Simple Equations Method (SEsM) Overview

• Simple Equations Method (SEsM) is a methodology that represents solutions of nonlinear PDEs via functions of simpler auxiliary differential equations.
• It employs transformations, structured ansätze, and balance procedures to convert complex nonlinear problems into solvable algebraic systems.
• SEsM unifies various techniques such as the tanh-method, Riccati-based approaches, and Fourier series expansions to solve a broad range of PDEs.

Simple Equations Method (SEsM) is a general methodology for constructing exact, and in principle approximate, solutions of nonlinear partial differential equations by representing the sought solution as a function of solutions of one or several simpler auxiliary differential equations, called simple equations. Its central operation is to replace direct treatment of a nonlinear PDE with a structured cascade: an optional transformation of the dependent variable, an ansatz in auxiliary functions, reduction through simple ODEs or PDEs, balance relations, and solution of a resulting nonlinear algebraic system. In the literature cited here, SEsM is presented both as a direct solution method and as a unifying framework that contains many named expansion, Riccati-based, elliptic-function, exponential, and Fourier constructions as particular cases (Vitanov et al., 2019, Vitanov et al., 2019, Vitanov, 2019, Dimitrova, 2024, Dimitrova, 23 Apr 2025).

1. Conceptual basis

SEsM starts from a nonlinear PDE, schematically

$DE(u,\ldots)=0,$

with $u=u(x,\ldots,t)$, and seeks a representation of the form

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$

or, in multi-component settings,

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$

The purpose of the transformation $T$ is to simplify the nonlinearity, to make it polynomial, or, in favorable cases, to remove it. The cited papers list Painlevé-type expansions, $u=4\arctan F$ for sine-Gordon, and $u=4\tanh^{-1}F$ for sinh-Gordon or Poisson–Boltzmann type equations among the admissible transformations (Vitanov et al., 2019, Dimitrova, 23 Apr 2025).

The transformed function $F$ is then expressed in auxiliary functions $f_1,\dots,f_N$, typically through a multivariate polynomial-type ansatz,

$$\begin{aligned} F &= a + \sum_{i_1=1}^N B_{i_1}f_{i_1} + \sum_{i_1=1}^N\sum_{i_2=1}^N Y_{i_1,i_2}f_{i_1}f_{i_2} + \cdots \ &\quad + \sum_{i_1=1}^N\cdots\sum_{i_N=1}^N \Theta_{i_1,\ldots,i_N}f_{i_1}\cdots f_{i_N}, \end{aligned}$$

although rational forms, finite series, and more general composite expressions are also explicitly allowed. Each auxiliary function is linked to a simple equation, meaning an ODE or PDE that is more simple than the original nonlinear PDE and whose solutions and algebraic or differential properties are known or manageable (Vitanov et al., 2019).

Within this framework, “simple” is relative rather than absolute. The admissible class includes first-order linear equations, Riccati and Bernoulli equations, equations generating trigonometric, hyperbolic, Jacobi elliptic, and Weierstrass elliptic functions, as well as, in the most general formulation, PDEs, ODEs with variable coefficients, stochastic differential equations, and equations with fractional derivatives. This breadth is one reason SEsM is described as a methodology rather than a single fixed ansatz (Vitanov et al., 2019).

2. Algorithmic structure

The literature presents SEsM in two closely related organizational forms: a detailed seven-step scheme and a more compressed four-step algorithm. Both descriptions encode the same core logic: transform, represent, choose simple equations, substitute, balance, and solve (Vitanov et al., 2019, Dimitrova, 23 Apr 2025).

The first stage is the optional transformation of the dependent variable. When the original PDE already has a tractable polynomial nonlinearity, this stage may be skipped and one sets $u=u(x,\ldots,t)$0. Otherwise, a nontrivial $u=u(x,\ldots,t)$1 is used to alter the nonlinearity or bilinearize the problem. The transformed PDE is then written in terms of $u=u(x,\ldots,t)$2, or of several transformed functions $u=u(x,\ldots,t)$3 in the multi-component case (Vitanov et al., 2019, Dimitrova, 2024).

The second stage is the structural ansatz. One chooses how $u=u(x,\ldots,t)$4 depends on the auxiliary functions. In single-function reductions this often becomes

$u=u(x,\ldots,t)$5

while in genuinely multivariate settings one may use mixed products, rational expressions, or composite functions such as

$u=u(x,\ldots,t)$6

The 2024 development places particular emphasis on quadratic composite forms in two simple functions, including

$u=u(x,\ldots,t)$7

and its specialization to a pure product $u=u(x,\ldots,t)$8 (Dimitrova, 2024).

The third stage is the choice of simple equations and, when needed, their reduction. Auxiliary PDEs may be reduced to ODEs through traveling-wave variables such as

$u=u(x,\ldots,t)$9

or analogous coordinates. The reduced functions may then be expressed again through still simpler functions,

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$0

often as finite series in powers of $u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$1 and $u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$2. In this sense SEsM is explicitly nested: the original PDE is rewritten through intermediate fields, which are themselves rewritten through solutions of simple equations (Vitanov et al., 2019, Dimitrova, 23 Apr 2025).

The final stage is substitution, balancing, and algebraic closure. After all replacements, the PDE is converted into a sum of linearly independent functions or monomials in the simple-function variables,

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$3

One then imposes balance conditions so that at least two different terms contribute to each dominant power, and sets all coefficients $u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$4 to zero. The resulting nonlinear algebraic system determines coefficients of the ansatz, parameters of the simple equations, and possibly wave numbers, frequencies, and PDE parameters. Exact parameter solutions yield exact PDE solutions; numerical parameter solutions yield numerical approximations in the same structural ansatz (Vitanov et al., 2019).

3. Mathematical structures and balance procedures

The choice of simple equation determines much of the analytic content of a SEsM construction. The cited papers repeatedly use Riccati-type equations,

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$5

Bernoulli-type equations,

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$6

first-order linear exponential equations,

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$7

and second-order oscillatory equations,

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$8

These choices respectively generate rational-hyperbolic, power-law, exponential, and trigonometric/Fourier structures (Vitanov et al., 2019).

A more general class used in earlier work has the schematic form

$u(x,\ldots,t)=T\big(F(x,\ldots,t)\big),$9

and the squared-derivative class

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$0

which encompasses Weierstrass and Jacobi elliptic functions. This is how SEsM produces elliptic and periodic wave families for generalized Kawahara-type equations and related higher-order models (Vitanov et al., 2019).

A distinctive feature of the 2019 multisoliton paper is the explicit use of more than one balance equation. For the generalized Kawahara family

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$1

one balance relation is reported as

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$2

and for the squared-derivative simplest equation formulation the balance changes to

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$3

These relations constrain truncation order and the degrees admissible in the simple equation (Vitanov et al., 2019).

The same paper singles out the fractional-power simple equation

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$4

with solution $$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$5. Its role is to handle PDEs containing fractional powers. For the traveling-wave treatment of

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$6

the balance procedure yields

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$7

and the resulting exact solution has the form

$$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$8

with $$u_i(x,\ldots,t)=T_i\big[F_i(x,\ldots,t),G_i(x,\ldots,t),\dots\big].$$9 determined by the corresponding algebraic parameter relations. This establishes a direct route from a fractional-power simplest equation to kink-type traveling waves with integer or fractional exponents (Vitanov et al., 2019).

4. SEsM as a unifying framework

The unifying claim of SEsM is explicit and cumulative across the cited papers. The 2019 overview identifies the Modified Method of Simplest Equation, the $T$0-method, the Exp-function method, the Tanh-method, and the Fourier-series method as particular cases. The 2025 paper adds the Jacobi Elliptic Function Expansion Method, the F-Expansion method, the Modified Simple Equation method, the Trial Function Method, the General Projective Riccati Equations Method, and the First Integral Method (Vitanov et al., 2019, Dimitrova, 23 Apr 2025).

Method	SEsM specialization	Source
Modified Method of Simplest Equation	One simple equation; polynomial ansatz in its solution	(Vitanov et al., 2019)
$T$1-method and $T$2-chain	One simple equation for $T$3; polynomial ansatz in $T$4	(Vitanov et al., 2019)
Exp-function method	Several first-order linear simple equations $T$5; rational ansatz in exponentials	(Vitanov et al., 2019)
Tanh-method	One simple equation $T$6; polynomial or modified polynomial ansatz in $T$7	(Vitanov et al., 2019)
Fourier-series method	Infinitely many linear second-order simple equations for trigonometric modes	(Vitanov et al., 2019)
Jacobi Elliptic Function Expansion Method	One Jacobi elliptic simple equation; traveling-wave polynomial expansion	(Dimitrova, 23 Apr 2025)
F-Expansion method	Polynomial ansatz in a function satisfying an elliptic-type ODE	(Dimitrova, 23 Apr 2025)
Trial Function Method	One implicit simple equation for the trial function; finite power-series structure	(Dimitrova, 23 Apr 2025)
General Projective Riccati Equations Method	Projective Riccati simple equation with projective-type ansatz	(Dimitrova, 23 Apr 2025)
First Integral Method	Restricted first-integral ansatz acting as an implicit simple-equation structure	(Dimitrova, 23 Apr 2025)

The identification is not merely terminological. In each case, the cited papers specify which SEsM step is skipped, how many simple equations remain, which ansatz is selected, and which auxiliary equation serves as the simple equation. For the classical $T$8-method, for example, one sets $T$9, obtains

$u=4\arctan F$0

and then uses

$u=4\arctan F$1

which is exactly the SEsM one-simple-equation construction specialized to a polynomial Riccati-type equation for $u=4\arctan F$2 (Vitanov et al., 2019).

The same logic applies to Jacobi elliptic expansion and F-expansion methods. Their distinction from SEsM lies not in a different reduction principle but in a narrower choice of simple equation and ansatz. This suggests that SEsM functions as an umbrella formalism for a large class of direct exact-solution techniques rather than as a competing isolated method.

5. Multisolitons, elliptic families, and composite-function extensions

One major application of SEsM is the reproduction of Hirota-type multisoliton structures. In the KdV case, the multisoliton papers use a logarithmic transformation of a tau-function-like quantity $u=4\arctan F$3 and the finite ansatz

$u=4\arctan F$4

with

$u=4\arctan F$5

so that each $u=4\arctan F$6 is exponential. Substitution yields the algebraic relations

$u=4\arctan F$7

and hence the classical two-soliton interaction coefficient emerges inside SEsM. The same line of argument is generalized to $u=4\arctan F$8-soliton tau functions built from many exponential simple equations (Vitanov et al., 2019, Vitanov, 2019).

The nonintegrable side is represented by generalized Kawahara-type equations and related higher-order nonlinear dispersive models. There SEsM combines Painlevé-derived transformations such as

$u=4\arctan F$9

single-function or polynomial ansätze, and Riccati or elliptic simple equations to obtain solitary, kink-type, and elliptic traveling-wave solutions. These examples are presented as evidence that the method retains the exact-solution capability of the Modified Method of Simplest Equation while extending it to more general constructions (Vitanov et al., 2019, Vitanov, 2019).

The 2024 paper extends SEsM beyond standard traveling-wave reductions by focusing on derivatives of composite functions of two simple-equation solutions. For

$u=4\tanh^{-1}F$0

it invokes the multivariate Faa di Bruno formula, following Constantine and Savits, to organize higher derivatives. In the two-variable, two-function case this yields explicit chain-rule formulas for $u=4\tanh^{-1}F$1, $u=4\tanh^{-1}F$2, $u=4\tanh^{-1}F$3, and $u=4\tanh^{-1}F$4, which are then inserted into the nonlinear PDE

$u=4\tanh^{-1}F$5

With simple equations of Jacobi elliptic type for $u=4\tanh^{-1}F$6 and $u=4\tanh^{-1}F$7, the paper derives explicit solution families including

$u=4\tanh^{-1}F$8

$u=4\tanh^{-1}F$9

$F$0

and

$F$1

together with trigonometric and hyperbolic limits such as $F$2 (Dimitrova, 2024).

6. Scope, limitations, and interpretation

SEsM is primarily a constructive method for exact particular solutions of nonlinear PDEs, especially traveling waves, kinks, solitary waves, periodic waves, and multisolitons. At the same time, the Fourier-series embedding shows that it also encompasses approximate constructions for linear PDEs, because truncated Fourier expansions fit the SEsM pattern of representing the solution through many simple equations. The cited literature further remarks that the method of orthogonal functions is likewise a particular case (Vitanov et al., 2019).

A recurrent misconception is to equate SEsM with a single finite power series in one auxiliary function. The cited papers explicitly reject that restriction: the one-simple-equation polynomial expansion is only the MMSE or modified-simple-equation corner of a much broader formalism. Another misconception is to treat SEsM as intrinsically a traveling-wave method. Traveling-wave reduction is common and often convenient, but the composite-function treatment of $F$3 shows that SEsM also operates directly with multivariable composite structures without an explicit traveling coordinate (Vitanov et al., 2019, Dimitrova, 2024).

Its limitations are also stated, or directly implied, in the cited work. There is no general algorithm for selecting the optimal transformation $F$4, the most effective ansatz for $F$5, or the most productive simple equations. The method searches within a chosen ansatz class, so solutions outside that class are missed. The algebraic systems that arise after substitution can become large and strongly nonlinear, often requiring computer algebra and sometimes yielding only numerical parameter values rather than closed forms. No completeness theorem is claimed: SEsM is a constructive framework for obtaining some exact or approximate solutions, not all possible solutions of a given PDE (Vitanov et al., 2019, Dimitrova, 23 Apr 2025).

The subsequent trajectory of the method broadens rather than narrows its scope. The 2019 papers emphasized unification, multisolitons, Painlevé-type transformations, more than one simple equation, and fractional-power nonlinearities. The 2024 paper formalized composite derivatives through multivariate Faa di Bruno machinery. The 2025 paper expanded the catalog of named methods that can be embedded into SEsM. Taken together, these developments portray SEsM as a flexible methodological architecture whose defining feature is not a particular special function or ansatz, but the systematic reduction of nonlinear PDEs to algebraic consistency conditions through appropriately chosen simple equations (Vitanov et al., 2019, Dimitrova, 2024, Dimitrova, 23 Apr 2025).