Methods for exact solutions of nonlinear ordinary differential equations\

Chengfa Wu; Micheline Musette; Robert Conte; Tuen Wai Ng

REVIEW

Methods for exact solutions of nonlinear ordinary differential equations\

Robert Conte, Micheline Musette, Tuen Wai Ng, Chengfa Wu

Exactly Solvable and Integrable Systems · Submitted 2 Mar 2025 · 30 references

Pith's verdict: Useful Eremenko-framework review with a clean Section 8 extension, but Section 5's headline elliptic solution does not verify as printed and needs corrected equations before the exhaustive claim is credible.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 28, 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 →

· 3 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2503.00834 (version 1)

Pith IDHMLI5I2H, Pith's permanent record of this paper and its review

CitationChengfa Wu, Micheline Musette, Robert Conte, Tuen Wai Ng. "Methods for exact solutions of nonlinear ordinary differential equations\." arXiv:2503.00834 (2025). https://pith.science/paper/HMLI5I2H

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

Two genuinely new items and one broken one. The Section 8 extension, removing Nisha et al.'s restriction (8.2) to get three Abel subequations instead of one, is new, explicit, and checkable; the authors rightly concede it is not exhaustive (their own remark says more general assumptions could yield more). The review's core message, turning Eremenko's theorem into a constructive "necessary" method via Briot-Bouquet subequations and Hermite decomposition, is a good corrective to the G'/G and Exp-method guessing literature. That part is sound and honestly referenced. They also admit Section 7 is a suggestion, not a result, which is the right way to write that section.

The broken one is Section 5, the advertised new fourth-order elliptic solution. As printed, u = a(℘(x;g2,g3) + 5b4/γ) does not satisfy the displayed F1. The leading balance of (2.1) gives a² = −5b4/γ, and cancelling the v² term in u = a(v + c) forces c = −b2/(5b4), not +5b4/γ — incompatible unless an unstated relation ties b2, b4, γ, and none appears. The first integral (5.3) is also not one as written: differentiating it yields +ωuu′ against the −ωu term in (2.1), and the leading term is missing the factor u′ multiplying u‴ (the correct form appears to be b4/24(u‴u′ − u″²/2) − b2/4 u′² + γu⁴/4 − ωu²/2 = K, with the ω sign flipped). Both fixes are local and are probably typos — the text is full of OCR-style garbles, e.g. "k² = b2/(5b3⁴)" for what must be b4 — but until the corrected equations are supplied, the "one and only one" conclusion is unverifiable, and the stress-test note holds up.

Secondary concern: the constructive algorithm is delegated wholesale to the authors' prior papers [23,4,8,9], and the exhaustiveness step for the fourth-order example is asserted rather than shown. A referee should ask for a corrected (5.3), an explicit substitution of the elliptic solution into the corrected first integral, and at least a sketch of why no other F1 coefficient sets survive.

Who it is for: people working on closed-form solutions of polynomial ODEs in optics or plasma physics who want certainty that no meromorphic solution is missed. The consolidation and Section 8 are genuinely useful, and the paper deserves a serious referee — but it should be revised, not accepted as is. I would send it back for correction and verification of Section 5.

Referee report

Significance

If the Section 5 result is correct, the paper delivers a genuinely useful review-level synthesis: the Eremenko completeness criterion, the Briot–Bouquet/Hermite constructive ingredients, and a worked example that upgrades a known sech² pulse to a nondegenerate elliptic solution. The framing 'necessary vs. sufficient' methods is well chosen and the two verifiable external pillars—Eremenko's theorem itself and the classical Briot–Bouquet characterization—are correctly quoted. The weaknesses, however, sit exactly in the paper's one new meromorphic example: the printed first integral (5.3) does not differentiate to the printed ODE (2.1), and the printed elliptic solution does not satisfy the printed subequation F1 because its constant shift is incompatible with the u² coefficient of F1. Since the abstract advertises 'new exact solutions' and Section 5 is the only place a new meromorphic solution is produced, these inconsistencies are load-bearing for the paper's novelty claim, even though each looks like a locally fixable typographical error.

Major objections (3)

[§5, Eq. (5.3)] As printed, Eq. (5.3) is not a first integral of Eq. (2.1). Differentiating (5.3) gives (b4/24)(u⁗ − u″u‴) − (b2/2)u′u″ + γu³u′ + ωuu′ = 0, which would correspond to (2.1) with the ω-term sign flipped (the correct integral of −ωu is −ωu²/2, not +ωu²/2) and with an extra −u″u‴ term. For (5.3) to be the first integral of (2.1) the bracket must read u′u‴ − u″²/2, whose derivative is exactly u′u⁗; the printed bracket u‴ − u″²/2 is missing the factor u′. Since the whole of Section 5—the reduction to Eremenko's class, the two-Laurent-series count, and the Briot–Bouquet subequations—rests on (5.3), the sign of the ωu²/2 term and the missing factor must be corrected and the consequences for F1 re-derived.
[§5, displayed F1 and elliptic solution] Substituting u = a(℘(x;g2,g3) + S) into the printed F1, the coefficient of ℘² is 36a⁴γ²(b2 − a²γS), which vanishes only when S = b2/(a²γ). With the paper's own a² = −5b4/γ, this requires S = −b2/(5b4), not the printed S = 5b4/γ. With the printed shift, the ℘² term survives unless the unstated relation b2 = −25b4²/γ holds, so the displayed solution does not satisfy the displayed F1 = 0 and hence cannot be verified against (2.1) as printed. The error is localized: the printed g2 is consistent with a² = −5b4/γ and S = b2/(a²γ) = −b2/(5b4), so the printed shift 5b4/γ is the apparent defect. Please correct the shift, re-derive g2, g3 and the K-condition, and verify the corrected solution by explicit substitution into F1, (5.3), and (2.1).
[§5, Step 4] The 'one and only one solution F1 = 0' claim is the output of a finite linear computation whose intermediate system is not shown; only the final F1 is displayed. Given that two sign/factor errors have been found in the same section, the exhaustive claim is not reproducible from the manuscript as it stands. Please state the linear system that determines the Briot–Bouquet coefficients (or otherwise give a machine-checkable verification) so that the 'one and only one' assertion can be checked independently.

Minor objections (4)

[Abstract and general] Typos to fix: 'physically relevent' (Abstract), 'vocabulory' (§1), 'particlular' (§6), 'respresents' (§8).
[§5, Remark 2] The claimed invariance of (5.3) under u→−u is true only for the corrected bracket u′u‴ − u″²/2 (both u′ and u‴ change sign, leaving the product invariant) together with the even term u²; the remark should be re-aligned with the corrected first integral.
[§6, Briot–Bouquet remark] The assertion that a first-order Briot–Bouquet ODE with a nondegenerate elliptic general solution must not contain the first power of M′ is attributed to [2, §181]; a one-sentence justification or a more precise citation would help the reader verify this classical fact.
[§6] The Maple algcurves computation is described only in words; stating what 'Weierstrassform' returned for the displayed example (e.g., the resulting rational functions before simplification) would make the third integration method reproducible.

How Pith checks papers → · Full audit of this paper → · Review history (2 revisions) →

The paper

The paper claims

For algebraic autonomous ODEs with one top-degree term and finitely many Laurent series, Eremenko's theorem guarantees that every meromorphic solution is elliptic or degenerate elliptic, so all such solutions can be found by a finite…

The reading

This paper makes a methodological claim about solving nonlinear algebraic autonomous ordinary differential equations without guesswork: if an ODE has exactly one term of maximal total degree and only finitely many Laurent series, then Eremenko's 2006 theorem guarantees that every solution meromorphic on the complex plane is elliptic or degenerate elliptic, so a finite check of Briot-Bouquet subequations can in principle find all such solutions rather than a lucky few. The authors contrast this 'necessary' route with the common 'sufficient' ansatz methods, such as tanh-sech expansions and polynomials in elliptic functions, which by construction cannot see solutions outside their assumed class. To show the method at work they derive a new exact solution of the fourth-order dispersion equation, exhibiting it as a Weierstrass elliptic combination that reduces to the known sech-squared pulse at a special frequency, and they revisit the quintic complex Ginzburg-Landau equation to show how a known nondegenerate elliptic solution arises from the same machinery. They also describe a separate trick for nonmeromorphic, multivalued solutions via Abel subequations, with solutions parametrized through the Lambert function or by direct quadrature once an unnecessary restriction is removed. A careful reader should care because the distinction between finding all solutions of a class and finding some solutions of a class is exactly what separates a method from a catalogue of tricks.

What carries the argument

The load-bearing pair is Eremenko's theorem together with the Briot-Bouquet characterization of elliptic functions. Eremenko's theorem asserts that any algebraic autonomous ODE with one top-degree term and finitely many Laurent series has every meromorphic solution elliptic or degenerate elliptic, so candidates outside that class need not be searched. Briot and Bouquet's characterization says an elliptic function of elliptic order $P$ obeys a first-order polynomial ODE $F(u', u) = 0$ whose degree in $u'$ is $P$ and whose degree in $u$ is the elliptic order of $u'$, which makes the space of candidate first-order subequations finite. Hermite's partial-fraction decomposition, polar part plus polynomial part, supplies the explicit form of the solution once a subequation is found, and the subequation method of Musette and Conte assembles these ingredients into the algorithm this paper reviews.

Extended reading notes

Core claim

The central claim is that for an autonomous algebraic ODE satisfying two easily checked conditions, exactly one top-degree term and finitely many Laurent series, the Eremenko theorem reduces the problem of listing all meromorphic solutions to a finite algebraic procedure. Constructively, the procedure consists of writing the Laurent expansions at all poles, restoring finiteness by passing to a first integral when a positive Fuchs index appears, writing the candidate Briot-Bouquet first-order polynomial subequations whose degrees are fixed by the elliptic order, and forcing the Laurent series to satisfy the subequation to solve for its coefficients, a system the authors emphasize is linear. Applied to the fourth-order dispersion ODE, the procedure yields one and only one solution, expressed as $u = a(\wp(x, g_2, g_3) + 5b_4/\gamma)$, a new elliptic generalization of the known sech-squared pulse (5.1), to which it reduces at the frequency $\omega = 24b_2^2/25b_4$. The same machinery recovers Vernov's nondegenerate elliptic solution of the quintic complex Ginzburg-Landau equation through Hermite decomposition and the identification of the four Laurent series.

Load-bearing premise

The Section 5 classification rests on (5.3) being a first integral of (2.1) under the intended sign convention, yet as printed the ODE has $-\omega u$ while differentiating (5.3) yields $+\omega u$, so a sign convention or correction must be supplied before the new elliptic solution can be verified against the printed equation.

Editorial extensions

If this is right

The new solution $u = a(\wp(x, g_2, g_3) + 5b_4/\gamma)$ of the fourth-order dispersion equation is the only meromorphic solution in its class with a single double pole; the two-double-pole configuration is checked and has no solution.
Because Eremenko's theorem covers all meromorphic solutions, the search for closed-form solutions of any ODE in the class reduces to a finite linear-algebra problem rather than an ansatz.
For the quintic complex Ginzburg-Landau equation, the 'meromorphic implies elliptic' conclusion holds for all parameter values, so any bounded closed-form traveling wave is among the solutions the method can enumerate.
Removing the restriction $\varphi' = \alpha_0 + \alpha_2 \rho^2$ in the nonmeromorphic Abel-subequation method yields three Abel subequations and four constraint sets, parametrizing multivalued solutions no longer expressible through the Lambert function.
Two further discriminant-vanishing cases of the elliptic solution hint at possible pulses on a nonzero background, although their frequencies are not real.

Reading between the lines

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

If the sign discrepancy between the printed ODE (2.1) and the asserted first integral (5.3) is resolved in favour of a consistent convention, the same exhaustive procedure could be re-run on parameter-indexed families of fourth-order dispersion generalizations, where elliptic solutions of higher order may appear.
The linearity of the subequation coefficient system suggests the method could be semi-automated, turning a per-equation derivation into a bulk classification of whole ODE families.
For the CGL5 equation with intrapulse Raman scattering, the three unconstrained Abel subequations are natural candidates to scan for bounded real-axis kinks beyond the Lambert-W one originally found.
A testable extension is to apply the same two-condition test to equations currently solved only by tanh/sech ansatz, to see whether the ansatz solutions saturate the exhaustive list or leave meromorphic siblings undiscovered.

Original abstract

In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by Eremenko, which can be turned into algorithms, thus avoiding \textit{ad hoc} assumptions, able to provide \textit{all} (as opposed to some) solutions in a precise class. To illustrate these methods, we present some new such exact solutions, physically relevent.

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, 2,980 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.36 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.67 · 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 order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by Eremenko, which can be turned into algorithms, thus avoiding \textit{ad hoc} assumptions, able to provide \textit{all} (as opposed to some) solutions in a precise class. To illustrate these methods, we present some new such exact solutions, physically relevent.

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 order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed.	background 1.00	0.04	0.04	asserted 1.00
s1	The goal of this short review is to recall classical, 19th-century results, completed in 2006 by Eremenko, which can be turned into algorithms, thus avoiding \textit{ad hoc} assumptions, able to provide \textit{all} (as opposed to some) solutions in a precise class.	contribution 0.85	0.14	0.51	asserted 0.99
s2	To illustrate these methods, we present some new such exact solutions, physically relevent.	contribution 0.89	0.91	0.14	asserted 1.00

What kind of paper the classifier reports this is: review 0.97 · evidence from proof or derivation 0.52.

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

Figures

**Figure 1.** Raman scattering, periodic solution u(x − x0), b4 = −1, b2 = 1, γ = 5, ω = −21/5, a = 1, g2 = 24/5, g3 = −4/5. The shift x0 is a half-period of ℘: x0 = 1.337 − 1.198i, and the real period is 2ℜ(x0). 6 Example CGL5. A nondegenerate elliptic solution The third order ODE (2.3) for M(ξ) admits for ei ̸= 0 (CGL5 case) exactly one top degree term −4ere 2 i M10 and exactly four Laurent series [9, Eq. (21)] M = A 2 0 ξ −1 1 + κi 4 + 2drA2 0 − 2eidiA6 0 4(1 + 4α2) ξ + O(ξ 2 ) , (6.1) 6

Figures reproduced from arXiv:2503.00834 by the authors.

References

Works cited (30)

[1]

M. Abramowitz, I. Stegun, Handbook of mathematical functions , Tenth printing (Dover, New York, 1972). https://kfk.pw/182101-uploads.pdf

View on Pith 1972
[2]

C. Briot et J.-C. Bouquet, Th´ eorie des fonctions elliptiques , 1` ere ´ edition (Mallet-Bachelier, Paris, 1859); 2i` eme ´ edition (Gauthier-Villars, Paris, 1875). https://gallica.bnf.fr/ark:/12148/bpt6k99571w?rk=21459;2

View on Pith
[3]

R. Conte and M. Musette, Link between solitary waves and projective Riccati equations, J. Phys. A 25 (1992) 5609–5623. http://dx.doi.org/10.1088/0305-4470/25/21/019

View on Pith doi:10.1088/0305-4470/25/21/019 1992
[4]

R. Conte and M. Musette, Elliptic general analytic solutions, Studies in ap- plied mathematics 123 (2009) 63–81. https://doi.org/10.1111/j.1467-9590.2009.00447.x http://arxiv.org/abs/0903.2009

View on Pith arXiv 2009
[5]

R. Conte and M. Musette, The Painlev´ e handbook, Mathematical physics studies, xxxi+389 pages (Springer Nature, Switzerland, 2020). https://doi.org/10.1007/978-3-030-53340-3

View on Pith doi:10.1007/978-3-030-53340-3 2020
[6]

R. Conte, M. Musette, Tuen Wai Ng and Chengfa Wu, New solutions to the complex Ginzburg-Landau equations, Physical review E 106:4 (2022) L042201. https://doi.org/10.1103/PhysRevE.106.L042201 https://arXiv.org/abs/2208.14945 https://hal.science/hal-04547537

View on Pith Pith review arXiv 2022
[7]

R. Conte, M. Musette, Tuen Wai Ng and Chengfa Wu, All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations, Physics letters A 481 (2023) 129024 (15 pp) https://doi.org/10.1016/j.physleta.2023.129024 http://arXiv.org/abs/2307.04220

View on Pith arXiv 2023
[8]

R. Conte and T.-W. Ng, Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation, Teoreticheskaya i Matematicheskaya Fizika 172 (2012) 224–235. Theor. math. phys.172 (2012) 1073–1084. http://dx.doi.org/10.1007/s11232- 012-0096-4 http://arXiv.org/abs/1204.3028

View on Pith Pith review arXiv 2012
[9]

R. Conte and T.W. Ng, Meromorphic traveling wave solutions of the complex cubic- quintic Ginzburg-Landau equation, Acta applicandae mathematicae 122 (2012) 153–166. http://dx.doi.org/10.1007/s10440-012-9734-y [Corrigenda: change dr to αdr, er to αer in (15), (16).] http://arXiv.org/abs/1204.3032 9

View on Pith Pith review arXiv 2012
[10]

R. Conte, Tuen Wai Ng and Chengfa Wu, Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations, Bulletin des sciences math´ ematiques174 (2022) 103096 (18 pp). https://doi.org/10.1016/j.bulsci.2021.103096 http://arXiv.org/abs/2112.15267

View on Pith arXiv 2022
[11]

R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W function, Advances in computational mathematics 5 (1996) 329–359. https://doi.org/10.1007/BF02124750

View on Pith doi:10.1007/bf02124750 1996
[12]

M.V. Demina and N.A. Kudryashov, Explicit expressions for meromorphic solu- tions of autonomous nonlinear ordinary differential equations, Commun. nonlinear sci. numer. simul. 16 (2011) 1127–1134. https://doi.org/10.1016/j.cnsns.2010.06.035 http://arXiv.org/abs/1112.5445

View on Pith Pith review arXiv 2011
[13]

A.E. Eremenko, Meromorphic traveling wave solutions of the Kuramoto-Sivashinsky equation, J. of mathematical physics, analysis and geometry 2 (3) (2006) 278–286. http://mi.mathnet.ru/eng/jmag/v2/i3/p278 http://arXiv.org/abs/nlin.SI/0504053

View on Pith arXiv 2006
[14]

C. Hermite, Remarques sur la d´ ecomposition en ´ el´ ements simples des fonctions double- ment p´ eriodiques,Annales de la facult´ e des sciences de Toulouse II (1888) C1–C12. O euvres d’Hermite, vol IV, pp 262–273. http://www.numdam.org/item/AFST_1888_1_2__C1_0/

View on Pith
[15]

Mark van Hoeij, package “algcurves”, Maple V (1997). http://www.math.fsu.edu/~hoeij/algcurves.html

View on Pith 1997
[16]

A. Jeffrey and Xu S., Travelling wave solutions to certain non-linear evolution equa- tions, Int. J. Non-Linear Mechanics 24 (1989) 425–429. https://doi.org/10.1016/0020- 7462(89)90029-2

View on Pith doi:10.1016/0020- 1989
[17]

Magnus Karlsson and Anders H¨ o¨ ok, Soliton-like pulses governed by fourth order dispersion in optical fibers, Optics communications 104 (1994) 303–307. https://doi.org/10.1016/0030- 4018(94)90560-6

View on Pith doi:10.1016/0030- 1994
[18]

A.V. Klyachkin, Modulational instability and autowaves in the active media described by the nonlinear equations of Ginzburg-Landau type, preprint 1339, Joffe, Leningrad (1989), unpublished

View on Pith 1989
[19]

N.A. Kudryashov, Exact solutions of the generalized Ginzburg-Landau equa- tion [in Russian], Matematicheskoye modelirovanie 1:9 (1989) 151–158. http://mi.mathnet.ru/eng/mm/v1/i9/p151 http://mi.mathnet.ru/mm2631

View on Pith 1989
[20]

N.A. Kudryashov, Seven common errors in finding exact solu- tions of nonlinear differential equations, Commun. nonlinear sci. nu- mer. simul. 14:9-10 (2009) 3507–3529. https://doi.org/10.1016/j.cnsns.2009.01.023. https://www.sciencedirect.com/science/article/pii/S1007570409000549

View on Pith doi:10.1016/j.cnsns.2009.01.023 2009
[21]

Maple, http://www.maplesoft.com/products/MAPLE/index.shtml

View on Pith
[22]

P. Marcq, H. Chat´ e and R. Conte, Exact solutions of the one-dimensional quintic com- plex Ginzburg-Landau equation, Physica D 73 (1994) 305–317. https://doi.org/10.1016/0167- 2789(94)90102-3 http://arXiv.org/abs/patt-sol/9310004

View on Pith Pith review arXiv 1994
[23]

M. Musette and R. Conte, Analytic solitary waves of nonintegrable equations, Physica D 181 (2003) 70–79. http://dx.doi.org/10.1016/S0167-2789(03)00069-1 http://arXiv.org/abs/nlin.PS/0302051

View on Pith arXiv 2003
[24]

Nisha, Neetu Maan, Amit Goyal, Thokala Soloman Raju, C.N. Kumar, Chirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scattering, Physics letters A 384:26 (2020) 126675 (5pp). https://doi.org/10.1016/j.physleta.2020.126675

View on Pith arXiv 2020
[25]

Ross Parker, Alejandro Aceves, Multi-pulse solitary waves in a fourth-order nonlinear Schr¨ odinger equation, Physica D: Nonlinear phenomena422 (2021) 132890 (12pp). 10

View on Pith 2021
[26]

R.O. Popovych and O.O. Vaneeva, More common errors in finding exact solutions of non- linear differential equations. I, Commun. nonlinear sci. numer. simul. 15 (2010) 3887-3899. http://dx.doi.org/10.1016/j.cnsns.2010.01.037 http://arXiv.org/abs/0911.1848v2

View on Pith Pith review arXiv 2010
[27]

A.M. Samsonov, Nonlinear strain waves in elastic waveguides, Nonlinear waves in solids, 349–382, eds. A. Jeffrey and J. Engelbrecht (Springer-Verlag, Wien, 1994). https://link.springer.com/chapter/10.1007/978-3-7091-2444-4_6

View on Pith doi:10.1007/978-3-7091-2444-4_6 1994
[28]

I.M. Uzunov, V.M. Vassilev, T.N. Arabadzhiev and S.G. Nikolov, Kink solutions of the complex cubic-quintic Ginzburg-Landau equation in the presence of intrapulse Raman scattering, Optik 286 (2023) 171033 (14 pp). https://doi.org/10.1016/j.ijleo.2023.171033 https://www.researchgate.net/publication/371232058

View on Pith arXiv 2023
[29]

V.M. Vassilev, Exact solutions to a family of complex Ginzburg-Landau equations with cubic- quintic nonlinearity, https://arxiv.org/abs/2304.07271 (6pp)

View on Pith Pith review arXiv
[30]

S.Yu. Vernov, Elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation, J. Phys. A 40 9833–9844 (2007). https://doi.org/10.1088/1751-8113/40/32/009 http://arXiv.org/abs/nlin.PS/0602060 11

View on Pith arXiv 2007

How the literature uses it

What this paper does with the arXiv papers it cites

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

What it cites · 17 sentences without a confident label

Uses the method or tool of 3 papers

* The construction of a method (subequation method) [ref] to find a closed form expression of all elliptic and degenerate elliptic solutions of any algebraic ODE.

Elliptic general analytic solutions · Constructive methods implementing the theorem of Eremenko · 0.90
* The construction of a method (subequation method) [ref] to find a closed form expression of all elliptic and degenerate elliptic solutions of any algebraic ODE.

nlin.PS/0302051 · Constructive methods implementing the theorem of Eremenko · 0.90
The technique to compute them efficiently has been explained by Demina and Kudryashov [ref] , this is to identify the four Laurent series ([ref]) to the four expansions of ([ref]) near $ξ=ξ_j - ξ_0$.

Explicit expressions for meromorphic solution of autonomous nonlinear ordinary differential equations · Example CGL5. A nondegenerate elliptic solution · 1.00

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. Abramowitz, I. Stegun, Handbook of mathematical functions , Tenth printing (Dover, New York, 1972). https://kfk.pw/182101-uploads.pdf

View on Pith 1972

[2] [2]

C. Briot et J.-C. Bouquet, Th´ eorie des fonctions elliptiques , 1` ere ´ edition (Mallet-Bachelier, Paris, 1859); 2i` eme ´ edition (Gauthier-Villars, Paris, 1875). https://gallica.bnf.fr/ark:/12148/bpt6k99571w?rk=21459;2

View on Pith

[3] [3]

R. Conte and M. Musette, Link between solitary waves and projective Riccati equations, J. Phys. A 25 (1992) 5609–5623. http://dx.doi.org/10.1088/0305-4470/25/21/019

View on Pith doi:10.1088/0305-4470/25/21/019 1992

[4] [4]

R. Conte and M. Musette, Elliptic general analytic solutions, Studies in ap- plied mathematics 123 (2009) 63–81. https://doi.org/10.1111/j.1467-9590.2009.00447.x http://arxiv.org/abs/0903.2009

View on Pith arXiv 2009

[5] [5]

R. Conte and M. Musette, The Painlev´ e handbook, Mathematical physics studies, xxxi+389 pages (Springer Nature, Switzerland, 2020). https://doi.org/10.1007/978-3-030-53340-3

View on Pith doi:10.1007/978-3-030-53340-3 2020

[6] [6]

R. Conte, M. Musette, Tuen Wai Ng and Chengfa Wu, New solutions to the complex Ginzburg-Landau equations, Physical review E 106:4 (2022) L042201. https://doi.org/10.1103/PhysRevE.106.L042201 https://arXiv.org/abs/2208.14945 https://hal.science/hal-04547537

View on Pith Pith review arXiv 2022

[7] [7]

R. Conte, M. Musette, Tuen Wai Ng and Chengfa Wu, All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations, Physics letters A 481 (2023) 129024 (15 pp) https://doi.org/10.1016/j.physleta.2023.129024 http://arXiv.org/abs/2307.04220

View on Pith arXiv 2023

[8] [8]

R. Conte and T.-W. Ng, Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation, Teoreticheskaya i Matematicheskaya Fizika 172 (2012) 224–235. Theor. math. phys.172 (2012) 1073–1084. http://dx.doi.org/10.1007/s11232- 012-0096-4 http://arXiv.org/abs/1204.3028

View on Pith Pith review arXiv 2012

[9] [9]

R. Conte and T.W. Ng, Meromorphic traveling wave solutions of the complex cubic- quintic Ginzburg-Landau equation, Acta applicandae mathematicae 122 (2012) 153–166. http://dx.doi.org/10.1007/s10440-012-9734-y [Corrigenda: change dr to αdr, er to αer in (15), (16).] http://arXiv.org/abs/1204.3032 9

View on Pith Pith review arXiv 2012

[10] [10]

R. Conte, Tuen Wai Ng and Chengfa Wu, Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations, Bulletin des sciences math´ ematiques174 (2022) 103096 (18 pp). https://doi.org/10.1016/j.bulsci.2021.103096 http://arXiv.org/abs/2112.15267

View on Pith arXiv 2022

[11] [11]

R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W function, Advances in computational mathematics 5 (1996) 329–359. https://doi.org/10.1007/BF02124750

View on Pith doi:10.1007/bf02124750 1996

[12] [12]

M.V. Demina and N.A. Kudryashov, Explicit expressions for meromorphic solu- tions of autonomous nonlinear ordinary differential equations, Commun. nonlinear sci. numer. simul. 16 (2011) 1127–1134. https://doi.org/10.1016/j.cnsns.2010.06.035 http://arXiv.org/abs/1112.5445

View on Pith Pith review arXiv 2011

[13] [13]

A.E. Eremenko, Meromorphic traveling wave solutions of the Kuramoto-Sivashinsky equation, J. of mathematical physics, analysis and geometry 2 (3) (2006) 278–286. http://mi.mathnet.ru/eng/jmag/v2/i3/p278 http://arXiv.org/abs/nlin.SI/0504053

View on Pith arXiv 2006

[14] [14]

C. Hermite, Remarques sur la d´ ecomposition en ´ el´ ements simples des fonctions double- ment p´ eriodiques,Annales de la facult´ e des sciences de Toulouse II (1888) C1–C12. O euvres d’Hermite, vol IV, pp 262–273. http://www.numdam.org/item/AFST_1888_1_2__C1_0/

View on Pith

[15] [15]

Mark van Hoeij, package “algcurves”, Maple V (1997). http://www.math.fsu.edu/~hoeij/algcurves.html

View on Pith 1997

[16] [16]

A. Jeffrey and Xu S., Travelling wave solutions to certain non-linear evolution equa- tions, Int. J. Non-Linear Mechanics 24 (1989) 425–429. https://doi.org/10.1016/0020- 7462(89)90029-2

View on Pith doi:10.1016/0020- 1989

[17] [17]

Magnus Karlsson and Anders H¨ o¨ ok, Soliton-like pulses governed by fourth order dispersion in optical fibers, Optics communications 104 (1994) 303–307. https://doi.org/10.1016/0030- 4018(94)90560-6

View on Pith doi:10.1016/0030- 1994

[18] [18]

A.V. Klyachkin, Modulational instability and autowaves in the active media described by the nonlinear equations of Ginzburg-Landau type, preprint 1339, Joffe, Leningrad (1989), unpublished

View on Pith 1989

[19] [19]

N.A. Kudryashov, Exact solutions of the generalized Ginzburg-Landau equa- tion [in Russian], Matematicheskoye modelirovanie 1:9 (1989) 151–158. http://mi.mathnet.ru/eng/mm/v1/i9/p151 http://mi.mathnet.ru/mm2631

View on Pith 1989

[20] [20]

N.A. Kudryashov, Seven common errors in finding exact solu- tions of nonlinear differential equations, Commun. nonlinear sci. nu- mer. simul. 14:9-10 (2009) 3507–3529. https://doi.org/10.1016/j.cnsns.2009.01.023. https://www.sciencedirect.com/science/article/pii/S1007570409000549

View on Pith doi:10.1016/j.cnsns.2009.01.023 2009

[21] [21]

Maple, http://www.maplesoft.com/products/MAPLE/index.shtml

View on Pith

[22] [22]

P. Marcq, H. Chat´ e and R. Conte, Exact solutions of the one-dimensional quintic com- plex Ginzburg-Landau equation, Physica D 73 (1994) 305–317. https://doi.org/10.1016/0167- 2789(94)90102-3 http://arXiv.org/abs/patt-sol/9310004

View on Pith Pith review arXiv 1994

[23] [23]

M. Musette and R. Conte, Analytic solitary waves of nonintegrable equations, Physica D 181 (2003) 70–79. http://dx.doi.org/10.1016/S0167-2789(03)00069-1 http://arXiv.org/abs/nlin.PS/0302051

View on Pith arXiv 2003

[24] [24]

Nisha, Neetu Maan, Amit Goyal, Thokala Soloman Raju, C.N. Kumar, Chirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scattering, Physics letters A 384:26 (2020) 126675 (5pp). https://doi.org/10.1016/j.physleta.2020.126675

View on Pith arXiv 2020

[25] [25]

Ross Parker, Alejandro Aceves, Multi-pulse solitary waves in a fourth-order nonlinear Schr¨ odinger equation, Physica D: Nonlinear phenomena422 (2021) 132890 (12pp). 10

View on Pith 2021

[26] [26]

R.O. Popovych and O.O. Vaneeva, More common errors in finding exact solutions of non- linear differential equations. I, Commun. nonlinear sci. numer. simul. 15 (2010) 3887-3899. http://dx.doi.org/10.1016/j.cnsns.2010.01.037 http://arXiv.org/abs/0911.1848v2

View on Pith Pith review arXiv 2010

[27] [27]

A.M. Samsonov, Nonlinear strain waves in elastic waveguides, Nonlinear waves in solids, 349–382, eds. A. Jeffrey and J. Engelbrecht (Springer-Verlag, Wien, 1994). https://link.springer.com/chapter/10.1007/978-3-7091-2444-4_6

View on Pith doi:10.1007/978-3-7091-2444-4_6 1994

[28] [28]

I.M. Uzunov, V.M. Vassilev, T.N. Arabadzhiev and S.G. Nikolov, Kink solutions of the complex cubic-quintic Ginzburg-Landau equation in the presence of intrapulse Raman scattering, Optik 286 (2023) 171033 (14 pp). https://doi.org/10.1016/j.ijleo.2023.171033 https://www.researchgate.net/publication/371232058

View on Pith arXiv 2023

[29] [29]

V.M. Vassilev, Exact solutions to a family of complex Ginzburg-Landau equations with cubic- quintic nonlinearity, https://arxiv.org/abs/2304.07271 (6pp)

View on Pith Pith review arXiv

[30] [30]

S.Yu. Vernov, Elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation, J. Phys. A 40 9833–9844 (2007). https://doi.org/10.1088/1751-8113/40/32/009 http://arXiv.org/abs/nlin.PS/0602060 11

View on Pith arXiv 2007