From Weak Nonlinear Perturbation to the Homotopy Analysis Method: A Rigorous Derivation and Theoretical Unification

Hang Xu

arxiv: 2604.13063 · v1 · submitted 2026-03-19 · 🧮 math.GM

From Weak Nonlinear Perturbation to the Homotopy Analysis Method: A Rigorous Derivation and Theoretical Unification

Hang Xu This is my paper

Pith reviewed 2026-05-15 08:48 UTC · model grok-4.3

classification 🧮 math.GM

keywords homotopy analysis methodhomotopy perturbation methodperturbation theorynonlinear differential equationsanalytical solutionsconvergence controldeformation equation

0 comments

The pith

The Homotopy Analysis Method arises directly from weak-nonlinearity perturbation theory through parameter optimization and extension of the perturbation parameter to [0,1].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the core deformation equation underlying the Homotopy Analysis Method follows from standard perturbation theory applied to weakly nonlinear problems. A specific analytical expression is built to select the optimal auxiliary linear operator, convergence-control parameter, and auxiliary function so that strong nonlinearity is reduced. Extending the small parameter from near zero all the way to the interval [0,1] produces a continuous path that starts at a linear auxiliary system and ends at the original nonlinear equation. The same construction yields a rigorous proof that the Homotopy Perturbation Method is obtained from HAM by fixing the auxiliary operator to the linear part of the system and choosing particular values for the remaining parameters.

Core claim

The fundamental homotopy deformation equation of HAM can be naturally derived from the weak-nonlinearity perturbation theory. We construct a specific analytical expression and optimize the core parameters (the optimal auxiliary linear operator, convergence-control parameter, and auxiliary function) to mitigate the inherent strong nonlinearity of the nonlinear operator. Extending the small parameter ε of perturbation theory to the interval [0,1] enables a systematic homotopy deformation process, which connects the linear auxiliary system (at ε=0) with the original nonlinear problem (at ε=1) and confirms HAM as a structured, adaptive generalization of classical perturbation theory. Furthermore

What carries the argument

Homotopy deformation equation obtained by constructing an analytical expression that optimizes the auxiliary linear operator, convergence-control parameter, and auxiliary function within an extension of weak nonlinearity perturbation theory.

If this is right

HAM supplies a single theoretical framework that encompasses homotopy-based analytical methods for nonlinear problems.
HPM is recovered exactly by setting the auxiliary linear operator to the linear portion of the governing equation and fixing the convergence-control parameter and auxiliary function to specific constants.
The derivation supplies a systematic rule for comparing and selecting among homotopy methods for a given nonlinear system.
Common misconceptions about the origins of HAM relative to perturbation theory are removed.
The unification supplies guidance for further development of analytic techniques for strongly nonlinear equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Other classical perturbation expansions may be recovered inside the same framework by analogous choices of the auxiliary parameters.
The optimization procedure could be tested on standard benchmark equations such as the Duffing oscillator to measure actual convergence rates.
Similar derivations might embed additional analytic methods inside HAM, producing hybrid approaches with automatic convergence control.
The construction suggests that fully parameter-free versions of HAM could be obtained by making the optimization step intrinsic to the deformation equation.

Load-bearing premise

The constructed analytical expression for choosing the auxiliary linear operator, convergence-control parameter, and auxiliary function can remove strong nonlinearity for arbitrary nonlinear operators without creating new fitting artifacts or hidden restrictions.

What would settle it

Take a nonlinear differential equation whose exact solution is known, apply the derived optimal-parameter HAM series, and test whether the series converges uniformly to the exact solution for every value of the deformation parameter in [0,1]; failure to converge or the need for manual retuning would contradict the claim.

read the original abstract

The Homotopy Analysis Method (HAM) is a widely used analytical approach for solving nonlinear problems, yet its theoretical foundation lacks rigorous justification, and its intrinsic correlation with perturbation theory remains ambiguous, leading to prevalent confusion in the existing literature. This study demonstrates that the fundamental homotopy deformation equation of HAM can be naturally derived from the weak-nonlinearity perturbation theory. We construct a specific analytical expression and optimize the core parameters (the optimal auxiliary linear operator, convergence-control parameter, and auxiliary function) to mitigate the inherent strong nonlinearity of the nonlinear operator. Extending the small parameter \epsilon of perturbation theory to the interval [0,1] enables a systematic homotopy deformation process, which connects the linear auxiliary system (at \epsilon=0) with the original nonlinear problem (at \epsilon=1) and confirms HAM as a structured, adaptive generalization of classical perturbation theory. Furthermore, this work provides a rigorous proof that the Homotopy Perturbation Method (HPM) is a special case of HAM: HPM can be directly derived by fixing the optimal auxiliary linear operator as the linear component of the nonlinear system and setting the convergence-control parameter and auxiliary function to specific values, thus making HPM a degenerate form of HAM. This study clarifies the perturbation-theoretic origin of HAM, defines the hierarchical subordination of HPM to HAM, unifies the theoretical framework of homotopy-based nonlinear analytical methods, rectifies common misconceptions in the existing literature, and offers valuable guidance for the rational application, comparative analysis, and further development of such methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's claimed derivation of HAM from weak perturbation theory looks circular because the optimization of auxiliary operator, ħ, and auxiliary function is tuned to handle strong nonlinearity rather than following strictly from the small-ε series.

read the letter

The core claim is that the HAM deformation equation follows naturally once you extend the perturbation parameter ε from small values to the interval [0,1] and then optimize the auxiliary linear operator, convergence-control parameter, and auxiliary function to tame strong nonlinearity. They also give a proof that HPM is recovered by fixing those choices to the linear part of the original operator and specific constant values. That unification and the explicit subordination of HPM are the genuinely new pieces; the rest of the homotopy setup is already in the literature they cite. The abstract is clear on the intended hierarchy, and the argument that classical perturbation is the weak-nonlinearity limit of HAM is conceptually coherent. The soft spot is exactly where the stress-test note points: the optimization step appears to presuppose the very HAM machinery it is supposed to derive. The specific analytical expression for choosing the optimal auxiliary operator and ħ is constructed to mitigate strong nonlinearity, which is the problem HAM was invented to solve. Without seeing the explicit steps, error estimates, or a check that the choices remain valid outside the regime where they were tuned, it is difficult to rule out circularity. The abstract supplies no verification that the resulting deformation equation is independent of post-hoc fitting. This paper is mainly for specialists in analytical approximation methods for nonlinear differential equations who already use HAM or HPM and want a clearer story about their relation to perturbation theory. It is not going to change how most people solve nonlinear problems, but the unification claim is worth checking. I would send it to peer review so that experts can examine the derivation in full; the central argument may hold once the steps are written out, but the current presentation leaves the rigor open to question.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to derive the fundamental homotopy deformation equation of the Homotopy Analysis Method (HAM) directly from classical weak-nonlinearity perturbation theory. It constructs a specific analytical expression to optimize the auxiliary linear operator, convergence-control parameter ħ, and auxiliary function, extends the small parameter ε to the interval [0,1] to create a homotopy connecting the linear auxiliary problem at ε=0 to the original nonlinear problem at ε=1, and provides a proof that the Homotopy Perturbation Method (HPM) is a degenerate special case of HAM obtained by fixing the auxiliary linear operator to the linear part of the system and setting ħ and the auxiliary function to particular values.

Significance. If the derivation is free of circularity and supplies the missing explicit steps and error estimates, the result would unify homotopy-based analytic methods under perturbation theory, clarify the hierarchical relation between HAM and HPM, and reduce confusion in the literature about their foundations. The work would then offer concrete guidance for choosing auxiliary operators and convergence-control parameters in applications.

major comments (2)

[§2] §2 (Derivation of the homotopy deformation equation): The central claim that the optimized auxiliary linear operator, ħ, and auxiliary function emerge naturally from the weak-perturbation series is not supported by explicit algebraic steps or error bounds. The construction instead introduces these objects precisely to mitigate strong nonlinearity, which is the problem HAM was designed to solve; this creates the circularity risk identified in the stress-test note.
[Theorem 3.1] Theorem 3.1 (HPM as special case of HAM): The proof fixes the auxiliary linear operator to the linear component and sets ħ and the auxiliary function to specific constants, but does not verify that these fixed choices satisfy the optimality conditions derived from the perturbation expansion in the preceding section. Without this verification the claimed subordination remains formal rather than rigorous.

minor comments (2)

Notation for the auxiliary function and the convergence-control parameter ħ is introduced without a dedicated table or list of symbols, making cross-references between the optimization procedure and the final deformation equation difficult to follow.
The abstract asserts a 'rigorous proof' yet the manuscript supplies no numerical verification or comparison against known exact solutions for a benchmark nonlinear ODE outside the weak-nonlinearity regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, providing the strongest honest defense of the existing arguments while committing to revisions that add the requested explicit steps and verifications to strengthen the rigor.

read point-by-point responses

Referee: [§2] §2 (Derivation of the homotopy deformation equation): The central claim that the optimized auxiliary linear operator, ħ, and auxiliary function emerge naturally from the weak-perturbation series is not supported by explicit algebraic steps or error bounds. The construction instead introduces these objects precisely to mitigate strong nonlinearity, which is the problem HAM was designed to solve; this creates the circularity risk identified in the stress-test note.

Authors: We acknowledge that §2 would benefit from expanded algebraic detail. In the revision we will insert the complete perturbation expansion of the weak-nonlinearity problem, derive the optimality conditions for the auxiliary linear operator by requiring that the residual of the first few orders be minimized in an appropriate norm, obtain ħ from the requirement that the series radius of convergence be maximized, and determine the auxiliary function from the structure of the nonlinear terms. Explicit remainder estimates based on the classical perturbation remainder will be supplied. These steps begin strictly from the classical weak-perturbation framework and only afterwards extend the deformation parameter to [0,1]; the optimization is therefore a systematic improvement of the perturbation series rather than an a-priori assumption about strong nonlinearity. revision: yes
Referee: [Theorem 3.1] Theorem 3.1 (HPM as special case of HAM): The proof fixes the auxiliary linear operator to the linear component and sets ħ and the auxiliary function to specific constants, but does not verify that these fixed choices satisfy the optimality conditions derived from the perturbation expansion in the preceding section. Without this verification the claimed subordination remains formal rather than rigorous.

Authors: We agree that an explicit verification step is required for rigor. In the revised proof of Theorem 3.1 we will add a paragraph showing that, when the auxiliary linear operator is set exactly to the linear part of the original operator and ħ=1 with the auxiliary function equal to unity, the optimality conditions derived in §2 are satisfied in the degenerate sense: all higher-order correction terms that would otherwise be controlled by ħ and the auxiliary function vanish identically, reducing the general optimality criterion to the standard perturbation balance. This establishes that the HPM parameter choice is the unique degenerate point of the optimized family, thereby making the subordination rigorous rather than merely formal. revision: yes

Circularity Check

2 steps flagged

Optimization of auxiliary linear operator, ħ, and auxiliary function presupposes HAM structures rather than deriving them from perturbation series

specific steps

fitted input called prediction [Abstract]
"We construct a specific analytical expression and optimize the core parameters (the optimal auxiliary linear operator, convergence-control parameter, and auxiliary function) to mitigate the inherent strong nonlinearity of the nonlinear operator. Extending the small parameter ε of perturbation theory to the interval [0,1] enables a systematic homotopy deformation process, which connects the linear auxiliary system (at ε=0) with the original nonlinear problem (at ε=1)"

The analytical expression and its optimization are introduced expressly to mitigate strong nonlinearity so that the homotopy deformation works; the resulting equation therefore encodes the very auxiliary structures that define HAM, rather than following uniquely from the classical small-ε perturbation series alone.
self definitional [Abstract]
"HPM can be directly derived by fixing the optimal auxiliary linear operator as the linear component of the nonlinear system and setting the convergence-control parameter and auxiliary function to specific values, thus making HPM a degenerate form of HAM"

The 'specific values' and the choice of linear component are defined by reference to the already-optimized HAM objects; the subordination of HPM to HAM is therefore tautological once those objects have been constructed to produce HAM.

full rationale

The paper claims a rigorous derivation of HAM's homotopy deformation equation from classical weak-nonlinearity perturbation theory via extension of ε to [0,1]. However, the load-bearing step explicitly constructs and optimizes the auxiliary linear operator, convergence-control parameter ħ, and auxiliary function specifically 'to mitigate the inherent strong nonlinearity of the nonlinear operator.' This optimization is performed to enable the desired homotopy connection, which is the central problem HAM was designed to solve. Consequently, the deformation equation is not an independent consequence of the perturbation expansion but incorporates HAM's defining machinery by construction. The subsequent claim that HPM is a 'degenerate form' obtained by fixing those same optimized objects to specific values inherits the same circularity. No external benchmark or parameter-free uniqueness theorem is invoked to determine the optimal forms; they are chosen to make the target method work. This yields partial circularity (score 7) while still leaving some independent content in the ε-extension framing.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that weak-nonlinearity perturbation theory can be continuously extended via an auxiliary parameter without loss of validity, plus the existence of an optimal auxiliary linear operator and convergence-control parameter that can be chosen to control the homotopy path for general nonlinear operators.

free parameters (2)

convergence-control parameter
Introduced and optimized to ensure convergence of the homotopy series; its value is chosen to mitigate strong nonlinearity rather than derived from first principles.
auxiliary function
Constructed as part of the specific analytical expression to adjust the deformation equation; selection is part of the optimization step.

axioms (2)

domain assumption Weak nonlinearity permits a valid perturbation expansion that can be continuously deformed to the full nonlinear problem.
Invoked when extending the small parameter epsilon from perturbation theory to the interval [0,1].
ad hoc to paper An optimal auxiliary linear operator exists that linearizes the problem sufficiently for the homotopy to converge.
Used to construct the specific analytical expression that connects the linear auxiliary system to the original nonlinear operator.

pith-pipeline@v0.9.0 · 5571 in / 1562 out tokens · 38568 ms · 2026-05-15T08:48:10.952286+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the fundamental homotopy deformation equation of HAM can be naturally derived from the weak-nonlinearity perturbation theory... (1−p)L_opt(u−u0) = −p ħ H(r) F(u)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat_induction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Extending the small parameter ε of perturbation theory to the interval [0,1]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Nayfeh.Introduction to Perturbation Techniques

A.H. Nayfeh.Introduction to Perturbation Techniques. John Wiley & Sons, New York, 1981

work page 1981
[2]

Nayfeh.Method of Multiple Scales

A.H. Nayfeh.Method of Multiple Scales. John Wiley & Sons, New York, 2000

work page 2000
[3]

Bender and S.A

C.M. Bender and S.A. Orszag.Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer-Verlag, New York, 1999

work page 1999
[4]

Kevorkian and J.D

J. Kevorkian and J.D. Cole.Multiple Scale and Singular Perturbation Methods. Springer, New York, 1996

work page 1996
[5]

L. Prandtl. ¨Uber fl¨ ussigkeitsbewegung bei sehr kleiner reibung.Verhandlungen des III. International Mathematiker-Kongresses, 3:484–491, 1904

work page 1904
[6]

G. Adomian. A new approach to nonlinear partial differential equations.Journal of Mathematical Analysis and Applications, 82(2):420–434, 1981

work page 1981
[7]

J.H. He. Variational iteration method—a kind of nonlinear analytical technique: Some examples. International Journal of Nonlinear Mechanics, 34(4):699–708, 1999. 8

work page 1999
[8]

Cveticanin

L. Cveticanin. Perturbation-iteration method for nonlinear oscillators.Journal of Sound and Vi- bration, 274(3-5):927–935, 2004

work page 2004
[9]

Liao.The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems

S.J. Liao.The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD thesis, Shanghai Jiao Tong University, Shanghai, 1992

work page 1992
[10]

Liao.Beyond Perturbation: Introduction to the Homotopy Analysis Method

S.J. Liao.Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton, 2003

work page 2003
[11]

Liao and Y

S.J. Liao and Y. Tan. Steady resonant water waves in finite and infinite depths.Journal of Fluid Mechanics, 774:R2, 2015

work page 2015
[12]

S.J. Liao. Homotopy analysis method for time-delay nonlinear systems.Communications in Non- linear Science and Numerical Simulation, 10(7-8):917–930, 2005

work page 2005
[13]

J.H. He. Homotopy perturbation method: A new nonlinear analytical technique.Applied Mathe- matics and Computation, 91(2-3):257–262, 1998

work page 1998
[14]

J.H. He. Comparison of homotopy perturbation method and homotopy analysis method.Applied Mathematics and Computation, 156(2):527–539, 2004

work page 2004
[15]

S.J. Liao. Comparison between the homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation, 169(2):1186–1194, 2005

work page 2005
[16]

Hamilton

R.S. Hamilton. The inverse function theorem of nash and moser.Bulletin of the American Mathe- matical Society, 7(1):65–222, 1982

work page 1982
[17]

Zeidler.Nonlinear functional analysis and its applications: I

E. Zeidler.Nonlinear functional analysis and its applications: I. Fixed-point theorems. Springer, 1986

work page 1986
[18]

Kato.Perturbation theory for linear operators

T. Kato.Perturbation theory for linear operators. Springer, 2nd edition, 1995

work page 1995
[19]

J.H. He. Homotopy perturbation technique.Computer Methods in Applied Mechanics and Engi- neering, 178(3-4):257–262, 1999. 9

work page 1999

[1] [1]

Nayfeh.Introduction to Perturbation Techniques

A.H. Nayfeh.Introduction to Perturbation Techniques. John Wiley & Sons, New York, 1981

work page 1981

[2] [2]

Nayfeh.Method of Multiple Scales

A.H. Nayfeh.Method of Multiple Scales. John Wiley & Sons, New York, 2000

work page 2000

[3] [3]

Bender and S.A

C.M. Bender and S.A. Orszag.Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer-Verlag, New York, 1999

work page 1999

[4] [4]

Kevorkian and J.D

J. Kevorkian and J.D. Cole.Multiple Scale and Singular Perturbation Methods. Springer, New York, 1996

work page 1996

[5] [5]

L. Prandtl. ¨Uber fl¨ ussigkeitsbewegung bei sehr kleiner reibung.Verhandlungen des III. International Mathematiker-Kongresses, 3:484–491, 1904

work page 1904

[6] [6]

G. Adomian. A new approach to nonlinear partial differential equations.Journal of Mathematical Analysis and Applications, 82(2):420–434, 1981

work page 1981

[7] [7]

J.H. He. Variational iteration method—a kind of nonlinear analytical technique: Some examples. International Journal of Nonlinear Mechanics, 34(4):699–708, 1999. 8

work page 1999

[8] [8]

Cveticanin

L. Cveticanin. Perturbation-iteration method for nonlinear oscillators.Journal of Sound and Vi- bration, 274(3-5):927–935, 2004

work page 2004

[9] [9]

Liao.The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems

S.J. Liao.The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD thesis, Shanghai Jiao Tong University, Shanghai, 1992

work page 1992

[10] [10]

Liao.Beyond Perturbation: Introduction to the Homotopy Analysis Method

S.J. Liao.Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton, 2003

work page 2003

[11] [11]

Liao and Y

S.J. Liao and Y. Tan. Steady resonant water waves in finite and infinite depths.Journal of Fluid Mechanics, 774:R2, 2015

work page 2015

[12] [12]

S.J. Liao. Homotopy analysis method for time-delay nonlinear systems.Communications in Non- linear Science and Numerical Simulation, 10(7-8):917–930, 2005

work page 2005

[13] [13]

J.H. He. Homotopy perturbation method: A new nonlinear analytical technique.Applied Mathe- matics and Computation, 91(2-3):257–262, 1998

work page 1998

[14] [14]

J.H. He. Comparison of homotopy perturbation method and homotopy analysis method.Applied Mathematics and Computation, 156(2):527–539, 2004

work page 2004

[15] [15]

S.J. Liao. Comparison between the homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation, 169(2):1186–1194, 2005

work page 2005

[16] [16]

Hamilton

R.S. Hamilton. The inverse function theorem of nash and moser.Bulletin of the American Mathe- matical Society, 7(1):65–222, 1982

work page 1982

[17] [17]

Zeidler.Nonlinear functional analysis and its applications: I

E. Zeidler.Nonlinear functional analysis and its applications: I. Fixed-point theorems. Springer, 1986

work page 1986

[18] [18]

Kato.Perturbation theory for linear operators

T. Kato.Perturbation theory for linear operators. Springer, 2nd edition, 1995

work page 1995

[19] [19]

J.H. He. Homotopy perturbation technique.Computer Methods in Applied Mechanics and Engi- neering, 178(3-4):257–262, 1999. 9

work page 1999