Modified Simplest Equation Method (MMSE)

• MMSE is a direct method that transforms nonlinear PDEs into ODEs by representing the solution as a finite series in an auxiliary function.
• It employs a balance equation to match dominant terms, leading to a nonlinear algebraic system for determining coefficients and admissible parameters.
• The method accommodates various simplest equations—such as Bernoulli, Riccati, and elliptic types—to generate diverse exact solutions like solitary and kink waves.

Modified Simplest Equation Method (MMSE) is a direct exact-solution methodology for nonlinear differential equations in which a nonlinear PDE is reduced, often by a traveling-wave ansatz, to an ODE whose solution is represented as a finite series in an auxiliary function satisfying a simpler differential equation. The coefficients of the series, together with admissible parameter values of the original equation, are determined by balancing dominant terms and solving the resulting nonlinear algebraic system. In the literature surveyed here, MMSE appears both as a development of the method of simplest equation rooted in Painlevé-type singularity analysis and as a broader constructive framework that later incorporates multiple simplest equations, Hirota-type structures, and other dependent-variable transformations (Dimitrova et al., 2013, Vitanov, 2019, Efimova, 2010).

1. Historical lineage and conceptual position

MMSE is historically tied to the method of simplest equation and, more distantly, to the Painlevé program. The 2013 integrability overview states that the method of simplest equation uses the first step of Painlevé analysis, while the modified method replaces that step by an explicit balance-equation procedure (Dimitrova et al., 2013). In that formulation, the motivation is analytic rather than purely formal: singularity structure suggests finite expansions, and dominant-balance reasoning guides the selection of a closed ansatz.

A later review places MMSE in a longer sequence running from transformation-based methods such as the Hopf–Cole transformation, through inverse scattering and Hirota’s direct method, to Kudryashov’s method of simplest equation and its subsequent refinements by Vitanov and collaborators (Vitanov, 2019). That review also introduces a useful terminological nuance: the earlier single-simplest-equation version, denoted MMSE1, is described as equivalent to MSE in the sense that both rely on a simplest equation, a finite expansion, a balance condition, and an algebraic consistency system. The distinctive enlargement of MMSE then lies in its later generalizations: multiple simplest equations, multiple balance equations, and broader transformations of the dependent variable (Vitanov, 2019).

An earlier 2010 formulation broadens the method in a different direction. There the modified simplest equation method is not restricted to classical traveling-wave reductions and allows the auxiliary function to satisfy a linear differential system in $x$ and $t$, such as

$$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$

This version is explicitly presented as capable of generating non-traveling-wave exact solutions and as amenable to symbolic automation (Efimova, 2010).

2. Canonical algorithm and the balance equation

In its standard traveling-wave form, MMSE begins with a reduction of a nonlinear PDE to an ODE,

$$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$

where $\xi$ is typically a traveling variable. The unknown profile is then sought in finite-series form

$F(\xi)=\sum_{i=0}^{r} a_i [\phi(\xi)]^i,$

with $\phi(\xi)$ satisfying a simpler auxiliary ODE, called the simplest equation (Dimitrova et al., 2013).

After substitution, all derivatives of $F$ are rewritten in terms of powers of $\phi$, yielding a polynomial identity of the form

$$P=\Omega_0+\Omega_1\phi+\Omega_2\phi^2+\cdots+\Omega_r\phi^r=0.$$

The conditions

$t$0

produce a nonlinear algebraic system for the coefficients of the ansatz, the parameters of the simplest equation, and often admissible parameter combinations of the original PDE (Dimitrova et al., 2013).

The defining technical step is the balance equation. Instead of inferring the truncation order solely from leading-order singularity analysis, MMSE balances the highest powers generated by different terms of the reduced ODE after substitution. This ensures that at least two terms contribute to the dominant order and that the truncation is nontrivial and self-consistent. One discussion explicitly notes that this role is analogous to leading-order analysis in Painlevé-type methods (Dimitrova, 2013).

Different implementations lead to different concrete balance relations. For example, the generalized exp-function analysis for a class of water-wave equations yields

$t$1

depending on which highest exponential powers are matched (Dimitrova, 2013). In the artery-with-aneurysm study, the Riccati-based construction produces the balance

$t$2

and with $t$3 this gives the quadratic ansatz $t$4 (Nikolova et al., 2017). In the generalized KdV treatment based on a polynomial simplest equation, the balance gives

$t$5

for the ansatz degree and simplest-equation degree (Vitanov et al., 2015).

A technically important refinement appears in the 2015 special-function formulation. If

$t$6

then repeated derivatives of $t$7, with $t$8 polynomial in $t$9, can be written as

$$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$0

where $$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$1 and $$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$2 are polynomials. The reduced equation therefore takes the structured form

$$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$3

and coefficient vanishing is imposed separately on $$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$4 and $$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$5 (Vitanov et al., 2015).

3. Simplest equations and generated solution classes

The simplest equation is the core modeling choice of MMSE. The Bernoulli equation,

$$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$6

and the Riccati equation,

$$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$7

are among the most frequently used auxiliary equations in the surveyed literature (Vitanov et al., 2012, Dimitrova et al., 2013). An extended tanh equation,

$$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$8

is used as a Riccati specialization with the same highest nonlinear power and therefore the same balance structure as the Riccati case (Vitanov et al., 2012).

A more general auxiliary class is

$$Z_x^{(l)} = b_1 Z_x^{(l-1)}+\cdots+b_{l-1}Z,\qquad Z_t = c_1 Z_x^{(l-1)}+\cdots+c_{l-1}Z.$$9

which is presented as containing trigonometric, hyperbolic, Jacobi elliptic, and Weierstrass elliptic cases (Vitanov et al., 2015). In that setting the paper introduces the special function

$$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$0

whose particular parameter choices reproduce familiar special functions, including $$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$1, $$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$2, $$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$3, and $$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$4 (Vitanov et al., 2015).

The choice of simplest equation strongly controls the qualitative solution family. Bernoulli-type constructions yield elementary exponential/rational expressions and are repeatedly associated with front or kink profiles (Jordanov et al., 2018). Riccati constructions generate hyperbolic or trigonometric forms, including $$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$5-type profiles (Vitanov et al., 2012). Polynomial and elliptic simplest equations extend the method to Jacobi and Weierstrass solutions (Vitanov et al., 2015, Vitanov et al., 2018).

This diversity is not merely formal. The literature explicitly characterizes MMSE solutions as rational, exponential, hyperbolic, trigonometric, elliptic, solitary-wave, kink-like, periodic, or composite, depending on the auxiliary equation and parameter branch (Dimitrova et al., 2013).

4. Relation to exp-function, Hirota-type, and multi-equation frameworks

One of the clearest structural claims in the MMSE literature is that the exp-function method is not fundamentally separate from MMSE. A 2013 analysis starts from the standard exp-function ansatz

$$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$6

and generalizes it, from the viewpoint of simplest-equation constructions, to

$$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$7

The paper’s stated conclusion is that exp-function constructions are special rational-exponential realizations of the broader MMSE philosophy (Dimitrova, 2013).

Later work extends MMSE well beyond the single auxiliary function. The nonlinear Schrödinger study introduces an extension using two simplest equations, motivated by the complex-valued factorization

$$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$8

with one simplest equation assigned to the phase factor,

$$\mathcal{P}\bigl(F(\xi),F'(\xi),F''(\xi),\dots\bigr)=0,$$9

and another to the real amplitude component (Vitanov et al., 2018). That extension yields Jacobi elliptic, Weierstrass elliptic, and solitary-wave solutions of the nonlinear Schrödinger equation and a more general Schrödinger-type equation (Vitanov et al., 2018).

The 2019 “new developments” paper formulates an extended MMSE in seven steps. Its key generalizations are: more than one simplest equation; a generalized polynomial-type relation for an auxiliary function $\xi$0 that includes both Hirota’s structure and earlier MMSE power-series forms; a transformation $\xi$1 broad enough to include Painlevé-type and sine-Gordon-type substitutions; and the possibility of more than one balance equation (Vitanov, 2019).

A closely related 2019 paper presents the Simple Equations Method (SEsM) as the encompassing framework and states explicitly that SEsM contains MMSE as a particular case when one uses one simple equation and searches for the solution as a power series of the solution of that simple equation (Vitanov, 2019). In that hierarchy, MMSE remains the one-simple-equation case, while SEsM accommodates multiple simple equations and Hirota-type multisoliton constructions.

5. Representative applications

The method is applied across a wide range of nonlinear evolution equations, especially where exact traveling-wave solutions are sought under polynomial nonlinearities or weakly nonintegrable structure.

PDE class or model	Simplest equation(s)	Reported solution type
Extended KdV and generalized Camassa–Holm	Bernoulli, Riccati, extended tanh	Rational-exponential, $\xi$2-type, $\xi$3-type waves
Hyperbolic reaction-diffusion equation with quartic nonlinearity	Bernoulli	Kink wave
Artery with aneurysm, reduced to variable-coefficient KdV–Burgers	Riccati	Exact travelling kink-like wave
Nonlinear Schrödinger equation	Two simplest equations; elliptic reductions	Jacobi, Weierstrass, solitary-wave solutions
Generalized and higher-order KdV equations	Polynomial, Riccati, elliptic simplest equations	Solitary, kink, Jacobi, Weierstrass solutions

In water-wave theory, MMSE is applied to the extended Korteweg–de Vries equation and a generalized Camassa–Holm equation using Bernoulli, Riccati, and extended tanh simplest equations. Some of the resulting exact traveling-wave profiles are stated to correspond to surface water waves (Vitanov et al., 2012). A related 2013 analysis applies an MMSE-inspired generalized exp-function ansatz to a PDE class containing the Camassa–Holm, Degasperis–Procesi, and Fornberg–Whitham equations, all linked in that paper to shallow-water wave theory and wave breaking (Dimitrova, 2013).

In reaction-diffusion theory, MMSE is used for the hyperbolic equation

$\xi$4

with quartic polynomial nonlinearity. The method yields an explicit kink-type traveling-wave solution based on a Bernoulli equation, together with parameter constraints such as

$\xi$5

and the paper emphasizes that the solution exists only on a constrained manifold in parameter space (Jordanov et al., 2018).

In biomechanics, an artery model with a localized axially symmetric dilatation is reduced, by reductive perturbation analysis, to a variable-coefficient KdV–Burgers equation. After removal of variable coefficients, MMSE with a Riccati simplest equation yields an exact travelling-wave solution. The study reports that a healthy artery supports a pure kink, while an aneurysmal artery produces a slight drop followed by a prompt jump, with the drop region shrinking as $\xi$6 decreases (Nikolova et al., 2017).

In dispersive wave theory, the two-simplest-equation extension is applied to the nonlinear Schrödinger equation

$\xi$7

and to a higher-order Schrödinger-type model with inverse and higher-power nonlinearities. The resulting exact solutions include $\xi$8-based and $\xi$9-based expressions, with the $F(\xi)=\sum_{i=0}^{r} a_i [\phi(\xi)]^i,$0 branch reducing to a $F(\xi)=\sum_{i=0}^{r} a_i [\phi(\xi)]^i,$1 solitary wave at $F(\xi)=\sum_{i=0}^{r} a_i [\phi(\xi)]^i,$2 (Vitanov et al., 2018).

6. Scope, limitations, and recurring misconceptions

MMSE is a constructive exact-solution technique, not a general integrability criterion. The Painlevé literature discussed alongside it explicitly warns that the Painlevé property is strong but not universal, and the MMSE itself works only when the chosen ansatz closes and the balance and algebraic constraints are solvable (Dimitrova et al., 2013). A plausible implication is that success depends less on a universal theorem than on the compatibility between the PDE structure and the selected simplest equation.

A recurring practical limitation is parameter restriction. The hyperbolic reaction-diffusion study states plainly that the exact kink solution does not exist for arbitrary coefficients (Jordanov et al., 2018). The 2010 and 2019 reviews likewise stress that substitution often produces substantial nonlinear algebraic systems and that computer algebra is frequently needed (Efimova, 2010, Vitanov, 2019).

Two misconceptions are explicitly corrected in the surveyed literature. First, the exp-function method is not treated as fundamentally distinct from MMSE; rather, it is presented as a special rational-exponential case of the broader simplest-equation framework (Dimitrova, 2013). Second, the modified method should not be conflated with a single immutable algorithm. The literature includes a non-traveling-wave, linear-auxiliary-equation version (Efimova, 2010), a Painlevé-motivated balance-equation version (Dimitrova et al., 2013), a special-function formulation with $F(\xi)=\sum_{i=0}^{r} a_i [\phi(\xi)]^i,$3 structure (Vitanov et al., 2015), and later multi-simplest-equation generalizations that overlap with Hirota-type and SEsM constructions (Vitanov, 2019, Vitanov, 2019).

Within that diversity, the common invariant is clear: MMSE replaces the original nonlinear differential problem by a finite algebraic closure built on an auxiliary solvable equation, and it uses dominant-balance reasoning to make that closure exact rather than merely asymptotic.