The generalized Darboux matrices with the same poles and their applications

Deng-Shan Wang; Yu-Yue Li

REVIEW 3 major objections 4 minor 30 references

The generalized Darboux matrices with the same poles and their applications

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Pith's one-line read Solving one linear system builds every n-fold Darboux matrix for 2x2 Lax pairs.

desk verdict Solid constructive Darboux-matrix paper with explicit new formulas, but the converse half of the central theorem has an omitted subcase that needs completing before the unification claim is fully backed. read the letter →

arxiv 2411.15599 v1 pith:2B32TCXI submitted 2024-11-23 nlin.SI math-phmath.MP

Yu-Yue Li , Deng-Shan Wang This is my paper · ORCID

classification nlin.SImath-phmath.MP MSC 37K1037K3535Q5135Q55

keywords Darbouxtransformationgeneralizedmatrixsame-poleLaxpairroguewavesmulti-polesolitonsKaup-BoussinesqequationnonlocalNLS

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that for a general $2\times2$ Lax pair, every Darboux matrix expressible as a product of $n$ first-order monic Darboux matrices can be constructed by a single algebraic recipe: solve linear equations that ask the matrix to annihilate chosen eigenfunctions and their spectral derivatives at prescribed points. Its central object is a 'same-pole' Darboux matrix, one whose pole form has a unique pole of order $n$; the first-order case is a single Jordan block, and the $n$-th-order case is written explicitly through a $\Gamma$ matrix built from derivatives of the eigenfunctions. If this is right, the classic Darboux transformation and the generalized Darboux transformation are two pole distributions of one construction, so constructing high-order solitons and rogue waves reduces to linear algebra rather than repeated limit processes. The paper demonstrates the recipe on the $x$-nonlocal focusing NLS equation, the focusing NLS equation, and the Kaup-Boussinesq equation.

What carries the argument

The load-bearing mechanism is the interpolation condition (132), which converts the transformation problem into linear algebra in the coefficients $D_i(x,t)$. For the same-pole case the paper's first-order building block is $S=H\Lambda H^{-1}$ with $H=(b_0,b_0^{(1)})$ and $\Lambda=\begin{pmatrix}\lambda_0&1\\0&\lambda_0\end{pmatrix}$, so its Jordan canonical form is a single block; when $n$ copies are folded, the product is rewritten in pole form (8)-(9) with coefficients assembled by the $\Gamma$ matrix (62). The $\Gamma$ matrix is what connects the chosen solutions $b(\lambda)$ of the Lax pair and $a(\lambda)$ of the adjoint Lax pair to the residues of $D(\lambda)$ and $D^{-1}(\lambda)$. Theorem 7 uses these ingredients to build an order-$s$ monic Darboux matrix by solving (132), and Theorem 8 closes the loop by showing that any decomposable monic Darboux matrix satisfies such conditions at the spectral values appearing in its factors.

What would settle it

Take any $2\times2$ Lax pair and a monic Darboux matrix known to factor as a product of first-order monic Darboux matrices; if at some real $(x,t)$ the coefficient matrix of the linear system (132) built from its factors has determinant zero while the product Darboux matrix is well defined, then Theorem 7's 'can be constructed by solving (132)' claim fails. A direct place to look is the focusing NLS example, where the paper shows $H_1$ in (126) has zeros that the iterated construction cancels, so one can test whether the coefficient matrix in (132) also degenerates exactly at those points.

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Extended reading notes

Core claim

The central claim is a characterization theorem. A monic polynomial $D(x,t;\lambda)=\lambda^s I+\lambda^{s-1}D_1+\cdots+D_s$ is a Darboux matrix if, for prescribed $\lambda_j$ and multiplicities $m_j$ with $2s=\sum_j m_j$, it satisfies the linear conditions $\partial^{k_j}/\partial\lambda^{k_j}(D(\lambda)b_j(\lambda))|_{\lambda=\lambda_j}=0$ for $k_j=0,\ldots,m_j-1$ (Theorem 7), and conversely any Darboux matrix that is a product of $n$ first-order monic Darboux matrices arises this way up to a scalar factor (Theorem 8). The proof shows that the classic first-order Darboux matrix, the same-pole first-order Darboux matrix, and their iterations all satisfy such interpolation conditions, so the unified theorem holds in this 'formal' regime, with invertibility of the linear system's coefficient matrix assumed. The same-pole specialization is made explicit in Theorem 5: $D(\lambda)=(\lambda-\lambda_0)^n(I-(b_0\ b_0^{(1)}\ \cdots\ b_0^{(n-1)})\Gamma^{-1}(a(\lambda_0)/(\lambda-\lambda_0),\ldots,\partial^{n-1}/\partial\lambda_0^{n-1}(a(\lambda_0)/(\lambda-\lambda_0)))^T)$, with $\Gamma$ given by a combinatorial sum of products of derivatives of $a$ and $b$. This formula is the paper's new tool for writing Darboux matrices of arbitrary pole order and their inverses in closed form.

Load-bearing premise

The construction assumes the coefficient matrix of the linear system (132) is invertible, and with it the $\Gamma$ matrix in (62); the paper labels this a formal assumption and notes in Remark 4 that the equivalence of the alternative invertibility conditions is not proved.

Editorial extensions

If this is right

To build an $n$-fold Darboux matrix, one solves the linear system (132) once; no repeated limit in $\lambda$ is needed, and the result automatically transforms the Lax pair into another Lax pair with polynomial coefficients.
For the focusing NLS equation, the same-pole construction reproduces the known first- and second-order rogue waves, and the pole-distribution theorem says these are equivalent to the standard generalized Darboux transformation with a different pole splitting.
For the $x$-nonlocal focusing NLS equation, the formulas produce global multiple-pole soliton solutions with spectral parameter $\lambda=0$; single-pole solitons are stationary, while multi-pole branches move with velocity of order $O(1/\sqrt{t})$.
For the good Kaup-Boussinesq equation, the unified theorem yields explicit multi-soliton, multi-pole, and rational solutions, including rational solutions that are singular at a point, and it also exhibits irreducible polynomial Darboux matrices that cannot be factored into first-order monic Darboux matrices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The derivative-with-respect-to-$\lambda$ device is likely a general principle: any construction that classically needs a limiting process at a degenerate spectral point can be replaced by Taylor coefficients of the eigenfunction, which may extend these formulas to Lax pairs of rank higher than 2.
The invertibility assumption in Theorem 7 is probably the same obstruction that makes the $\Gamma$-matrix and the interpolation matrix fail simultaneously; a systematic classification of singular cases could yield a minimal-degree regularized construction, in the spirit of the paper's focusing NLS example where a singular intermediate factor cancels.
The pole-distribution invariance means Darboux transformations form equivalence classes under redistributing pole order between $D(\lambda)$ and $D^{-1}(\lambda)$; if this holds beyond $2\times2$ systems, it would let soliton and rogue-wave solutions be transferred between different dressing conventions without changing the solution.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real news here is the explicit n-th order same-pole Darboux matrix, formulas (61)-(66), plus a unified linear-system recipe (132) that recovers classic and generalized DTs as special cases. The derivative-based construction replacing the usual limit process is a genuinely useful trick, and the classification of first-order monic DTs is clean. The applications to the x-nonlocal NLS, focusing NLS, and Kaup-Boussinesq equations produce concrete solutions, some of which appear to be new. The algebra is mostly direct and checkable, and there is no circularity: the Darboux matrices are built from explicit solutions of the Lax pair and its adjoint, not from the answer.

The soft spots are real but localized. The load-bearing one is Theorem 8, the converse half of the paper's central claim that every decomposable Darboux matrix is obtained from (132). The induction explicitly omits Subcase 2.3, where both eigenvalues of the final first-order factor already lie in the set of previous poles. The authors say it is 'similar to Subcase 2.2,' but that is not a proof: in this case two multiplicities must be adjusted simultaneously and the redefined seed vectors can interact. Until that subcase is handled, the claim 'All Darboux matrices expressible as the product of n first-order monic Darboux matrices can be constructed in this way' is not fully established. The paper's own 'formal' invertibility assumption on the coefficient matrix of (132) is flagged honestly, but it is load-bearing because Theorem 8 invokes uniqueness of the linear system. The authors even show a case where the intermediate matrix H1 degenerates and the singularity cancels only after iteration, so the genericity assumption is not trivial. A second, minor soft spot: some combinatorial steps in the proofs of Theorem 5 and Proposition 2 are sketched rather than fully shown; they look plausible, but a referee would want those details.

Who should read this: anyone working on Darboux transformations for 2x2 Lax pairs, especially on higher-order poles and rogue-wave or multi-pole soliton constructions. The constructive part is solid and the formulas are usable as-is. The converse theorem is the part that needs work before the strongest advertised claim is accepted.

Recommendation: send to peer review. The paper deserves referee time, and the gap in Theorem 8 is fixable - either by completing Subcase 2.3 or by weakening the claim to what is actually proved.

Referee Report

3 major / 4 minor

Summary. The paper develops an algebraic construction of Darboux matrices for 2x2 Lax pairs whose coefficient matrices are polynomials in the spectral parameter. It treats first-order monic Darboux matrices with a single pole, gives an explicit n-th order pole formula using solutions of the Lax pair and its adjoint, states an invariance theorem for pole distributions, and proposes a unified construction (Theorem 7) by which any monic Darboux matrix that is a product of first-order monic Darboux matrices can be obtained by solving the linear system (132)/(145). The theoretical results are illustrated on the x-nonlocal focusing NLS equation, the focusing NLS equation, and the Kaup-Boussinesq equation, with explicit multi-pole soliton and rational solutions. The central claim is that the linear equations determine all decomposable Darboux matrices, up to scalar factors.

Significance. If the main theorems are fully correct, the paper offers a useful unification of classic and generalized Darboux transformations, together with explicit pole-form formulas that are not available in this form elsewhere. The classification of first-order monic Darboux matrices, the explicit inverse formula in Proposition 2, the pole-distribution invariance theorem, and the large collection of explicit solutions are all concrete contributions. I also credit the paper for being explicit about its limitations: Remark 4 and the opening of Section 5 state that the relevant invertibility conditions are assumed formally and that their equivalence is not proven, and the examples include cases where a matrix H1 becomes singular. The main reserve is that the converse half of the central claim, Theorem 8, contains an omitted subcase, and the construction therefore remains conditional on a nondegeneracy assumption that is not established.

major comments (3)

[Section 5, Theorem 8 (Subcase 2.3)] The induction proof of the converse half omits Subcase 2.3, where both eigenvalues µ1 and µ2 of the final first-order factor already lie in the set {λ1,...,λp} used by the lower-order factor. The text says only that the proof is similar to Subcase 2.2 and omits it. This omission is load-bearing: the case requires increasing both multiplicities m1 and m2 at once, so the redefinition of the seed vector in (166) cannot be performed for one pole without simultaneously controlling the condition at the other pole. The simultaneous adjustment of the two seed vectors, and the invertibility of the enlarged linear system (145) after both multiplicities are increased, need to be proved. Until this subcase is completed, Theorem 8 does not establish that every decomposable monic Darboux matrix can be constructed from (132)/(145).
[Section 5, opening paragraph and Eq. (132); Remark 4] The paper explicitly labels as formal the assumption that the coefficient matrix of the linear equations (132) is invertible, and Remark 4 concedes that the equivalence of this condition with invertibility of the Γ matrix and of the matrices H[k] is not proven. This assumption is used in Theorem 8 through the uniqueness of solutions of (145); without it, the constructed G may differ from the target Darboux matrix even when the induction subcases are completed. The manuscript should either prove the relevant invertibility under the theorem hypotheses or state Theorems 7 and 8 as conditional on this genericity assumption. The focusing NLS example in Section 4, where H1 becomes singular yet the iterated DT yields a global solution, shows that the invertibility failure is not merely hypothetical.
[Section 3, Theorem 5 proof, step (2)] The central identity for the vanishing of the derivatives (D(λ)b(λ))_0^{(s)} at λ=λ0 for s=0,...,n−1 is stated as "the second equality comes from" an unproved combinatorial sum involving arrangement numbers; no derivation or reference is supplied. Since this identity is exactly what verifies the defining condition (59), the proof of Theorem 5 should include the combinatorial calculation or an explicit reference. The identity may be correct, but as written the proof is incomplete at a load-bearing point.

minor comments (4)

[Throughout] There are several typos and infelicities: "multiple-ploe" in the introduction, "loos of generality" in Proposition 1, "dose not change" in Remark 3, and a duplicated phrase in the quadruple-pole sentence of Section 4. Please proofread carefully.
[Section 4, after Eq. (116)] The text refers to "Table 2(a)" and "Table 2(b)" when the quoted tables are numbered Table 3 and Table 4. Fix the cross-references.
[Lemma 3 and Section 4] The phrase "conjugate transport" is used for what appears to be the conjugate transpose operation (∗). Please standardize the terminology with a definition at first use.
[Notation, Eqs. (69), (129)] The notation D[λ0, n, b(λ)] introduced in (69) is later used in different forms such as D[1][λ0, 2, h1(x,t;λ)] in (129). The meaning of the bracketed arguments and the order of the superscripts should be explained consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Darboux-matrix construction is derived from explicit Lax-pair data; the omitted subcase is a proof gap, not a circular step.

full rationale

This paper does not derive its target by fitting or by self-referential definition. The generalized Darboux matrices are constructed explicitly from Lax-pair eigenfunctions and adjoint eigenfunctions (Theorems 1, 3, and 5), and the 'same pole' first-order case is classified by direct coefficient comparison in the defining polynomiality conditions (Theorem 4). The unified Theorem 7 is a linear-algebra construction: equations (132) form a system of 2s linear equations for the s coefficient matrices of a monic degree-s polynomial, and the proof exhibits a solution as a product of previously constructed first-order Darboux matrices; Corollary 2 then reads off the factorization. The converse Theorem 8 proceeds by induction using the classification and the induction hypothesis; no step is shown to be equivalent to the conclusion by construction. The explicitly omitted 'Subcase 2.3' and the explicit 'formal' invertibility assumption in Section 5 are genuine proof gaps or regularity conditions rather than circular reductions. Cited external results such as [GHZ04], [Cie09], and [GLL12] are background or comparison material; the only self-citation ([GSW22]) concerns stability of Kaup-Boussinesq solutions and is not load-bearing for the Darboux construction. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard mathematical facts, existence and analyticity of Lax pair solutions, and invertibility conditions for matrices arising in the construction. No new physical entities are introduced. The free parameters are the usual arbitrary constants in soliton solutions, not fitted parameters.

free parameters (2)

ξ^{(j)}_0 (coefficients in ξ(λ)) = Specified in examples, e.g., (1,-i), (0,-1), (1,0), (0,0), (-3,0), (0,0), (45,0), (0,0), (-1575,0), (0,0) for n≤5 in…
These vector parameters are freely chosen to satisfy the symmetry conditions (78) and to generate specific soliton solutions. They are not fitted to external data; they parameterize the family of solutions.
Seed solutions u0, w0 and spectral parameter λ0 in examples = u0=0, λ0=0 for x-nonlocal NLS; u0=(1/2)e^{-it/2} for focusing NLS; u0=c, w0=d for Kaup-Boussinesq
These are standard choices for constructing exact solutions; they are inputs, not fitted quantities.

assumptions (4)

standard math Liouville's theorem and the fundamental theorem of algebra are used to show that holomorphic functions with certain growth or zero counts vanish.
Used in the proof of Theorem 1 (entire function G satisfies G(∞)=0) and in Theorem 6 (polynomial of degree s-1 with s zeros).
domain assumption The Lax pair (1) admits a fundamental solution matrix Φ(x,t;λ) that is analytic at the used spectral points λ0, and the adjoint Lax pair has corresponding row solutions a(x,t;λ).
The constructions in Theorems 1, 5, and the examples require analyticity of Φ and a at the pole. In the focusing NLS example with plane wave seed, analyticity of Φ at α=0 is argued separately.
domain assumption The derivative conditions (60), namely ∂^s(a(λ)b(λ))=0 at λ0 for s=0,...,n-1, can be satisfied for the chosen seed and auxiliary solutions.
These conditions are necessary for the pole form of the Darboux matrix in Theorem 5. In applications they are enforced by choosing η(λ) and ξ(λ) appropriately (e.g., Lemma 3).
ad hoc to paper The Γ matrix in (62) and the coefficient matrix of the linear system (132) are invertible for the problems considered.
The paper explicitly calls the invertibility of (132) a 'formal' assumption in Section 5. Remark 4 notes that equivalence of different invertibility conditions is not proven, and the focusing NLS example shows that H1 can be singular while the final solution remains global.

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Pith. "Pith review of The generalized Darboux matrices with the same poles and their applications." pith.science (2026). https://pith.science/paper/2B32TCXI

@misc{pith2026241115599,
  author       = {Pith},
  title        = {Pith review of: The generalized Darboux matrices with the same poles and their applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2B32TCXI}},
  note         = {Machine review of arXiv:2411.15599}
}

abstract

Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$ Lax pair, in which the coefficient matrices are polynomials of spectral parameter. The first-order monic Darboux matrix is constructed explicitly and its classification theorem is presented. Then by using the solutions of the corresponding adjoint Lax pair, the $n$-order monic Darboux matrix and its inverse, both sharing the same unique pole, are derived explicitly. Further, a theorem is proposed to describe the invariance of Darboux matrix regarding pole distributions in Darboux matrix and its inverse. Finally, a unified theorem is offered to construct formal Darboux transformation in general form. All Darboux matrices expressible as the product of $n$ first-order monic Darboux matrices can be constructed in this way. The nonlocal focusing NLS equation, the focusing NLS equation and the Kaup-Boussinesq equation are taken as examples to illustrate the application of these Darboux transformations.

Figures

Figures reproduced from arXiv: 2411.15599 by the authors.

**Figure 1.** The single-pole soliton (97) in the x-nonlocal focusing NLS equation (72), whose velocity is zero and the corresponding spectral parameter is λ = 0. For n = 2, the double-pole soliton solution is u2(x, t) = 4(6ix + 12x 2 − 8ix 3 + 24t(i + 2x) − 3) 3 + 192t 2 + 24x 2 − 32ix 3 − 16x 4 , (98) whose density distribution |u2| is shown in [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗

**Figure 2.** The global double-pole soliton (98) in the [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

**Figure 3.** The global triple-pole soliton (99) in the [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: The global quadruple-pole and quintuple-pole solitons in the [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]

**Figure 5.** Figure 5: (a) Contour map of the density distribution [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

**Figure 6.** Figure 6: (a) Contour map of the density distribution [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

**Figure 7.** Figure 7: (a) The first-order rogue wave (114). (b) The second-order rogue wave [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]

**Figure 8.** Figure 8: (a) The triple-pole soliton. (b) The quadruple-pole soliton. The DT [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]

**Figure 9.** Figure 9: The multi-pole soliton (193) of the KB equation (180) obtained by [PITH_FULL_IMAGE:figures/full_fig_p048_9.png]

**Figure 10.** Figure 10: The multi-soliton solution (194) of the KB equation (180) obtained [PITH_FULL_IMAGE:figures/full_fig_p049_10.png]

**Figure 11.** Figure 11: The rational solution of the good KB equation (180) for different [PITH_FULL_IMAGE:figures/full_fig_p050_11.png]

**Figure 12.** Figure 12: Rational solutions of the good KB equation (180) by the second [PITH_FULL_IMAGE:figures/full_fig_p053_12.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

The inverse scattering transform-Fourier analysis for nonlinear problems Stud

Ablowitz MJ, Kaup DJ, Newell AC and Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems Stud. Appl. Math. 1974; 53: 249-315

work page 1974

[2] [2]

Integrable nonlocal nonlinear Schr \"o dinger equation

Ablowitz MJ and Musslimani ZH. Integrable nonlocal nonlinear Schr \"o dinger equation. Phys. Rev. Lett. 2013; 110: 064105

work page 2013

[3] [3]

A binary Darboux transformation for the Toda lattice

Babich VM, Matveev VB and Sail' MA. A binary Darboux transformation for the Toda lattice. J. Soviet Math. 1986; 35: 2582-2589

work page 1986

[4] [4]

Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background

Chen J, Pelinovsky DE. Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background. J. Nonlinear Sci. 2019;29: 2797-2843

work page 2019

[5] [5]

Algebraic construction of the Darboux matrix revisited

Cie \'s li \'n ski JL. Algebraic construction of the Darboux matrix revisited. J. Phys. A: Math. Theor. 2009;42: 404003

work page 2009

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