Modified Simplest Equation Method for PDEs

• Modified Simplest Equation Method is a systematic ansatz approach that represents solutions as finite series in an auxiliary function solving a simpler ODE.
• It employs a balance principle to determine the truncation order and uses simplest equations such as Riccati or Bernoulli to match nonlinearities.
• The method’s algorithmic structure enables explicit derivation of traveling and solitary-wave solutions for both integrable and nonintegrable nonlinear PDEs.

The Modified Simplest Equation Method (MMSE) is a systematic ansatz-based methodology for obtaining exact particular solutions—typically traveling-wave and solitary-wave profiles—of nonlinear partial differential equations (PDEs). As a well-defined specialization of the broader Simple Equations Method (SEsM), MMSE proceeds by representing the unknown field as a finite series in powers of a function solving a simpler ordinary differential equation (termed the "simplest equation"), which itself is chosen to structurally match the nonlinearities in the reduced ODE stemming from the original PDE. The method's technical structure enables algorithmic reduction of nonlinear PDEs (including both integrable and nonintegrable classes) to nonlinear algebraic systems, yielding explicit solutions parameterized by the simplest equation's constants.

1. Foundations and Relationship to SEsM

The MMSE is the case of SEsM where the searched solution of a nonlinear PDE is constructed via a polynomial (or occasionally rational) function of a single auxiliary function, the latter solving a "simplest equation" which is an ordinary differential equation of lower order and complexity than the PDE itself (Vitanov et al., 2019, Dimitrova, 23 Apr 2025). SEsM, introduced by Vitanov et al., serves as a unifying meta-framework within which numerous particular ansatz methodologies (e.g., Hirota’s bilinear method, G′/G, Tanh-method, Exp-function method, Jacobi elliptic expansion, Fourier-spectral series) are recognized as specializations. In this taxonomy, MMSE corresponds to the N=1 case: one building-block function and one simplest ODE, no nonlinear transformations beyond variable reduction.

Canonical form under MMSE/SEsM:

$$u(x,t) = \sum_{i=0}^{N} H_i\, [v(\xi)]^{i}, \qquad \xi = \alpha x + \beta t + \theta$$

where $v(\xi)$ satisfies a simplest equation (e.g., Riccati, Bernoulli, or elliptic).

This method was developed to overcome the limited solution structures generated by the original Method of Simplest Equation (MSE), which is largely restricted to single-soliton or single-kink waveforms and relatively rigid ansatz spaces (Vitanov, 2019, Dimitrova et al., 2013).

2. Algorithmic Procedure

The methodology of the MMSE consists of the following canonical steps (Vitanov et al., 2019, Vitanov, 2019, Dimitrova, 23 Apr 2025):

1. Reduction to an ODE: Apply a traveling-wave or similarity ansatz ($\xi = \alpha x + \beta t$, etc.), reducing the PDE to an ODE for $u(\xi)$:

$\mathcal{D}_\xi[u(\xi)] = 0$

2. Choice of Simplest Equation and Ansatz Construction: Select an ODE for $v(\xi)$ with known closed-form solutions, e.g., Riccati,

$v'(\xi) = a + b v(\xi) + c v(\xi)^2$

or Bernoulli,

$v'(\xi) = p v(\xi) + q v(\xi)^m$

Represent $u(\xi)$ as a finite series in $v(\xi)$:

$u(\xi) = \sum_{i=0}^N a_i [v(\xi)]^i$

3. Balance Principle: Determine the maximal series order $N$ by equating the highest exponents of $v(\xi)$ arising from the nonlinear and dispersive terms of the ODE—so at least two terms contribute to the leading order (ensuring a nontrivial algebraic system).

$$P_{\text{nl}}(N) = P_{\text{disp}}(N) \implies N = \text{balance solution}$$

1. Substitution and Elimination: Substitute the ansatz and derive expressions for $u'(\xi), u''(\xi), \ldots$ in terms of $v(\xi)$ and its derivatives, which are themselves reduced using the simplest equation. This produces a sum over monomials $v(\xi)^k$ with parametric coefficients.
2. Algebraic System Solution: Set each coefficient of $v(\xi)^k$ to zero, yielding a nonlinear algebraic system for the unknowns ($a_i, \alpha, \beta$, and parameters of the simplest equation). This is typically solved via symbolic computation.
3. Integration and Construction of Final Solution: Integrate the simplest equation for $v(\xi)$, and substitute the result and solved constants back into the ansatz to produce an explicit solution for $u(x,t)$.

3. Simplest Equations and Key Ansätze

Supported simplest equations include:

Equation Type	General ODE Form	Solution Type
Bernoulli	$v' = p v + q v^m$	Exponential/logistic/hyperbolic
Riccati	$v' = a + b v + c v^2$	Rational/hyperbolic (tanh/coth)
Elliptic function	$v'^2 = 4 v^3 - g_2 v - g_3$ (Weierstrass)	Elliptic (sn, cn, dn, wp, etc.)

The MMSE ansatz is typically a finite polynomial series: $u(\xi) = \sum_{i=0}^{N} a_i [v(\xi)]^{i}$ but rational or negative-power expansions (reminiscent of Laurent expansions in Painlevé analysis) have also been used in generalized MMSE (Vitanov, 2019).

4. Representative Applications and Explicit Examples

KdV Equation (integrable case):

The MMSE applied to the Korteweg–de Vries (KdV) equation

$u_t + 6u u_x + u_{xxx} = 0$

via traveling-wave reduction ($\xi = x - ct$), yields an ODE. Taking the Riccati equation as simplest equation ($v' = v^2 - k^2$), the solution ansatz is

$u(\xi) = A_0 + A_1 v(\xi) + A_2 v^2(\xi)$

The balance procedure fixes $N=2$; solving the algebraic system provides

$u(\xi) = 2 k^2 \sech^2(k \xi)$

recovering the classical KdV soliton (Vitanov et al., 2019, Vitanov, 2019).

Nonintegrable fifth-order PDE:

For the nonintegrable case ($$u_t + (a_0 + a_1 u + a_2 u^2) u_x + b u_{xxx} + d u_{xxxxx} = 0$$), the MMSE yields polynomial-exponential traveling-wave solutions of degree up to $N=4$ in $w(\xi) = e^{\lambda \xi}$, as fixed via the balance principle (Vitanov, 2019).

5. Structural Advantages and Limitations

Advantages

• Algorithmic and systematic: Reduces PDEs to nonlinear algebraic systems, suitable for symbolic and computer-algebraic implementation (Vitanov et al., 2019).
• Unified framework: Embeds and generalizes many special ansätze (e.g., Exp-function, Tanh-method), situating them as cases of a single underlying algebraic procedure (Dimitrova, 23 Apr 2025).
• Applicability: Admits both integrable and nonintegrable PDEs, with flexibility in the choice of simplest equations to match nonlinearities (e.g., polynomial, rational) (Vitanov, 2019, Vitanov, 2019).

Limitations

• Algebraic complexity: For large series truncations ($N \gg 1$) or high-parameter PDEs, the resulting algebraic system can be computationally intensive and may admit multiple—or no—nontrivial solutions.
• Solution class: Typically yields only classes of particular solutions (e.g., traveling waves, solitary or periodic waves), not the full general solution.
• Ansatz selection: Success depends on an informed choice of simplest equation and truncation indices; there is no fully automated or universal rule for optimal selection (Vitanov, 2019).
• Limited to one-wave structure (in basic MMSE): Only extended MMSE (multi-equation SEsM) systematically constructs multi-soliton or multi-phase solutions (Vitanov, 2019).

6. Generalizations and Extensions

Extensions of MMSE, as structured within SEsM (Vitanov, 2019, Vitanov et al., 2019):

• Multi-simplest equation expansion (MMSEₙ):

Construction of solutions as compositions or combinations of multiple simplest equation solutions, enabling multi-soliton, breather, and more diverse wave interactions.

$$u(x, t) = F(v_1(\xi_1), v_2(\xi_2), \ldots, v_N(\xi_N))$$

• Nonlinear transformations:

Pre- or post-application transformations (e.g., Painlevé expansions, bilinearizing maps) permitting richer ansatz classes and connection to integrability analyses.

• Ansatz enrichment:

Inclusion of negative or fractional powers, exponential or elliptic combinations, rational ansätze, or even variable-coefficient simplest equations.

Notable recent developments:

• Use of simplest equations tailored to yield explicit $\sech^n$- or $\tanh^n$-type solitary waves, including treatment of fractional-power nonlinearities and “grade” matching for monomial derivative structures (Vitanov et al., 2017, Vitanov et al., 2019).
• Explicit extension to nonlinear Schrödinger-type PDEs, requiring multi-function ansätze to decouple amplitude and phase solution components (Vitanov et al., 2018).

7. Comparative Context and Practical Impact

The MMSE, as a special case of SEsM, stands in contrast to methods such as Hirota’s direct bilinear approach, Painlevé truncations, or the Exp-function method, all of which are now recognized as specializations or limiting instances within the SEsM architecture (Vitanov et al., 2019, Dimitrova, 23 Apr 2025). The systematic enforcement of the balance principle, explicit coefficient elimination, and symbolic computability are the distinguishing features that facilitate both the breadth and feasibility of the method for practitioners addressing broad classes of nonlinear PDEs in mathematical physics, fluid dynamics, population dynamics, and related fields.

In sum, the Modified Method of Simplest Equation provides a transparent and algebraically grounded framework for constructing explicit, exact solutions to nonlinear PDEs. Its strengths lie in unification, algorithmic structure, and direct algebraic reducibility; its principal obstacles are the complexity of resultant algebraic systems and the inherent limitations of the ansatz-driven paradigm in capturing the full solution space (Vitanov, 2019, Vitanov, 2019, Vitanov et al., 2019, Dimitrova, 23 Apr 2025).