Simple equations method (SEsM) and some of its numerous particular cases

Nikolay K. Vitanov; Zlatinka I. Dimitrova

REVIEW 3 major objections 4 minor 1 cited by

Simple equations method (SEsM) and some of its numerous particular cases

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Simple Equations Method claims to contain the Modified Method of Simplest Equation, the G'/G-method, the Exp-function method, the Tanh-method, and the method of Fourier series for linear equations as particular cases.

desk verdict A correct but mostly definitional taxonomy of exact-solution methods; useful as a survey, not as a new result. read the letter →

arxiv 1908.07459 v1 pith:CPETYDG4 submitted 2019-08-19 nlin.SI

Nikolay K. Vitanov , Zlatinka I. Dimitrova This is my paper · ORCID

classification nlin.SI MSC 35C0535C07

keywords simpleequationsmethodexactsolutionsnonlinearpartialdifferentialofsimplestequationG'/GExp-functionTanhFourierseries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper presents the Simple Equations Method (SEsM), a seven-step recipe for constructing exact solutions of nonlinear partial differential equations by writing the unknown solution as a function of solutions of one or several simpler equations. The central claim is that this single recipe contains the Modified Method of Simplest Equation, the G'/G-method, the Exp-function method, the Tanh-method, and the Fourier-series method for linear equations as particular cases. A final proposition goes further: any method that builds solutions from solutions of simple differential equations, using an arbitrary combination of them, is a particular case of SEsM. If the claim holds, the practical payoff is that many apparently distinct exact-solution techniques become choices inside one machinery, differing only in the chosen transformation, simple equations, and solution form.

What carries the argument

The central object is the seven-step SEsM template, especially Step 2's representation of the transformed unknown as a function of solutions of simple equations and Step 4's representation of the reduced functions as $A[v(\xi)]$, $B[w(\zeta)]$, and so on. The template is deliberately open-ended: the forms of $T(F)$ and of $A,B$ are not prescribed, and $F$ can be the polynomial combination (4) or another form. That open-endedness is what makes the containment arguments work. The balance procedure in Step 6 and the coefficient-zeroing of Step 7 form the shared final step: after substitution, the nonlinear PDE becomes a sum of terms whose coefficients are set to zero, converting the problem into a system of nonlinear algebraic equations for the parameters of the solution and of the underlying simple equations.

What would settle it

To settle the claim, take a published method for exact solutions of nonlinear PDEs that constructs the solution by a route not based on auxiliary differential equations, say a direct integral representation with no auxiliary ODE, and check whether the method can still be re-expressed through SEsM's seven steps; if it cannot, the universal Sec. 3.5 proposition is limited to the family of trial forms built from auxiliary equations. A sharper check is to rerun the paper's proofs under a restricted version of SEsM in which Step 2 is limited to the finite polynomial (4) and Step 4 to the finite series (7); the Exp-function and Fourier-series methods would then fail to be contained, demonstrating that the unrestricted forms are what carry the result.

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Extended reading notes

Core claim

The paper's central claim is that SEsM is an umbrella methodology. In its seven steps one transforms the unknown function $u(x,t)$ by an arbitrary transformation $T(F)$; represents $F$ as a function, typically a polynomial combination, of functions $f_1,\dots,f_N$ connected to simpler differential equations; allows each reduced function $a(\xi)$ to be written as a function $A[v(\xi)]$ of a solution of a simple ordinary differential equation; imposes a balance procedure to keep the final expression non-degenerate; and sets every resulting coefficient to zero to obtain a system of algebraic equations. Because the forms of $T(F)$, $F(f_1,\dots,f_N)$, and $A,B$ are deliberately left unrestricted, each named method is recovered by making the appropriate choices: the G'/G chain follows from one simple equation of the form $dw/d\xi = -w^2 + \sum_{j=0}^N \beta_j w^j$; the Exp-function method follows from $k$ simple equations $df_l/d\xi = l f_l$; the Tanh-method follows from $dv/d\xi = 1 - v^2$; and Fourier series follows from $d^2 v_k/d\xi^2 = -k^2 v_k$. The paper states these as propositions with proofs, and then proves a broader proposition: any method that searches a solution as an arbitrary combination of solutions of $n$ simple differential equations is a particular case of SEsM.

Load-bearing premise

The containment claim rests on the template imposing no restrictions on the admissible transformations, trial functions, or auxiliary functions; if SEsM were limited to fixed finite polynomial forms, most of the named methods would no longer be particular cases.

Editorial extensions

If this is right

The Modified Method of Simplest Equation, the G'/G-method, the Exp-function method, the Tanh-method, and the Fourier-series method for linear partial differential equations are all recoverable from the same SEsM template, so the paper's seven steps give a common language for them.
The G'/G-method is not limited to its standard second-order linear equation: the paper's chain of $(G'/G)_N$ methods corresponds to choosing the simple equation $dw/d\xi = -w^2 + \sum_{j=0}^N \beta_j w^j$, so each $N$ gives a legitimate variant.
A future method built from any collection of simple differential equations, with the solution taken as an arbitrary combination of their solutions, is automatically a particular case of SEsM by the proposition in Sec. 3.5.
The Fourier-series method for linear equations and, by the same argument, methods based on orthogonal functions are particular cases because sine and cosine satisfy $d^2 v_k/d\xi^2 = -k^2 v_k$.
SEsM also generates extensions of the classical methods; for example, the paper exhibits a generalized Exp-function trial form that reduces to the standard method when $K=0$, $A_0=1$, and $B_0=1$.

Reading between the lines

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

The editorial reading is that the containment claim is formal rather than computational: because Steps 2 and 4 allow arbitrary functions and forms, every named method is included by definition, and the genuine question is whether the resulting algebraic systems are solvable for the specific PDE under study.
A testable next step would be to restrict SEsM to finite polynomial trial forms of fixed degree and then ask which of the named methods remain representable; under that restriction the Exp-function and Fourier-series methods would drop out, showing how much of the unification is carried by the unrestricted forms.
The same template could be used proactively: instead of asking whether a known method fits SEsM, one could enumerate simple equations by their special-function solutions and generate new hybrid trial forms that mix, say, a Riccati factor with an exponential factor, without devising a new method from scratch.
The paper's Sec. 3.5 proposition implies that the umbrella covers methods that build the solution as a function of auxiliary ODE solutions; a method that constructs solutions by a fundamentally different route, say a direct integral transform with no auxiliary differential equations, would be outside the claimed scope.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a well-organized summary of known reductions, but the flagship containment claim is true by definition. The paper's own references [105–107] already introduced SEsM, so the genuinely new material is the systematic list of particular cases, not the framework.

What it does well: the algebraic steps in Sec. 3 are correct. Setting w = G'/G turns the simple equation into the G'/G equation; the exp-function ansatz follows from d f_l / dξ = l f_l; tanh follows from d v/dξ = 1 − v²; MMSE is literally the one-simple-equation case. The paper is upfront about the unconstrained nature of the template: Step 2 says no general form of F is known, and Step 4 says the kinds of A, B are not prescribed. That openness is exactly why the later propositions hold. The prose is direct, and the references to prior descriptions of SEsM are present.

Soft spots: the central claim is mostly a definitional absorption. Proposition 3.5 says any method based on simple equations and arbitrary combination is a particular case—which is the same as saying SEsM is the class of all such methods. The abstract says 'we show' SEsM contains these methods, but the containment is achieved by choosing the simple equations and solution forms to match the target method. No new solution, no new measurable prediction, no sharpened generality. Also, the Fourier-series proposition only covers traveling-wave solutions of linear equations, not the boundary/initial-value procedure that gives Fourier's method its name; the abstract overstates this. Minor point: the paper has many self-citations, but they point to the actual prior work, so that is not a flaw.

Bottom line: this is a bibliographic-organizational contribution for people who work with exact solution methods and want a common vocabulary. It does not need fixing in its algebraic content; it needs a more honest framing of what 'particular case' means. A serious referee could push for that framing. I would send it to peer review in a specialist journal with the expectation of a major framing revision—it deserves referee time because the reductions are checkable and the literature organization is useful.

Referee Report

3 major / 4 minor

Summary. The manuscript describes a 'Simple Equations Method' (SEsM) as a general multi-step template for constructing exact solutions of nonlinear PDEs: transform the unknown, represent it as a function of solutions of simpler equations, choose specific functional forms, apply a balance procedure, and solve the resulting algebraic system. It then claims, in a series of propositions, that this template contains as particular cases the Modified Method of Simplest Equation, the G'/G-method, the Exp-function method, the Tanh-method, and the method of Fourier series for linear equations. Section 3.5 further claims that essentially any method based on solutions of simple differential equations and an arbitrary combination of those solutions is a particular case of SEsM.

Significance. If the claimed unification were a substantive mathematical statement, the paper would provide a useful organizing framework for a large family of ansatz-based methods. The specific algebraic reductions that are actually checked in the text are correct: the substitution w=G'/G transforms the Riccati-type simple equation into the G''/G relation, the exp-function construction follows from f_l=exp(l xi), and v=tanh(xi) solves dv/dxi=1-v^2. The paper also gives credit to prior work, e.g., the Riccati relation for the G'/G method is attributed to Kudryashov. However, the central containment claim is not a theorem with mathematical content; it is true by construction because the SEsM template in Section 2 imposes no restrictions on the function F(f_1,...,f_N) or on the functions A, B, ... in Step 4. The paper's value is therefore taxonomic and expository rather than a new proof of inclusion of the listed methods.

major comments (3)

[Sec. 2, Steps 2 and 4; Sec. 3.5] The central claim in the abstract and in Proposition 3.5 that SEsM contains all methods based on solutions of simple equations is a definitional tautology rather than a substantive containment statement. Step 2 states that 'No general form of the function F(f1,...,fN) is known up to now,' and Step 4 states that 'the kinds of the functions A, B, ... are not prescribed.' Given this unrestricted template, the proof of Proposition 3.5, which chooses the same simple equations and the same arbitrary combination inside SEsM, is circular in the sense that the class of methods characterized is exactly the class of SEsM instances. The authors should either impose a nontrivial restriction on the admissible forms of F and A, B, ... so that the containment claims become substantive, or explicitly state that SEsM is an unrestricted framework and that the 'propositions' are definitional identifications. As written, the abstract's claim that the paper 'shows' these containments overstates the mathematical content.
[Sec. 3.5, Fourier-series proposition] The proposition on the method of Fourier series is narrower than the abstract promises. The proof only considers a traveling-wave reduction xi = alpha x + beta t and represents u(xi) as a trigonometric series in that single variable; the same restriction appears in the sentence following Eq. (24). This does not cover the standard Fourier-series method for linear PDEs, which typically proceeds by separation of variables and superposition to satisfy initial and boundary conditions. The abstract's claim that SEsM contains 'the method of Fourier series for obtaining exact and approximate solutions of linear differential equations' is therefore not established. The authors should either restrict the claim to traveling-wave Fourier series or give a proof that actually handles the initial/boundary-value procedure.
[Sec. 3.2, Eq. (19)] The proposition for the Exp-function method states that there are 'k simple equations' with l = 0, 1, ..., k, but this is a list of k+1 equations. This is a minor notational slip, but it also affects the statement of the proposition: the denominator in Eq. (19) includes the j=0 term, so the simple equation for f_0 is needed. The proof itself is correct once this is understood, but the proposition should be stated cleanly.

minor comments (4)

[Eq. (10)] Equation (10) contains a typographical error: the right-hand side should be a polynomial in G'/G, not in G''/G, as the surrounding text and the subsequent chain of equations make clear.
[Throughout] The text contains numerous typographical errors and OCR artifacts, including 'Scienecs', 'Aca d.', 'convetional', 'diﬀerentioa l', 'cosider', 'Diﬀerential', and inconsistent capitalization in the references. A careful copyedit is needed.
[Sec. 3.1] The proposition in Section 3.1 is not a proposition in the usual mathematical sense; it is an identity check. Rewording it as a remark or example would better reflect the level of formality and avoid overstating the result.
[Sec. 3.4] The proof of the Modified Method of Simplest Equation proposition consists only of choosing the same simple equation and the same solution form, which is the same definitional circularity noted above. This should be flagged as a remark about the framework rather than presented as a substantive proof.

Circularity Check

2 steps flagged · score 8.0 of 10

Central containment claim is true by construction: the unrestricted Step 2/Step 4 definitions make any 'simple equation + arbitrary combination' method an SEsM instance, so Proposition 3.5 is a tautology.

self definitional [Section 3.5, Proposition ('A very general case that is particular case of the SEsM methodology') and its proof]
"Let us consider any method of solving the equation DE u(x, t) = 0 that is based on solutions v1(ξ1), ..., vn(ξn), ξ1 = αix + βit + γi (αi, βi, γi: parameters) of n simple differential equations Oiui(ξi) = 0, i = 1, ..., n and let the function u(x, t) be searched as an arbitrary combination of these simple equations. Then this method is a particular case of the SEsM methodology."

The proof immediately 'consider[s] the SEsM methodology based on n simple differential equations Oiui(ξi)=0 ... and on the function u(x,t) that can be any combination of these simple equations.' Since Step 2 says 'No general form of the function F(f1,...,fN) is known up to now' and Step 4 says 'the kinds of the functions A, B, ... are not prescribed,' the hypothesis of the proposition is exactly the SEsM template. The proposition thus asserts that every instance of the unrestricted SEsM template is an SEsM instance. No method is shown to satisfy any condition beyond the definition of SEsM, so the claimed containment is true by definition rather than by a substantive derivation.
self definitional [Section 3.1, Proposition ('The chain of the (G'/G)_N-methods is particular case of the SEsM...') and its proof]
"Let the simple equation be dw/dξ = −w^2 + Σ_{j=0}^N β_j w^j, and let us search for the solution ... u(ξ) = Σ_{m=0}^M α_m w^m ... Let us now set G'/G = w."

This specific reduction is forced by the same open-ended template: the paper chooses the SEsM simple equation to be exactly the Riccati-type equation whose solutions are G'/G, and chooses u to be exactly the G'/G polynomial. Step 2 permits any F and Step 4 permits any A, so the selection is allowed by construction. The subsequent identity G''/G = w' + w^2 is simply the definition of w; it verifies an algebraic equivalence but provides no independent argument that SEsM constrains or predicts the G'/G method. The Exp-function and Tanh propositions are of the same form: pick the same simple equations and the same u and declare containment.

full rationale

The paper's central claim is that SEsM contains MMSE, G'/G, Exp-function, Tanh, and Fourier as particular cases. The proof strategy in every case is to instantiate the SEsM Steps 2-4 with exactly the ansatz and simple equations of the target method. Because the paper explicitly leaves F unrestricted ('No general form of the function F(f1,...,fN) is known up to now') and the functions A,B unrestricted ('the kinds of the functions A, B, ... are not prescribed'), the containment is true by construction: any method of the form 'solutions of simple equations combined arbitrarily' is an SEsM instance by definition. Proposition 3.5 makes this explicit by defining the method class to coincide with the SEsM template. The algebraic identities in Sections 3.1-3.4 are correct, but they never test a substantive restriction of the framework. The Fourier-series proposition is additionally narrower than the abstract: the proof only treats traveling-wave representations, not the initial/boundary-value superposition procedure associated with the Fourier method. Self-citations in the introduction are historical and are not load-bearing for the containment proofs, so the circularity is definitional rather than citation-based. Because the central claim reduces to the definition of SEsM, the score is 8; it is not 10 only because the individual algebraic correspondences (e.g., G''/G = w' + w^2) are genuine identities rather than pure relabelings.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper introduces no new physical entities and no fitted parameters. Its contribution is a naming and taxonomic framework; the axioms are mostly structural assumptions about the method's applicability.

assumptions (4)

domain assumption The transformed equation left-hand side can be expressed as a sum of terms whose coefficients can be set to zero.
Step 6 of Sec. 2 assumes this form; it is not guaranteed for all NPDEs.
domain assumption The balance procedure yields a system of nonlinear algebraic equations that has a nontrivial solution.
Step 7 admits this may fail and suggests numerical solving, so a nontrivial analytic solution is not guaranteed.
ad hoc to paper Any method based on solutions of simple equations can be described by the SEsM template.
Proposition 2 in Sec. 3.5 makes this true by definition of the template, which is the source of the circularity.
standard math Standard calculus and ODE solution existence for the simple equations.
Used throughout the proofs in Sec. 3.

invented entities (1)

SEsM methodology
purpose: Unified framework naming the approach of representing solutions as functions of simple-equation solutions.
The paper introduces this name and claims it subsumes many methods, but no independent falsifiable handle is provided beyond restating existing methods.

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Pith. "Pith review of Simple equations method (SEsM) and some of its numerous particular cases." pith.science (2026). https://pith.science/paper/CPETYDG4

@misc{pith2026190807459,
  author       = {Pith},
  title        = {Pith review of: Simple equations method (SEsM) and some of its numerous particular cases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPETYDG4}},
  note         = {Machine review of arXiv:1908.07459}
}

read the original abstract

We discuss a new version of a method for obtaining exact solutions of nonlinear partial differential equations. We call this method the Simple Equations Method (SEsM). The method is based on representation of the searched solution as function of solutions of one or several simple equations. We show that SEsM contains as particular case the Modified Method of Simplest Equation, G'/G - method, Exp-function method, Tanh-method and the method of Fourier series for obtaining exact and approximate solutions of linear differential equations. These methods are only a small part of the large amount of methods that are particular cases of the methodology of SEsM.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Simple Equations Method (SEsM) and the use of exponential functions for obtaining simple and multisoliton solutions of some nonlinear partial differential equations
nlin.SI 2019-09 conditional novelty 3.0 of 10

A generalized ansatz method (SEsM) is shown to reproduce Hirota's KdV multisoliton solutions and to yield an exact solution of a nonintegrable fifth-order KdV-type equation.

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