Xian-long Sun, Da-jun Zhang, Xiao-ying Zhu, Deng-yuan Chen
By introducing suitable non-isospectral flows we construct two sets of symmetries for the isospectral differential-difference Kadomstev-Petviashvili hierarchy. The symmetries form an infinite dimensional Lie algebra.
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In this paper, a variable coefficient Bilinear neural network method is proposed to deal with the analytical solutions of variable coefficient nonlinear partial differential equations. As an example, a Kadomstev-Petviashvili equation with variable coefficients is investigated by using the variable coefficient Bilinear neural network method. By establishing "3-2-2-1" and "3-3-2-1" models respectively, rich analytical solutions of the variable coefficient Kadomstev-Petviashvili equation are obtained. By choosing some special values of the parameters, the dynamics properties are demonstrated in s…
In the paper we investigate integrability characteristics for the dispersionless Kadomtsev-Petviashvili hierarchy. These characteristics include symmetries, Hamiltonian structures and conserved quantities. We give a Lax triad to construct a master symmetry and a hierarchy of non-isospectral dispersionless Kadomtsev-Petviashvili flows. These non-isospectral flows, together with the known isospectral dispersionless Kadomtsev-Petviashvili flows, form a Lie algebra, which is used to derive two sets of symmetries for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy. By means of the m…
We employ a Lax pair representation of the two-component BKP hierarchy and construct its bihamiltonian structure with R-matrix techniques.
In this note we consider a two-component extension of the Kadomtsev-Petviashvili (KP) hierarchy represented with two types of pseudo-differential operators, and construct its Hamiltonian structures by using the $R$-matrix formalism.
Based on the idea of symmetric constraint, we apply the Gesztesy-Holden's method to derive explicit representations of the Baker-Ahkiezer function $ψ_1$ of the KP hierarchy, from which we provide theta function representations of algebro-geometric solutions for the whole Kadomtsev-Petviashvili (KP) hierarchy. This provides a approach to obtain some special subclasses of algebro-geometric solutions for the KP hierarchy and other high dimensional hierarchy of equations.
In this paper, the compatibility between the gauge transformations and the additional symmetry of the constrained discrete Kadomtsev-Petviashvili hierarchy is given, which preserving the form of the additional symmetry of the cdKP hierarchy, up to shifting of the corresponding additional flows by ordinary time flows.
An extension of the Kadomtsev-Petviashvili (KP) hierarchy defined via scalar pseudo-differential operators was studied in [16, 20]. In this paper, we represent the extended KP hierarchy into the form of bilinear equation of (adjoint) Baker-Akhiezer functions, and construct its additional symmetries. As a byproduct, we derive the Virasoro symmetries for the constrained KP hierarchies.
We extend Gesztesy-Holden's method to 2+1 dimensional case to obtain a unified construction to the algebro-geometric solutions of the whole modified Kadomtsev-Petviashvili (mKP) hierarchy. Our tools include the relations between solutions of the Gerdjikov-Ivanov (GI) and mKP hierarchy, the Baker-Akhiezer functions in 2+1 dimensions, a special function $ψ_1(P)ψ_2(P^*)$ on $X\times \mathbb{R}^3$ and Dubrovin-type equations for auxiliary divisors.
The paper reveals clear links between the differential-difference Kadomtsev-Petviashvili hierarchy and the (continuous) Kadomtsev-Petviashvili hierarchy, together with their symmetries, Hamiltonian structures and conserved quantities. They are connected through a uniform continuum limit. For the differential-difference Kadomtsev-Petviashvili system, we introduce Lax triads to generate isospectral and non-isospectral flows. This approach provides an integrable master symmetry and simple zero curvature representations of flows. The obtained flows are then proved to generate a Lie algebra w.r.t. …
In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows (2008, Phys. Lett. A, 372: 3819). By introducing an auxiliary parameter (denoted by $z$), whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of K…
Principal hierarchy of Infinite-dimensional Frobenius Manifold Underlying the Extended Kadomtsev-Petviashvili Hierarchy
本文构造了扩展KP层级对应的无穷维Frobenius流形的主层级，并证明它是带一个可移动极点的零亏格Whitham层级的扩展。
We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together with they scalar and three-dimensional reductions. Variants of the Kadomtsev-Petviashvili type and Korteweg-de Vries type equations are represented among them. Our algorithm is based on the combination of …
We derive a particular solution of the extended $r$-reduced KP hierarchy, which is specified by a generalized string equation. The work is a generalization to arbitrary $r\geq 2$ of Buryak's recent results of a solution to the extended open KdV hierarchy which corresponds to $r=2$.
A characterization of the Kadomtsev-Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic-geometrical solutions and in particular elliptic solutions are discussed in detail. The new identity for theta-functions of curves with holomorphic involution having fixed points is obtained.
Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy
我们考虑的向量方法的二进制达布变换的Kadomtsev-Petviashvili系列在其Zakharov-Shabat制定。我们得到明确的公式达布转换潜力的Grammian型行列式。我们还研究了$n$ -th Gel'fand-Dickey族，引入谱算子，得到了类似的结果.将上述结果化为Kadomtsev-Petviashvili I和II实形式，得到相应的矢量达布变换.特别是对于Kadomtsev-Petviashvili I族，我们得到了线孤子、块解和Johnson-Thompson块，以及它们的非线性叠加的行列式公式。对于Kadomtsev-Petviashvili II，除了线孤子，我们得到了奇异有理解，其奇异集描述了弦在平面内的运动。我们还考虑了Gel'fand-Dickey族的I和II实形式，在这两种情况下都得到了矢量达布变换.
On dispersionless BKP hierarchy and its reductions
考虑B型可积无色散Kadomtsev-Petviashvili（KP）族.导出了$τ$ -函数的加法公式和无色散BKP（dBKP）族的共形不变方程.对称约束的dBKP层次进行了研究。
In this paper squared eigenfunction symmetry of the differential-difference modified Kadomtsev-Petviashvili (D$Δ$mKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the D$Δ$mKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R-Toda) hierarchy. An invertible transformation is given to connect the positive and negative R-Toda hierarchies. The positive R-Toda hierarchy is reduced to the differential-difference Burgers hierarchy. We also consider another D$Δ$mKP hierarchy and show that its squared eigenfunction symm…
Lagrangian multiforms for Kadomtsev-Petviashvili (KP) and the Gelfand-Dickey hierarchy
本文首次为完整 KP 层次构造拉格朗日多形式，并通过约化得到 Gelfand-Dickey 层次（含 KdV 与 Boussinesq）的多形式，统一生成层次并体现可积性。
The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI
建立了一类特殊的Painleve VI方程之间的等价性，参数化的共形维数$μ$，时间相关的Euler top方程，isomonodromic变形和三维Frobenius流形。通过Grassmannian公式，利用2-向量1-约束辛Kadomtsev-Petviashvili（CKP）族的Wronskian解，构造了Euler顶方程的等序τ函数和解.这些朗斯基解给出了Painleve VI方程$μ=1,2,{...} $的有理解。
In this paper, we construct the bilinear identities for the wave functions of an extended B-type Kadomtsev-Petviashvili (BKP) hierarchy, which contains two types of (2+1)-dimensional Sawada-Kotera equation with a self-consistent source (2d-SKwS-I and 2d-SKwS-II). By introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the tau-function and the bilinear identities for this extended BKP hierarchy. The bilinear identities can generate all the Hirota's bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, the Hiro…
We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator for our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral oper…
We study the existence and uniqueness of the Kadomtsev-Petviashvili (KP) hierarchy solutions in the algebra of $\F Cl(S^1,\K^n)$ of formal classical pseudo-differential operators. The classical algebra $ΨDO(S^1,\K^n)$ where the KP hierarchy is well-known appears as a subalgebra of $\F Cl(S^1,\K^n).$ We investigate algebraic properties of $\F Cl(S^1,\K^n)$ such as splittings, r-matrices, extension of the Gelfand-Dickii bracket, almost complex structures. Then, we prove the existence and uniqueness of the KP hierarchy solutions in $\F Cl(S^1,\K^n)$ with respect to extended classes of initial val…
In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili(KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra $W_{1+\infty}$. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative KP hierarchy with self-consistent sources and the constrained noncommutative KP hierarchy. The above results will be further generalized to the noncommutative Gelfand-Dickey hierarchies which co…
We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also generalize these results to a KP hierarchy modelled on formal pseudo-differential operators with coefficients wh…
We consider a Frobenius structure associated with the dispersionless Kadomtsev-Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic potential with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.
We review the integration of the KP hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation; an algebra of functions equipped with a derivation that generalizes the gradient operator; an algebra of quaternion-valued functions; a differential Lie algebra; an algebra of polynomials equipped with the Pincherle differential; a Moyal algebra. In all these cases we can formulate and solve the Cauchy problem of the KP hierarchy. Also, in each of these cases we derive different Zakharov…
A KP-mKP Hierarchy via Operators with Two Derivations
本文通过含两个导子的伪微分算子将KP层次推广为KP-mKP层次，并推导出其Bäcklund变换、双线性方程和Hirota方程，证明其等价于带标记点的色散Whitham层次的子层次。
Some Remarks on the KP System of the Camassa-Holm Hierarchy
我们研究了Kadomtsev-Petviashvili系统的局部Camassa-Holm层次获得候选人的Baker-Akhiezer函数的第一个减少推广当地Camassa-Holm。我们把注意力集中在与标准KdV-KP的情况下的差异。
Constraints of the D$Δ$KP hierarchy to the semi-discrete AKNS and Burgers hierarchies
本文研究了D$Δ$KP层级对半离散AKNS和Burgers层级的约束，通过特征函数约束和递归代数结构证明了相关结果。
We construct a class of infinite-dimensional Frobenius manifolds on the spaces of pairs of meromorphic functions with a pole at infinity and a movable pole. Such Frobenius manifolds are shown to be underlying the universal Whitham hierarchy, which is an extension of the dispersionless Kadomtsev-Petviashvili hierarchy.
Reduction properties of the KP-mKP hierarchy
本文研究 KP-mKP 梯队的 $(n,m)$-约化及其向扩展 $r$-约化 KP 梯队的约化，并由 mKP 的 Hirota 方程导出后者方程，证实 Alexandrov 关于开放 KdV 梯队的猜想。
Functional representation of the Ablowitz-Ladik hierarchy
Ablowitz-Ladik层次结构（ALH）被认为是在框架中的逆散射方法。在建立了辅助线性问题的解的结构之后，ALH，它最初被引入作为一个无限差分微分方程组，被提出作为一个有限差分泛函方程组。所得到的表示，当重写广田的双线性形式主义，是用来证明ALH和其他一些可积系统，特别是Kadomtsev Petviashvili层次之间的关系。
Symmetries of the D$Δ$mKP hierarchy and their continuum limits
该研究引入平方本征函数对称约束的新自变量表述，推导D$Δ$mKP系统对应的离散谱问题与Chen-Lee-Liu型微分-差分导数非线性薛定谔方程层级，还得到两组非等谱对称性及代数结构，并通过统一连续极限方案建立mKP与D$Δ$mKP层级可积结构的对应关系。
Quasi-periodic solutions to hierarchies of nonlinear integrable equations and bilinear relations
本文综述了在 Kyoto 学派框架下构造非线性可积方程族拟周期解的方法，并通过 KP 族、2D Toda 族及 BKP 族三个代表性例子详细说明了该理论的应用。
Q-deformed KP hierarchy: Its additional symmetries and infinitesimal Bäcklund transformations
我们研究了与$q$ -变形Kadomtsev-Petviashvili（$q$ -KP）层次结构相关的额外对称性。在识别预解算子作为附加对称的生成元之后，$q$ -KP层次可以一致地简化为所谓的$q$ -变形约束KP（$q$ -cKP）层次。然后，我们表明，额外的对称性作用于波函数可以被看作是无穷小的贝克隆变换通过作用的顶点算子的tau-功能的$q$ -KP的层次。这为$q$ -KP层级建立了Adler-Shiota-van Moerbeke公式。
Based on the analytic property of the symmetric $q$-exponent $e_q(x)$, a new symmetric $q$-deformed Kadomtsev-Petviashvili ($q$-KP) hierarchy associated with the symmetric $q$-derivative operator $\partial_q$ is constructed. Furthermore, the symmetric $q$-CKP hierarchy and symmetric $q$-BKP hierarchy are defined. Here we also investigate the additional symmetries of the symmetric $q$-KP hierarchy.
Solutions to the KP hierarchy with an elliptic background
本文利用伪微分算子表述，通过Darboux变换从平稳解构造KP层次方程的椭圆孤子解，涵盖Li-Zhang行列式解，并展示实值网状图案示例。
Quasi-symmetric functions and the KP hierarchy
拟对称函数出现在求解Kadomtsev-Petviashvili（KP）族的方法中。这还具有一个新的非结合产品的拟对称函数，满足简单的关系，普通产品和外余积。特别是，提供了这个新的产品和外余积，代数的拟对称函数成为一个无穷小双代数。利用这些结果，我们推导出一个序列的身份在代数的拟对称函数，在正式对应的方程的KP层次。
The explicit expression of the flow equations of the noncommutative Kadomtsev-Petviashvili(ncKP) hierarchy is derived. Compared with the flow equations of the KP hierarchy, our result shows that the additional terms in the flow equations of the ncKP hierarchy indeed consist of commutators of dynamical coordinates \{$u_i$\}. The recursion operator for the flow equations under $n$-reduction is presented. Further, under 2-reduction, we calculate a nonlocal recursion operator $Φ(2)$ of the noncommutative Korteweg-de Vries(ncKdV) hierarchy, which generates a hierarchy of local, higher-order flows. …
Fredholm determinant solutions to the modified KP hierarchy and 2D Toda lattice: isospectral and non-isospectral reductions
本文研究修正KP层次和二维Toda晶格的Fredholm行列式解，通过等谱约化构造了mKdV、修正Camassa-Holm和聚焦非线性薛定谔方程的解，并展示了非等谱约化得到变形Bessel行列式解。
A new approach to deformation equations of noncommutative KP hierarchies
部分灵感来自佐藤的理论的Kadomtsev Petviashvili（KP）层次，我们开始与一个相当普遍的层次的线性常微分方程的矩阵空间，并从中推导出一个矩阵Riccati层次。后者则表现出一个潜在的“弱非结合”（WNA）的代数结构，从中我们可以得出结论，与以前的工作，Riccati系统的任何解决方案也解决了潜在的KP层次（在相应的矩阵代数）。然后，我们转向矩阵的分量使用（广义）星积相乘的情况。与变形参数相关联，存在扩大相应KP层级的额外对称性（流动方程）。它们在WNA结构方面有一个紧凑的提法。我们还提出了一个制定的KP层次方程本身的变形流方程。
A Lorentzian SU(3)-covariant noncommutative KP hierarchy and hypercomplex gauge fields
本文提出一个洛伦兹协变的非交换KP层级框架，结合扭曲四元数与SU(3)对称，用于描述非阿贝尔gauge理论的可积结构。
Nonassociativity and Integrable Hierarchies
设A是一个非结合代数，使得结合子（A，A^2，A）为零.若A由元素f自由生成，则存在交换导子delta_n，n= 1，2，.，使得delta_n（f）是f中n+1次非线性齐次多项式。我们证明了表达式delta_{n_1}. delta_{n_k}（f）满足与Kadomtsev-Petviashvili（KP）族的方程相对应的恒等式。结果，“非结合层次”的解partial_{t_n}（f）= delta_n（f），n= 1，2，.，的常微分方程导致的KP系列的解决方案。通过引入A-模的概念，并借助导子delta_n，零曲率联络和线性系统，对框架进行了扩展.
We introduce integrable multicomponent non-commutative lattice systems, which can be considered as analogs of the modified Gel'fand-Dikii hierarchy. We present the corresponding systems of Lax pairs and we show directly multidimensional consistency of these Gel'fand-Dikii type equations. We demonstrate how the systems can be obtained as periodic reductions of the non-commutative lattice Kadomtsev-Petviashvilii hierarchy. The geometric description of the hierarchy in terms of Desargues maps helps to derive non-isospectral generalization of the non-commutative lattice modified Gel'fand-Dikii sys…
The noncommutative KP hierarchy and its solution via descent algebra
该研究通过含GLM方程的直接线性化给出完整非交换KP层级的解，结合Sato-Wilson装扮变换与Poppe首创的半加性散射数据方法，借助增广前Poppe代数与下降代数完成证明，凸显其构造性与组合结构价值。
Virasoro Symmetry of Constrained KP Hierarchies
额外的非等谱对称性制定的约束Kadomtsev-Petviashvili（\cKP）可积族。额外的对称性与基本的约束的兼容性的问题是明确解决的Virasoro部分的额外的对称性，通过适当的修改标准的额外的对称流一般（无约束）KP层次。我们还讨论了\cKP的特殊情况-截断KP族，作为初始纯微分Lax算子的Darboux-Bäcklund轨道。后者产生与离散（多）矩阵模型相关的Toda晶格状结构。我们的构造建立了附加对称流与\cKP族的离散贝克隆变换的交换性条件，从而导出了矩阵模型中的弦方程约束。
We present recent developments on geometric theory of the Hirota system and of the non-commutative discrete Kadomtsev-Petviashvili (KP) hierarchy adding also some new results which make the picture more complete. We pay special attention to multidimensional consistency of the Desargues maps and of the resulting non-linear non-commutative systems. In particular, we show three-dimensional consistency of the non-commutative KP map in its edge formulation. We discuss also relation of Desargues maps and quadrilateral lattice maps. We study from that point of view reductions of the Hirota system to …
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