MMSEₙ: Multi-Simplest Equation Expansion

• MMSEₙ is a multi-simplest equation expansion method that transforms nonlinear PDEs into exact solutions using multiple auxiliary functions and a truncated multivariate series.
• It extends classical approaches by incorporating several simplest equations and balance conditions to derive multi-soliton solutions for both integrable and nonintegrable PDEs.
• Applications include nonlinear Schrödinger, Korteweg–de Vries, and Maxwell–Cataneo equations, showcasing its versatility in modeling complex wave interactions.

Multi-simplest equation expansion, commonly described in the later review literature as the $\mathrm{MMSE}_n$ extension of the Modified Method of Simplest Equation, is an ansatz-based methodology for exact solutions of nonlinear partial differential equations in which the solution is constructed from more than one auxiliary function, each governed by its own simplest equation. In its generalized form, the dependent variable is first transformed as $u(x,\dots,t)=H(F(x,\dots,t))$, and $F$ is expanded as a multivariate algebraic expression in $f_1,\dots,f_N$; one or more balance equations determine admissible truncations, and coefficient cancellation yields a nonlinear algebraic system whose nontrivial solutions produce exact solutions of the original equation. The method is presented as capable of obtaining multi-soliton solutions of integrable nonlinear partial differential equations if such solutions do exist, and of obtaining particular solutions of nonintegrable nonlinear partial differential equations; a concrete two-simplest-equation realization was developed for nonlinear Schrödinger-type equations (Vitanov, 2019, Vitanov, 2019, Vitanov et al., 2018).

1. Historical emergence and conceptual scope

The review literature places MMSE within a broader sequence of exact-solution methods for nonlinear PDEs that includes the Hopf–Cole transformation, the inverse scattering transform, Hirota’s direct method, and Painlevé truncation or expansion methods. Within that trajectory, the Method of Simplest Equation emerged from the observation that truncated Painlevé-type expansions could be adapted by keeping the constant term and representing the solution as a finite series in powers of the solution of a simpler ODE. The Modified Method of Simplest Equation then developed first in a one-simplest-equation form, denoted in the review as MMSE1, and later in an explicitly multi-simplest-equation form, denoted MMSE$n$ (Vitanov, 2019).

In the generalized interpretation, the decisive conceptual step is the transition from a single auxiliary branch to several auxiliary branches. The earlier single-simplest-equation version was described as connected to the use of a single simplest equation, which is why it was not suitable for bisoliton, trisoliton, and multisoliton solutions. The extended framework therefore introduces multiple simplest equations and multiple auxiliary functions $f_1,\dots,f_N$, allowing cross terms and more complicated interaction structures than a single solitary wave (Vitanov, 2019).

The 2010 modified simplest equation method is an important antecedent but not yet a formal $\mathrm{MMSE}_n$ hierarchy. There the field $u(x,t)$ is expressed through an auxiliary function $Z$ satisfying a coupled linear system in $x$ and $u(x,\dots,t)=H(F(x,\dots,t))$0, and the method is applied directly to nonlinear PDEs of polynomial form. This is conceptually close to a multi-equation closure for an auxiliary function, but it is not a multivariate expansion in several independent simplest equations (Efimova, 2010).

2. General algebraic and differential architecture

The generalized framework begins from a nonlinear PDE written schematically as

$u(x,\dots,t)=H(F(x,\dots,t))$1

followed by a transformation

$u(x,\dots,t)=H(F(x,\dots,t))$2

The transformed quantity $u(x,\dots,t)=H(F(x,\dots,t))$3 is then expanded as a generalized polynomial in several functions,

$u(x,\dots,t)=H(F(x,\dots,t))$4

This is the characteristic multi-simplest-equation expansion: $u(x,\dots,t)=H(F(x,\dots,t))$5 depends on multiple functions, cross terms such as $u(x,\dots,t)=H(F(x,\dots,t))$6 appear, the Hirota-like form is recovered as a special case, and the earlier one-function power series is also recovered as a special case (Vitanov, 2019).

The same review tradition also formulates a reduction hierarchy for the auxiliary branches. The functions $u(x,\dots,t)=H(F(x,\dots,t))$7 may satisfy PDEs which, after reductions such as

$u(x,\dots,t)=H(F(x,\dots,t))$8

become ODE systems. Each reduced auxiliary function can then be represented through another function, for example

$u(x,\dots,t)=H(F(x,\dots,t))$9

and, in finite-series form,

$F$0

The functions $F$1 then satisfy the simplest equations proper. In this sense, $F$2 is not merely a larger truncation of MMSE1; it is a multibranch construction in which different auxiliary components may have different reductions and different simplest equations (Vitanov, 2019, Vitanov, 2019).

A closely related two-branch formulation appears in the nonlinear Schrödinger application, where after reduction the PDE is written schematically as

$F$3

with

$F$4

and finite truncations

$F$5

The truncation orders $F$6 are not arbitrary; they are fixed by balance conditions (Vitanov et al., 2018).

3. Balance equations, truncation, and algebraic closure

The balance procedure is the mechanism that determines whether a chosen ansatz closes consistently. In the two-simplest-equation nonlinear Schrödinger construction it is described as the MMSE replacement for the Painlevé-type leading-order test: one substitutes the finite-series ansatz into the reduced equation, collects terms in powers of the auxiliary functions, and requires compatibility between the highest-order derivative terms and the highest-order nonlinear terms. This yields one or more balance equations for polynomial degrees and solution parameters (Vitanov et al., 2018).

A central extension of $F$7 is precisely that more than one balance equation may be needed. The review of the extended method states that after substitution the left-hand side becomes a sum of terms, each being a function multiplied by a coefficient depending on PDE parameters and solution parameters; comparison of highest-order terms may therefore produce one or more balance equations, in contrast to earlier MMSE settings where typically only one balance equation was used (Vitanov, 2019).

Once a consistent truncation has been identified, all coefficients in the reduced expression are set to zero. The resulting nonlinear algebraic system involves PDE parameters, ansatz parameters, parameters of the simplest equations, and coefficients in the multivariate expansion. Any nontrivial solution of this algebraic system gives an exact solution of the original nonlinear PDE. This coefficient-annihilation step is common to MMSE1 and MMSE$F$8, but in the multi-simplest setting the algebraic system reflects several auxiliary branches rather than a single one (Vitanov, 2019, Vitanov, 2019).

Single-simplest-equation papers illustrate the same logic in a narrower setting. In the artery-with-aneurysm study, the reduced generalized KdV–Burgers equation is solved by a Riccati simplest equation,

$F$9

and the balance relation is

$f_1,\dots,f_N$0

Because $f_1,\dots,f_N$1, the truncation becomes

$f_1,\dots,f_N$2

This is a standard MMSE instance rather than a genuine multi-simplest-equation derivation, but it shows the same balance-and-substitution mechanism in a one-branch form (Nikolova et al., 2017).

4. Two-simplest-equation realization for nonlinear Schrödinger equations

A concrete realization of the multi-simplest-equation idea was given for the classical nonlinear Schrödinger equation

$f_1,\dots,f_N$3

The field is factorized as

$f_1,\dots,f_N$4

with $f_1,\dots,f_N$5 real and $f_1,\dots,f_N$6 complex. This factorization is the first explicit appearance of the two-simplest-equation structure: one simplest equation determines the complex phase factor $f_1,\dots,f_N$7, and the second simplest equation determines the real amplitude $f_1,\dots,f_N$8 (Vitanov et al., 2018).

For the phase branch, the imposed simplest equation is

$f_1,\dots,f_N$9

with solution

$n$0

This choice makes the phase factor exponential and ensures that $n$1, so the nonlinear term remains compatible with the factorized ansatz. After substitution, the nonlinear Schrödinger equation reduces to

$n$2

Since $n$3 is assumed real, the imaginary part must vanish, yielding

$n$4

Multiplication by $n$5 and integration then produce

$n$6

with integration constant $n$7 (Vitanov et al., 2018).

For the amplitude branch, the substitution

$n$8

is introduced, with $n$9 fixed by consistency. The reduced equation becomes

$f_1,\dots,f_N$0

The admissible truncation exponents are

$f_1,\dots,f_N$1

For $f_1,\dots,f_N$2, the amplitude equation has the elliptic form

$f_1,\dots,f_N$3

leading to a Jacobi elliptic-function family and, when $f_1,\dots,f_N$4, to the bright-soliton form with $f_1,\dots,f_N$5. For $f_1,\dots,f_N$6, the reduced equation is of Weierstrass type and yields a Weierstrass elliptic family; the paper takes the special case $f_1,\dots,f_N$7 to obtain an explicit solution (Vitanov et al., 2018).

The same two-simplest-equation procedure is then applied to the generalized nonlinear Schrödinger-type equation

$f_1,\dots,f_N$8

With the same phase ansatz $f_1,\dots,f_N$9 and the amplitude choice $\mathrm{MMSE}_n$0, the simplest equation for $\mathrm{MMSE}_n$1 becomes

$\mathrm{MMSE}_n$2

Specializing to $\mathrm{MMSE}_n$3 reduces the problem again to an elliptic form and yields a Jacobi $\mathrm{MMSE}_n$4-based exact solution. The structural content of the solution is explicit, although the printed formula is reported as somewhat typographically corrupted (Vitanov et al., 2018).

5. Representative applications beyond nonlinear Schrödinger equations

The Korteweg–de Vries bisoliton example is the clearest demonstration of the multibranch expansion mechanism in an integrable setting. Starting from

$\mathrm{MMSE}_n$5

the method uses $\mathrm{MMSE}_n$6 and

$\mathrm{MMSE}_n$7

which yields a Hirota-like equation for $\mathrm{MMSE}_n$8. The multi-function ansatz is

$\mathrm{MMSE}_n$9

and the auxiliary equations are

$u(x,t)$0

Their solutions are exponential, and coefficient matching gives

$u(x,t)$1

The final formula is explicitly identified as a bisoliton solution. The decisive structural feature is the cross term $u(x,t)$2, which is the interaction term absent from the earlier single-simplest-equation setting (Vitanov, 2019).

A different application concerns the generalized Maxwell–Cataneo equation

$u(x,t)$3

used to illustrate the method for a nonintegrable family. With $u(x,t)$4, $u(x,t)$5, and a finite expansion

$u(x,t)$6

the chosen simplest equation is

$u(x,t)$7

whose solution is

$u(x,t)$8

The balance step yields two balance equations,

$u(x,t)$9

and the resulting algebraic system gives an exact particular solution. The paper then lists the special cases $Z$0, corresponding to $Z$1, $Z$2, and $Z$3 profiles (Vitanov, 2019).

Earlier MMSE work supplies additional background applications without yet adopting the full MMSE$Z$4 structure. The 2010 paper obtains exact solutions for a generalized Korteweg–de Vries equation with cubic source and for a third-order Kudryashov–Sinelshchikov equation by using an auxiliary function $Z$5 governed by linear differential equations in $Z$6 and $Z$7. The 2017 artery-with-aneurysm paper derives an exact travelling-wave solution of a variable-coefficient KdV–Burgers-type equation via a single Riccati simplest equation. These studies belong to the modified simplest-equation lineage, but they remain one-branch or one-auxiliary-function constructions rather than full multi-simplest-equation expansions (Efimova, 2010, Nikolova et al., 2017).

6. Relation to adjacent methods, limits, and common interpretive issues

A defining property of $Z$8 is that it absorbs several earlier exact-solution constructions at the level of transformation and expansion. The generalized transformation $Z$9 is broad enough to include Painlevé-type expansions and the kind of transformation used in Hirota’s method as special cases; the literature also lists

$x$0

for the sine-Gordon equation and

$x$1

for the sinh-Gordon or Poisson–Boltzmann equation. This places MMSE$x$2 not as a replacement for those methods, but as a framework in which their transformation structures may be embedded (Vitanov, 2019, Vitanov, 2019).

A common misconception is that every paper using the phrase “modified method of simplest equation” is already multi-simplest-equation in the strict sense. The literature itself distinguishes these levels. The artery-with-aneurysm study uses only one simplest equation,

$x$3

and is therefore best viewed as a standard MMSE instance rather than a full MMSE$x$4 scheme. The 2010 paper likewise does not present a formal MMSE$x$5 hierarchy; it employs a coupled simplest-equation system for a single auxiliary function $x$6, which is conceptually close to multi-equation closure but not a multivariate expansion in several auxiliary functions (Nikolova et al., 2017, Efimova, 2010).

Another interpretive point concerns scope. The extended method is presented as allowing multi-soliton solutions of nonlinear partial differential equations if such solutions do exist, while also yielding particular exact solutions of nonintegrable nonlinear partial differential equations. This suggests that $x$7 is not an integrability test and not a single fixed ansatz; it is a flexible framework whose outcome depends on the chosen transformation, the auxiliary branches, the simplest equations, and the balance equations (Vitanov, 2019).

The literature also leaves the method open-ended. In the nonlinear Schrödinger study, the two-simplest-equation construction is introduced specifically because one simplest equation is often not enough for deep-water-wave or complex NLS-type problems, and it is noted that the approach should extend to more than two simplest equations in principle, though that development is left for future work. In the review formulation, the number of auxiliary functions $x$8 can be $x$9, possibly even infinite. A plausible implication is that the term “multi-simplest equation expansion” denotes a family of constructions rather than a single canonical ansatz, unified by multibranch auxiliary structure, balance conditions, and algebraic closure (Vitanov et al., 2018, Vitanov, 2019).