and $\tanh$ terms, and a final substitution step is said to fix the free parameter $a$ to $\pm 1/\sqrt{6}$. A sympathetic reader would care because explicit closed-form traveling waves of this standard reaction-diffusion model provide concrete benchmarks for numerical and asymptotic work.

What carries the argument

The load-bearing object is the homogeneous balance ansatz $u = f''\phi_x^2 + f'\phi_{xx} + a f'\phi_x + b$, combined with the special functions $f(\phi) = -6\ln\phi$ and $\phi(x,t) = 1 + e^{kx+ct}$. Balancing the highest powers of $\phi_x$ forces $f''^2 = f^{(4)}$, and the resulting identities $f'f'' = 3f^{(3)}$ and $f'^2 = 6f''$ reduce the PDE to a small algebraic system. Hyperbolic identities then convert the rational exponential expression into $\operatorname{sech}^2$ and $\tanh$ terms, which is what makes the solution read as a solitary wave.

Extended reading notes

Core claim

The central claim is that equation (26), namely $u(x,t) = -\frac{3}{2}k^2\operatorname{sech}^2\!\left(\frac{kx+ct}{2}\right) + 3ak\tanh\!\left(\frac{kx+ct}{2}\right) - 3ak + b$ with $k = (2b-1)/(6a)$ and $c = -5(2b-1)/6$, is an exact solution of the FKPP equation. For $b=0$ the solution takes the form (27), and for $b=1$ it takes the form (28). The paper states that substituting these expressions back into the PDE restricts $a$ to $\pm 1/\sqrt{6}$, so the family collapses to two explicit solitary waves. This is the conclusion a fair reader should test.

Load-bearing premise

The load-bearing premise is the unstated algebraic substitution that restricts $a$ to $\pm 1/\sqrt{6}$; the balance equations derived earlier leave $a$ free, and if that substitution is wrong or incomplete the explicit solutions (27) and (28) do not satisfy the FKPP equation.

Editorial extensions

If this is right

If (26) is exact, the FKPP equation has traveling waves with fixed speed $c = 5/6$ for $b=0$ and $c = -5/6$ for $b=1$, with the wave number tied to the amplitude parameter by $k = (2b-1)/(6a)$.
The restriction $a = \pm 1/\sqrt{6}$ leaves exactly two solitary waves, so the balance method plus the substitution step is claimed to select discrete solutions rather than a continuous family.
The same HB construction, with $f = -m\ln\phi$ and a rebalanced highest power, provides a template for deriving sech-tanh solutions of other reaction-diffusion and convection-diffusion equations.
The paper's plots of the solution surface give a direct visual consistency check that the constructed waves have the expected bell-and-kink shape.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The step that fixes $a = \pm 1/\sqrt{6}$ is asserted, not derived; the honest reading is that the real content of the paper is that single substitution check, and the balance equations alone do not determine $a$.
If the residual vanishes only at those two $a$ values, the ansatz is rigid; if it vanishes for every nonzero $a$, then the stated restriction is unnecessary and the family is continuous.
The hyperbolic identities used here are transferable: any HB solution with $f = -m\ln\phi$ will convert to $\operatorname{sech}^2$ plus $\tanh$, so the calculation can be rerun mechanically on related equations.
A concrete next step would be to run the same two-term ansatz on a related convection-reaction-diffusion equation with an added $u u_x$ term, to see whether a similar discrete parameter selection appears.

Original abstract

In this paper, we use the homogeneous balance(HB) method is used to construct function transformation to solve the nonlinear development equation||Fisher-Kolomogror-Pertrovskii-Piskmov equation (FKPP equation), then the exact solution of FKPP equation is obtained, and the solutionis converted into the form of isolated wave solution. Finally, we have shown the solution of FKPPequation in some picture forms, and the rationality of the solution in this paper can be verified byusing the pictures.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,400 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this paper, we use the homogeneous balance(HB) method is used to construct function transformation to solve the nonlinear development equation||Fisher-Kolomogror-Pertrovskii-Piskmov equation (FKPP equation), then the exact solution of FKPP equation is obtained, and the solutionis converted into the form of isolated wave solution. Finally, we have shown the solution of FKPPequation in some picture forms, and the rationality of the solution in this paper can be verified byusing the pictures.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper, we use the homogeneous balance(HB) method is used to construct function transformation to solve the nonlinear development equation\|\|Fisher-Kolomogror-Pertrovskii-Piskmov equation (FKPP equation), then the exact solution of FKPP equation is obtained, and the solutionis converted into the form of isolated wave solution.	method 0.59	0.95	0.64	asserted 1.00
s1	Finally, we have shown the solution of FKPPequation in some picture forms, and the rationality of the solution in this paper can be verified byusing the pictures.	result 0.46	0.93	0.16	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (5)

[1]

M. Wang, Phys. Lett. A 199, 169 (1995); 213, 279 (1996)

View on Pith 1995
[2]

Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308,31 ( 2003)

View on Pith 2003
[3]

Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 ( 2004)

View on Pith 2004
[4]

Wang, M. and Li, X. (2014), Journal of Applied Mathematic s and Physics, 2, 823-827. doi: 10.4236/jamp.2014.28091

View on Pith arXiv 2014
[6]

Finally, we get the following ﬁgures as FIG. 1 are discussed for the solutions. IV. SUMMAR Y In this paper, the FKPP equation is solved mainly by the HB method, an d the structural form of eq.(2) is mainly adopted for the transformation of the nonlinear function of the c onstructive solution. On this basis, the function transformation form of various solu...

View on Pith

How the literature uses it

4 arXiv papers cite this in 7 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Uses its method or tool 3 papers

To solve the FKPP equation [ref] , we begin with a heuristic solution as, [equation] According to the partial balance principle, the power of $∂ω/∂s$ should be balanced, and the power of $∂ω/∂t$ should also be balanced.

An Analytical Solution of the Balitsky-Kovchegov Equation with the Homogeneous Balance Method · Solutions of FKPP equation with Homogeneous Balance Method · 0.94
The analytical solution is given by homogeneous balance method [ref] .

Page Entropy of Proton System in Deep-Inelastic-Scattering at Small-$x$ · PDFs from evolution equation · 0.97
According to the homogeneous balance method, we can get the solution of Eq.([ref]) through starting with a heuristic solution as [ref]

Analytic solution of Balitsky-Kovchegov equation with running coupling constant using homogeneous balance method · Analytical solution of BK equation with running coupling constant · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

M. Wang, Phys. Lett. A 199, 169 (1995); 213, 279 (1996)

View on Pith 1995

[2] [2]

Y. B. Zhou, M. L. Wang, and Y. M. Wang, Phys. Lett. A 308,31 ( 2003)

View on Pith 2003

[3] [3]

Y. B. Zhou, M. L. Wang, and T. D. Miao, Phys. Lett. A 323, 77 ( 2004)

View on Pith 2004

[4] [4]

Wang, M. and Li, X. (2014), Journal of Applied Mathematic s and Physics, 2, 823-827. doi: 10.4236/jamp.2014.28091

View on Pith arXiv 2014

[5] [6]

Finally, we get the following ﬁgures as FIG. 1 are discussed for the solutions. IV. SUMMAR Y In this paper, the FKPP equation is solved mainly by the HB method, an d the structural form of eq.(2) is mainly adopted for the transformation of the nonlinear function of the c onstructive solution. On this basis, the function transformation form of various solu...

View on Pith