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Da-jun Zhang

Publications and source records attributed to Da-jun Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Elliptic soliton solutions of lattice eigenfunction equations of the KdV type

It is known that some integrable lattice equations can be formulated by the eigenfunctions of the Lax triplet of the lattice potential Korteweg-de Vries (lpKdV) equation; the eigenfunctions satisfy a discrete Abel formula in which the `integration' constant defines the $δ$ parameter in some equations in the Adler-Bobenko-Suris (ABS) classification. In this paper, we present a full Darboux transformation for the Lax triplet of the lpKdV equation, and eigenfunctions are formulated in terms of Casoratians. When the eigenfunctions take the elementary discrete Lamé functions, we get the elliptic $N$-soliton solution for the lpKdV equation after $N$-fold Darboux transformation. We then determine the constant in the discrete Abel formula in this case. The eigenfunctions are used to formulate elliptic $N$-soliton solutions for various equations in the ABS list, such as H3($δ$), $Q1(Δ)$, Q2 and Q3($Δ$) equations.

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arXiv2026

Realization of several related $(2+1)$-dimensional integrable systems via an anti-self-dual Yang--Mills reduction

We derive several integrable equations in $(2+1)$ dimensions from the anti-self-dual Yang--Mills (ASDYM) equations with gauge group $G=\mathrm{GL}(2,\mathbb C)$ by imposing a single one-dimensional constraint on the ASDYM matrices. The resulting systems include the $(2+1)$-dimensional nonlinear Schrödinger (NLS) equation, a $(2+1)$-dimensional generalized derivative NLS family encompassing the Gerdjikov--Ivanov, Chen--Lee--Liu, and Kaup--Newell equations, and the $(2+1)$-dimensional Heisenberg ferromagnetic (HF) equation. The $(1+1)$-dimensional Wadati--Konno--Ichikawa equation is also recovered through a reciprocal transformation of the HF equation. The nonlinear variables and the associated gauge transformations are expressed explicitly in terms of the entries of the ASDYM matrices. The same one-dimensional constraint is implemented in the Cauchy matrix framework, yielding explicit solutions of the reduced integrable equations. In this formulation, the ASD Yang--Mills Lagrangian density is expressed in terms of a $τ$-function; for suitable exponential data, this $τ$-function becomes a Grammian-type $τ$-function of the Kadomtsev--Petviashvili equaton. The studies are extended to $\mathrm{GL}(M,\mathbb C)$ at the matrix level.

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arXiv2026

Lattice KP type equations arising from eigenfunctions and Dbar problem

In this paper, we construct the lattice Kadomtsev-Petviashvili (KP) type eigenfunction equations. A homogeneous nonlocal $\bar{\partial}$ problem is considered, from which we are able to define the eigenfunction of the Lax pair of the lattice KP equation. The eigenfunction together with its expansions at infinity and at a finite analytic point provide formulations of the lattice modified KP equation, the lattice Schwarzian KP equation and the Nijhoff-Quispel-Capel KP (NQC-KP) equation. We also consider an inhomogeneous nonlocal $\bar{\partial}$ problem. It defines the eigenfunction of the Lax pair of the lattice modified KP equation. This eigenfunction generates a direct formulation for the NQC-KP equation, which is different from the previous ones. Explicit solutions of these equations are obtained, from which we can see the difference of the different formulations for same equations.

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arXiv2026

Sato-theoretic construction of the anti-self-dual Yang-Mills hierarchy and the Ward conjecture

We develop the Sato-theoretic dressing framework for the anti-self-dual Yang-Mills (ASDYM) hierarchy based on a normalized Riemann-Hilbert decomposition. A four-sector expansion of the generating function yields a bi-infinite matrix array of relative coordinates, extending the affine coordinates on the big cell of the Sato Grassmannian. We derive the Sato-Wilson equations, continuous coordinate flows, and their discrete analogues. Several classical integrable hierarchies are recovered under dimensional reduction constraints, with their nonlinear variables identified as specific relative coordinates.

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arXiv2026

Nonlinearization of bilinear equations of the sine-Gordon type, nonlinear Schrödinger type and Benjamin-Ono type

This is a continuation of the paper [Commun. Theor. Phys., 77 (2025) 115006] on the nonlinearization of bilinear equations. The sine-Gordon type and nonlinear Schrödinger type bilinear equations are introduced by Jarmo Hietarinta during his search for integrable bilinear equations. In this paper, we provide a formulation to convert these two types of bilinear equations into nonlinear forms. In addition, the nonlinearization related to the equations involving the Hilbert transformations is also considered. Bell polynomials are employed in the nonlinearization and illustrative examples are provided.

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arXiv2026

On the relation between the dual AKP equation and an equation by King and Schief, and its N-soliton solution

The Dual of the lattice AKP (DAKP) equation [P.H. van der Kamp et al., J. Phys. A 51, 365202 (2018)] is equivalent to a 14-point equation related to the lattice BKP equation, found by King and Schief (KS), and hence it is integrable. We show that the KS equation is dual to Hirota's version of AKP (DAGTE), and we settle the conjectured $N$-soliton solution of DAKP, for all values of the parameters.

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arXiv2026

A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction

We show a unified approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the unreduced derivative nonlinear Schrödinger (DNLS) hierarchies (including the Kaup-Newell, Chen-Lee-Liu, Gerdjikov-Ivanov and a generalized DNLS), together with their multi-component extensions, in the framework of the anti-self-dual Yang-Mills (ASDYM) reduction. By restricting the gauge group to GL(2), the Kadomtsev-Petviashvili (KP) and modified KP (mKP) hierarchies are formulated in the ASDYM reduction via squared eigenfunction symmetry constraints. In this case, the bilinearization of the generalized DNLS equations can also be understood through this reduction. Finally, Gram-type exact solutions for the relevant equations are presented in terms of quasi-determinants.

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arXiv2026

$\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators

In this paper, we present a unified approach to constructing continuous and discrete $\mathrm{PGL}(3)$-invariant integrable systems, formulated in terms of the common dependent variables $z_1,z_2$, from linear spectral problems and their factorisation. Starting from third-order spectral problems, we first provide explicit forms of the differential and difference invariants, generalising the Schwarzian derivative and cross-ratio to the rank-$3$ setting. The factorisation induces dualities among linear spectral problems, underlying the exact discretisation and multi-dimensional consistency of the associated Boussinesq systems. Then, we derive both continuous and discrete $\mathrm{PGL}(3)$-invariant Boussinesq systems, representing natural rank-$3$ generalisations of the Schwarzian KdV and cross-ratio equations. A geometric lifting-decoupling mechanism is developed to explain the reduction of these systems to the $\mathrm{PGL}(2)$-invariant Boussinesq equations. Finally, we derive a ${\mathrm{PGL}}(3)$-invariant system of generating PDEs together with its Lagrangian structure, in which the lattice parameters serve as independent variables, providing the generating PDE system for the Boussinesq hierarchy.

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arXiv2026

Bilinear structures of the fourth-order lattice Gel'fand-Dikii equations

In this paper we derive bilinear forms and present their solutions in Casoratians for several fourth-order lattice Gel'fand-Dikii (lattice GD-4) equations. These equations were recently formulated from the direct linearization approach and exhibit the multidimensionally consistent property in multi-component form. Based on the obtained soliton solutions, we are able to extend these equations by introducing a parameter $δ$. These $δ$-extended lattice GD-4 type equations are still consistent around the cube, and their bilinear forms together with Casoratian solutions are provided.

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arXiv2026

Constraints of the D$Δ$KP hierarchy to the semi-discrete AKNS and Burgers hierarchies

The paper investigates three eigenfunction constraints of two (2+1)-dimensional differential-difference integrable systems. First, we revisit the known squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (D$Δ$KP) hierarchy, which gives rise to a semi-discrete Ablowitz-Kaup-Newell-Segur hierarchy. Second, we introduce a linear eigenfunction constraint for the D$Δ$KP system and obtain a combined semi-discrete Burgers (sdBurgers) hierarchy. In the third one, we consider another linear eigenfunction constraint for the modified D$Δ$KP system and obtain the same combined sdBurgers hierarchy. All these constraint results are proved by using recursive algebraic structures of the involved integrable hierarchies generated by their master symmetries.

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arXiv2026

Discretization of the Mikhailov model

In this paper the Mikhailov model is discretized by means of the Cauchy matrix approach. A pair of discrete Miura transformations are constructed. The discrete Mikhailov model is a coupled system, in which one equation comes from the compatibility of the two Miura transformations, the other is transformed from the discrete negative order Ablowitz-Kaup-Newell-Segur system by using the Miura transformations. Explicit solutions, including solitons and multiple-pole solutions, are presented via two Cauchy matrix schemes respectively, namely, the Ablowitz-Kaup-Newell-Segur type and the Kadomtsev-Petviashvili type. By straight continuum limits, semi-discrete and continuous Mikhailov models together with their Cauchy matrix structures and solutions are recovered.

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arXiv2025

On the formulation of the NQC variable

The Nijhoff-Quispel-Capel (NQC) equation is a general lattice quadrilateral equation presented in terms of a function $S(a,b)$ where $a$ and $b$ serve as extra parameters. It can be viewed as the counterpart of Q3 equation which is the second top equation in the Adler-Bobenko-Suris list. In this paper, we review some known formulations of the NQC variable $S(a,b)$, such as the Cauchy matrix approach, the eigenfunction approach and via a spectral Wronskian. We also present a new perspective to formulate $S(a,b)$ from the eigenfunctions of a Lax pair of the lattice (non-potential) modified Korteweg de Vries equation. A new Dbar problem is introduced and employed in the derivation.

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arXiv2025

Elliptic solutions of the lattice CKP equation and its elliptic direct linearisation scheme

A direct linearisation scheme, based on an elliptic Cauchy kernel, is set up for the lattice CKP equation. This leads to an elliptic parametrisation of the lattice CKP equation, together with its Lax triplet, which allows us to perform appropriate continuum limits and construct elliptic solutions. By selecting appropriate integration measures and domains for the singular linear integral equation in the scheme, elliptic multi-soliton solutions of the lattice CKP equation are found.

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arXiv2025

The direct linearization scheme with the Lamé function: The KP equation and reductions

The paper starts from establishing an elliptic direct linearization (DL) scheme for the Kadomtsev-Petviashvili equation. The scheme consists of an integral equation (involving the Lamé function) and a formula for elliptic soliton solutions, which can be confirmed by checking Lax pair. Based on analysis of real-valuedness of the Weierstrass functions, we are able to construct a Marchenko equation for elliptic solitons. A mechanism to obtain nonsingular real solutions from this elliptic DL scheme is formulated. By utilizing elliptic $N$th roots of unity and reductions, the elliptic DL schemes, Marchenko equations and nonsingular real solutions are studied for the Korteweg-de Vries equation and Boussinesq equation. Illustrations of the obtained solutions show solitons and their interactions on a periodic background.

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arXiv2025

On the Dbar method and direct linearization approach of the lattice KdV type equations

The purpose of this paper is to bridge the gap between the Dbar method and the direct linearization approach for the lattice Korteweg-de Vries (KdV) type equations. We develop the Dbar method to study some discrete integrable equations in the Adler-Bobenko-Suris list. A Dbar problem is considered to define the eigenfunctions of the Lax pair of the lattice potential KdV equation. We show how an extra parameter is introduced in this approach so that the lattice potential modified KdV equation and lattice Schwarzian KdV equation are derived. We also explain how the so-called spectral Wronskians make sense in constructing the H3$(δ)$, Q1$(δ)$ and Q3$(δ)$ equations. Explicit formulae of multi-soliton solutions are given for the derived equations, from which one can see the connections between the direct linearization variables ($S^{(i,j)}$ and $V(p)$) and the eigenfunctions and their expansions respectively at infinity and a finite point.

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arXiv2025

Nonlocal massive Thirring model and its solutions

A nonlocal version of the massive Thirring model (MTM) and its solutions are presented. We start from a 4-component system that can be reduced to the classical MTM and nonlocal MTM. Bilinear form of the 4-component system and general double Wronskian solutions are derived. By utilizing reduction technique we obtain solutions of the nonlocal MTM. Relations between the nonlocal MTM and the nonlocal Fokas-Lenells equation is discussed. Some solutions of the nonlocal MTM, such as solitons, double-pole solution, algebraic solitons and high order algebraic solitons are analyzed and illustrated.

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arXiv2025

Nonlinearization of the KdV-type and mKdV-type bilinear equations

In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations (in the sense having 3-soliton solutions) of the KdV type and mKdV type that were found by Jarmo Hietarinta in 1980s. Examples of non-integrable bilinear equations of the KdV type are also given.

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