The Simple Equations Method (SEsM) and the use of exponential functions for obtaining simple and multisoliton solutions of some nonlinear partial differential equations

Nikolay K. Vitanov

REVIEW 3 major objections 5 minor 118 references

The Simple Equations Method (SEsM) and the use of exponential functions for obtaining simple and multisoliton solutions of some nonlinear partial differential equations

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read One method unifies many exact-solution recipes for nonlinear PDEs

desk verdict SEsM paper has a correct KdV re-derivation but the nonintegrable example is internally inconsistent (γ3 = 48 vs 28) and the 13-equation system is absent, so the main demonstration is incomplete. read the letter →

arxiv 1909.00330 v1 pith:37TJVQXJ submitted 2019-09-01 nlin.SI

Nikolay K. Vitanov This is my paper · ORCID

classification nlin.SI MSC 35Q5135Q5335C08

keywords SimpleEquationsMethodexactsolutionsnonlinearpartialdifferentialmultisolitonKorteweg-deVriesequationModifiedofSimplestexponentialfunctionsnonintegrable

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 argues that the Simple Equations Method (SEsM), which builds exact solutions of nonlinear partial differential equations from solutions of several simpler equations, is general enough to contain many established exact-solution methodologies as particular cases. A reader should care because this turns a scattered collection of ansatz techniques into one schema and because the schema reaches both integrable and nonintegrable equations: SEsM reproduces the two-soliton solution of the Korteweg-de Vries equation and produces a traveling-wave exact solution of a nonintegrable fifth-order equation. The working assumption is that the unknown solution can be written as a finite polynomial or finite sum of products of exponentials or other special functions, with the coefficients fixed by setting an algebraic system to zero.

What carries the argument

The machinery is the ansatz (4): $F=\alpha+\sum_i \beta_i f_i+\sum_{i,j}\gamma_{ij}f_i f_j+\cdots$, expressing the transformed unknown as a finite polynomial in functions $f_i$, each governed by its own simple equation (commonly $df_i/d\xi_i=f_i$ for exponentials). The rest is algorithmic: substitute the ansatz into the transformed equation, impose a balance equation such as $N(l-m)=4(p-1)$ to fix the polynomial degree, then set every coefficient of the resulting polynomial to zero, giving algebraic systems such as (23). The balance equation and the coefficient system are what convert a choice of simple equations into a concrete exact solution.

What would settle it

Apply the SEsM balance equation (33) to a nonlinear PDE whose known exact solution is not of finite polynomial-in-exponential form, such as a KdV-type equation with a rational or higher-order pole solution: if no integer $N$ satisfies the balance condition, or if the resulting algebraic system has only the trivial solution, the claimed universality fails for that equation. A concrete version is to run the full procedure on a standard integrable equation with a known three-soliton solution and check whether the ansatz with product ratios (12) reproduces the known tau function.

Watch

Extended reading notes

Core claim

The central claim is that SEsM is a general container for ansatz-based exact-solution methods: with one simple equation and a power-series representation it reduces to the Modified Method of Simplest Equation; with the relationship (4) it contains the bilinear ansatz of the direct method used for soliton equations; and with the simple equation $dv/d\xi = v^2 - v$ plus a polynomial ansatz it reduces to the 2012 polynomial method based on the kink-shaped solution $1/(1+\exp(\xi))$. The paper demonstrates this reach by two constructions. For the Korteweg-de Vries equation $u_t+\sigma uu_x+u_{xxx}=0$ with $\sigma=-6$, the ansatz $F=1+f_1+f_2+c f_1 f_2$ with $\partial f_i/\partial x=\alpha_i f_i$, $\partial f_i/\partial t=\beta_i f_i$, and exponential $f_i$, leads through the algebraic system (23) to the two-soliton solution (25) with $c=(\alpha_1-\alpha_2)^2/(\alpha_1+\alpha_2)^2$ and $\beta_i=-\alpha_i^3$. For a nonintegrable equation of the form (37), the same machinery with one simple equation yields the exact solution (41). The assertion is therefore that one method, by choosing the number and kind of simple equations, spans both integrable multisoliton problems and nonintegrable equations.

Load-bearing premise

The construction assumes the target equation's solution can be captured by a finite polynomial or finite product-sum in solutions of chosen simple equations, and that the resulting algebraic coefficient system has a nontrivial solution; the paper provides no theorem guaranteeing either.

Editorial extensions

If this is right

If SEsM is as general as claimed, results proved for the Modified Method of Simplest Equation, including balance equations and truncation rules, carry over automatically to the multi-equation setting.
Adding more simple equations for exponentials should produce N-soliton solutions of integrable equations such as KdV without a separate bilinearization step.
For nonintegrable equations, SEsM inherits the ability to find exact traveling-wave polynomial-in-exponential solutions, as shown by solution (41).
Because the simple-equation class (1) includes trigonometric, hyperbolic, Jacobi elliptic, and Weierstrass functions, solutions built from these special functions all fall inside the same framework.
The method gives a uniform, potentially automatable workflow: choose transformation, choose simple equations, balance powers, and solve an algebraic system.

Reading between the lines

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

The paper leaves implicit that the same exponential-product ansatz should reproduce known N-soliton tau functions for other integrable hierarchies, such as the sine-Gordon or modified KdV equations; testing that would confirm the claimed universality beyond KdV.
The method's practical boundary is set by the algebraic system in step 7: for most equations that system is overdetermined, and the paper gives no existence theorem, so mapping which equations admit nontrivial solutions would define the true scope of SEsM.
The balance equation (33) doubles as a quick filter: if it forces a non-integer $N$ or $N<0$, no solution of the assumed polynomial form exists, so the method can be ruled out for an equation before any algebra is attempted.
One could extend SEsM beyond exact solutions by using the same multi-simple-equation decomposition as a numerical or asymptotic ansatz when the exact coefficient system has no solution.

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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

Colleague, here is my read. The KdV two-soliton derivation checks out: the bilinearization, the coefficient c = (α1−α2)^2/(α1+α2)^2, and the final solution (25) are all correct. It is a clear, honest re-derivation of Hirota's 1971 result using exponential simple equations. The reduction of SEsM to Kudryashov's method in Section 3 is also legitimate; the paper explicitly identifies that prior art. Those parts deserve credit.

The soft spots are real but localized. First, the claim that SEsM 'contains as particular cases many other methodologies' is largely definitional, because steps 2, 4, and 5 allow arbitrary solution forms and arbitrary simple equations. That is a naming convention, not a theorem. It is fine as a survey statement, but it should not be marketed as a new result.

Second, and more serious: the nonintegrable example in Section 3 is not verifiable as printed. The paper says a system of 13 nonlinear algebraic equations arises, but never prints the system. The parameter assignments in Eq. (40) are asserted without derivation. And Eq. (41) has a concrete internal inconsistency: the cubic term has coefficient 28·C, while Eq. (40) sets γ3 = 48·C (C being the common prefactor). The other coefficients match. So either the displayed solution does not satisfy the stated parameter relations, or there is a typo. Either way, a reader cannot check the claimed nonintegrable solution, and the abstract's claim about SEsM retaining the MMSE property for nonintegrable equations rests entirely on this example. This is not a fatal flaw in the SEsM concept, but it is a load-bearing defect in the paper's demonstration. The general ansatz-existence concern raised elsewhere is inherent to all ansatz methods; the concrete coefficient mismatch is the actual problem.

Minor issues like 'nonitegrable' and 'bisoliton' are cosmetic.

Who is this for? Applied mathematicians and physicists who use ansatz-based exact solution methods and want a unified taxonomy of the author's program. It is a useful overview, but as a standalone paper it needs the nonintegrable example fixed and the algebraic system shown before its claims can be taken at face value.

My recommendation: send it to peer review, not desk reject. The KdV core is correct and the framework, while not deeply novel, is of practical interest. A serious referee can require the missing algebra and a corrected, verified solution. That is a standard path for a repairable paper.

Referee Report

3 major / 5 minor

Summary. The paper describes the Simple Equations Method (SEsM), a seven-step ansatz framework for constructing exact solutions of nonlinear partial differential equations. The method generalizes the Modified Method of Simplest Equation by allowing several simple equations and flexible functional forms for the solution ansatz, and the authors claim that many existing methods are particular cases of SEsM. As demonstrations, the paper re-derives the two-soliton solution of the Korteweg-de Vries equation through an exponential ansatz, and presents a traveling-wave solution for a nonintegrable fifth-order equation with quadratic nonlinearity. The KdV calculation is carried out in detail through bilinearization and reproduces the standard Hirota interaction coefficient. The nonintegrable example, however, is asserted through an unshown system of 13 algebraic equations, and the printed solution is internally inconsistent, undermining the claim that SEsM retains the ability of the MMSE to produce exact solutions of nonintegrable equations.

Significance. If fully substantiated, the framework could serve as a unifying description of many ansatz-based exact-solution methods, and the KdV example is a genuinely useful illustration that a Hirota-type exponential combination fits naturally inside the SEsM steps. The verification of the KdV two-soliton coefficient c = (alpha1 - alpha2)^2 / (alpha1 + alpha2)^2 is correct and checkable, and this part of the paper is a strength. However, the broader claims of the abstract are not yet supported: the containment claim is underspecified to the point of being nearly tautological, and the only nonintegrable example contains a definite coefficient mismatch. As it stands, the paper is best viewed as an expository contribution whose nonintegrable demonstration must be repaired before the central claims can be accepted.

major comments (3)

[§3, Eqs. (40) and (41)] The printed solution is not internally consistent. Eq. (40) sets gamma3 = 48 * C, where C = 6^{1/2} 35^{3/4} beta gamma / (alpha2^3 beta^3 gamma^2)^{1/4}, whereas the cubic term in Eq. (41) has coefficient 28 * C. Since the system of 13 algebraic equations is not shown, the reader cannot determine which coefficient is correct or verify that Eq. (41) satisfies Eq. (37) with the stated parameter relations. This example is the sole demonstration that SEsM retains the MMSE property for nonintegrable equations, so the coefficient must be corrected and the final solution verified by direct substitution before the abstract's claim can be accepted.
[§3, paragraph following Eq. (39)] The application of the method for p = 2 is said to lead to a system of 13 nonlinear algebraic equations, but this system is never printed, and the parameter solution in Eq. (40) is asserted without derivation. Because this computation carries a central claim of the paper, the authors should provide the complete algebraic system and the derivation or computer-algebra verification of Eqs. (40) and (41), either in the main text or in a supplement.
[§2, Steps 2–5 and §3, first paragraph] The statement that SEsM contains many other methodologies is made at a level of generality where the functional forms in Eqs. (4) and (7) and the choice of simple equations in Step 5 are left essentially unrestricted. As formulated, the containment claim is not a falsifiable mathematical assertion, because no precise class of methods or ansatze is delimited. The paper demonstrates two particular reductions (a Hirota-type exponential combination and Kudryashov's polynomial ansatz), but it does not prove membership for a nontrivial family of methods. The authors should either specify the class of methods covered or soften the abstract's claim accordingly.

minor comments (5)

[§3, Step 4] The text says that a(ξ) is a solution of a simple equation 'of the class (8)', but Eq. (8) is the KdV equation; the intended reference is Eq. (1) or Eq. (32).
[Eq. (14)] The summation notation ‘(l n)’ is not defined and is confusing; standard binomial notation or an explicit explanation of the sums over combinations would improve readability.
[Abstract and throughout] There are several typographical errors, including 'nonitegrable' in the abstract, 'Scienecs' in the affiliation, and 'Mor more applications' in the introduction; these should be corrected.
[§2.1, text after Eq. (25)] The sentence claiming that use of more simple equations 'will lead to solutions containing more solitons' is stated without proof of the general N-soliton reduction in Eq. (14); the two-soliton case is verified, but the general case should be proved, sketched, or attributed to a standard result.
[References] The reference list contains formatting errors such as '92014)' in reference [16] and 'Spronger' in reference [73], and many preprints are cited without arXiv identifiers or DOIs; normalizing these would improve the final version.

Circularity Check

1 steps flagged · score 4.0 of 10

Definitional containment: SEsM's claim to contain many methods follows from its own unrestricted Step 2/4 definitions; the KdV and nonintegrable solution derivations are self-contained.

renaming known result [Abstract and Section 2 Steps 2 and 4; Section 3 Kudryashov-inclusion argument]
"The form of the function F(f1, . . . , fN) is not prescribed and can be given by different relationships, e.g., ... The kinds of the functions A, B, . . . are not prescribed. ... Then the SEsM methodology is reduced to the method of Kudryashov. Hence the method of Kudryashov is particular case of the SEsM methodology."

SEsM is defined in Step 2 and Step 4 with the forms of F(f1,...,fN) and of A[v], B[w], ... explicitly 'not prescribed.' Consequently, the abstract's central claim that 'SEsM contains as particular cases many other methodologies' is true by construction: any ansatz-based method can be described as a particular case by choosing the unconstrained forms. The Section 3 demonstration for Kudryashov does the same thing: it selects 'lack of transformation, one simple equation (Eq.(26)) and the polynomial representation (28)' and then concludes 'the SEsM methodology is reduced to the method of Kudryashov.' The claimed unification is thus a definitional renaming rather than a derived result.

full rationale

The abstract's strongest claim is that SEsM contains many existing exact-solution methodologies. That claim is not derived from a theorem; it is built into the 7-step schema, because Step 2 leaves F(f1,...,fN) 'not prescribed' and Step 4 leaves the auxiliary functions A, B, ... 'not prescribed.' The only explicit inclusion proof, for Kudryashov's method, is an instantiation of these free choices, so the unification is a definitional taxonomy rather than an independent mathematical result. By contrast, the KdV two-soliton derivation in Section 2.1 is self-contained: the ansatz (16) and exponential simple equations (18)-(22) lead to the algebraic system (23), whose nontrivial solution (24) yields the standard two-soliton solution (25), checked against an external known result. Section 3 similarly sets up a polynomial ansatz (30) in the solution of an explicit simple equation (26), derives the balance condition (33), and states a solution of the resulting algebraic system; no fitted parameter is renamed as a prediction. The many self-citations document the method's history but are not load-bearing for the derivations shown. Two non-circular correctness defects should be flagged: the 13-equation algebraic system behind Eq. (40) is never displayed, and Eq. (41) uses coefficient 28 for the cubic term while Eq. (40) sets gamma3 = 48 times the common factor, making the nonintegrable example internally inconsistent and unverifiable as printed. These are verifiability/correctness problems, not circularity. On the circularity axis, the only real issue is the definitional character of the 'contains many methods' umbrella claim, which is partial and does not infect the concrete solution derivations.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The framework's generality is purchased by allowing arbitrary transformations, arbitrary solution forms, and arbitrary simple equations. The price is that existence of a finite ansatz and solvability of the resulting algebraic system are assumed, not proved. The examples depend on several hand-chosen parameters, and the nonintegrable example in Section 3 relies on an algebraic system that is not shown.

free parameters (4)

alpha1, alpha2 and gamma1, gamma2 in the KdV two-soliton solution = arbitrary real constants (with omega_i = -alpha_i^3)
Enter through f_i = exp(alpha_i x - alpha_i^3 t + gamma_i); they parameterize amplitude, speed, and phase of each soliton and are not fixed by the method.
c, the interaction coefficient in F = 1 + f1 + f2 + c f1 f2 = (alpha1 - alpha2)^2 / (alpha1 + alpha2)^2
Selected so that the algebraic system Eq (23) is satisfied; not independently predicted.
Truncation order N and simple-equation degree p in Section 3 = N = 4(p-1)/(l-m); the example takes p = 2, l = 2, m = 1, so N = 4
The balance equation Eq (33) restricts the integers; the particular values are chosen by hand for the example.
Coefficients gamma0 to gamma4, mu, nu, and alpha1 in the nonintegrable solution = Expressions given in Eq (40)
Listed as one nontrivial solution of an unshown 13-equation algebraic system; the derivation of these values is not available in the paper.

assumptions (5)

ad hoc to paper A target PDE solution can be represented as a finite polynomial or series in solutions of simpler equations (Eqs (7), (28), (30), (35)).
This is the defining ansatz of SEsM; no theorem guarantees such a representation for a given PDE.
domain assumption The algebraic system obtained by zeroing coefficients has a nontrivial solution (Step 7).
The paper notes the system is often too complicated for computer algebra systems, so solvability is not guaranteed.
domain assumption The chosen simple equations have known closed-form solutions (exponential and logistic in the examples).
SEsM requires available solutions of the simple equations; for many PDEs such equations may not be known.
domain assumption A suitable transformation T r(F) exists and is invertible for the solutions sought (Step 1).
The paper states no general form of T r(F) is known; the KdV example uses the standard logarithmic transformation.
standard math The bilinear equation Eq (15) is equivalent to the KdV equation under u = (12/sigma)(ln F)_xx.
This is the standard Hirota bilinearization of KdV, treated as a known result.

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Cite this review

Pith. "Pith review of The Simple Equations Method (SEsM) and the use of exponential functions for obtaining simple and multisoliton solutions of some nonlinear partial differential equations." pith.science (2026). https://pith.science/paper/37TJVQXJ

@misc{pith2026190900330,
  author       = {Pith},
  title        = {Pith review of: The Simple Equations Method (SEsM) and the use of exponential functions for obtaining simple and multisoliton solutions of some nonlinear partial differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37TJVQXJ}},
  note         = {Machine review of arXiv:1909.00330}
}

read the original abstract

We discuss the last version as well as applications of a method for obtaining exact solutions of nonlinear partial differential equations. As this version is based on more than one simple equation we call it Simple Equations Method (SEsM). SEsM contains as particular case the Modified Method of Simplest Equation (MMSE) for the case when we use one simple equation and the solution is searched as power series of the solution of the simple equation. SEsM contains as particular cases many other methodologies for obtaining exact solutions of non-linear partial differential equations. We demonstrate that SEsM can lead to multisoliton solutions of integrable nonlinear partial differential equations and in addition we demonstrate that SEsM keeps the property of the Modified Method of Simplest Equation to lead to exact solutions of nonitegrable nonlinear partial differential equations.

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