Metin Gürses, Aslı Pekcan
We study the AKNS($N$) hierarchy for $N=3,4,5,6$. We give the Hirota bilinear forms of these systems and present local and nonlocal reductions of them. We give the Hirota bilinear forms of the reduced equations. The compatibility of the commutativity diagrams of the application of the recursion operator, reductions of the AKNS($N$) systems, and Hirota bilinearization is also studied.
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In this work we continue to study negative AKNS($N$) that is AKNS($-N$) system for $N=3,4$. We obtain all possible local and nonlocal reductions of these equations. We construct the Hirota bilinear forms of these equations and find one-soliton solutions. From the reduction formulas we obtain also one-soliton solutions of all reduced equations.
We first construct a $(2+1)$-dimensional negative AKNS hierarchy and then we give all possible local and (discrete) nonlocal reductions of these equations. We find Hirota bilinear forms of the negative AKNS hierarchy and give one- and two-soliton solutions. By using the soliton solutions of the negative AKNS hierarchy we find one-soliton solutions of the local and nonlocal reduced equations.
An Efficient Algorithm of Logarithmic Transformation to Hirota Bilinear Form of KdV-type Bilinear Equation
本文建立了KdV型双线性方程的Hirota双线性型对数变换的一个有效算法。该算法成功地应用了Hirota算子和对数变换的一些性质，证明了非线性偏微分方程的线性项在求Hirota双线性型中起着至关重要的作用.通过对各种积分微分方程的实验，证明了该算法在求Hirota双线性型，特别是求对数变换系数方面，比Zhou，Fu，Li等人的算法更有效.
Finite-genus solutions for the Hirota's bilinear difference equation
利用紧致Riemann曲面上θ函数的Fay恒等式，构造了Hirota双线性差分方程的有限亏格解。
Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method to these equations one obtains the same $N$-soliton solutions of the generating equation which differ only by the dispersion relations. Similar discussions can be made for the locally and nonlocally reduced equations as well. We give, as an example, AKNS system of equations in $(1+1)$-dimensions.
Periodic wave solutions of nonlinear equations by Hirota's bilinear method
本文利用Hirota的双线性方法直接构造非线性方程的周期波解。分析了周期波解的渐近性质。结果表明，已知的孤子解可以从周期波解约化。
Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy
在本文中，我们构造了一个扩展的Kadomtsev-Petviashvili（KP）族的波函数的双线性恒等式，该KP族是具有特定扩展流的KP族（2008，Phys. Lett. A，372：3819）。通过引入一个辅助参数（记为$z$），它的流对应于KP族的所谓平方特征函数对称性，我们找到了这个扩展KP族的τ-函数.证明了双线性恒等式将生成扩展KP族的零曲率形式的所有Hirota双线性方程，其中包括两类具有自洽源的KP方程（KPSCS）.与Hu和Wang（2007，Inverse Problems，23：1433）的结果相比，本文得到的KPSCS的Hirota双线性方程组形式更为简单.
Multi-place shifted nonlocal reductions of a multi-component AKNS system
本文基于多组分AKNS系统推导出新的移位非局部非线性薛定谔方程，通过 Hirota 方法获得单孤子解，并讨论了非平凡解的奇异性及参数取值下的非奇异解。
Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to…
The Method of Hirota Bilinearization
本文系统研究(2+1)维非线性偏微分方程的Hirota双线性化，给出最一般的六阶局域双线性方程及其12参数形式，并构造孤子、lump和混合解。
Hirota bilinear forms with 2-toroidal symmetry
本文计算了$A_l, D_l, E_l$型2-环面李代数顶点表示的齐次实现和主实现所产生的Hirota双线性型。
A Symmetric Generalization of Linear Bäcklund Transformation associated with the Hirota Bilinear Difference Equation
将Hirota双线性差分方程推广到任意维数的离散空间。非线性差分方程的解可通过相应线性问题的贝克隆变换得到。
Construction of Nth-order rogue wave solutions for Hirota equation by means of bilinear method
本文利用双线性方法构造了Hirota方程的N阶Rouge波解。这个公式可以用行列式来表示.此外，一些有趣的动力学模式的流氓波展出。
Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2+1 dimensions
利用Riemann θ函数构造了一类（2+1）维Hirota双线性方程的一周期和二周期波解.所涉及的解决方案分析的基础是广田双线性制定，并特别依赖于独立变量的方程保证存在一个周期和两个周期波解决方案，涉及一个任意的纯虚黎曼矩阵。将所得理论应用于具有Hirota双线性形式的两个非线性方程：$u_t+u_{xxy}-3uu_y-3u_xv=0$和$u_t+u_{xxxxy}-(5u_{xx}v+10u_{xy}u-15u^2v)_x=0$，其中$v_x=u_y$，从而产生描述一维波传播的一周期和二周期波解。
Multilinear Operators: The Natural Extension Of Hirota's Bilinear Formalism
基于$τ$函数的规范不变性原理，我们引入了多线性算子，推广了Hirota的双线性$D$算子。我们表明，这些运营商可以构造系统使用双线性$D$的积木。我们特别集中在三线性的情况下，并研究可能的可积性方程与一个因变量。Lax族的五阶方程和萨摩最低阶规范不变方程都具有简单的三线性表达式。形式主义可以扩展到任意程度的多重线性。
A Classification of Hirota-Integrable Supersymmetric Bilinear KdV-Type Equations
本文推广Hirota三孤子判据至超对称情形，证明超对称双线性KdV型方程存在三超孤子解需满足八个可积性条件，并据此完成此类方程的分类，揭示Hietarinta分类在超对称框架下的有效子集。
A Search Method for Hirota Bilinear Systems of Nonlinear Evolution Equations
本文提出一种基于已知种子解识别耦合项来推导广田双线性系统的系统搜索方法，证明同一解类共享同一双线性系统，并在非可积NLSE上验证了其有效性。
Introduction to the Hirota bilinear method
本文初步介绍了Hirota构造可积非线性发展方程多孤子解的直接方法。我们将详细讨论这是如何在Korteweg-de Vries类方程。我们还展示了如何Hirota的方法可以用来搜索新的可积的发展方程通过测试3-和4-孤子解的存在性，并列出的结果，已经获得这种方式的KdV，mKdV/SG和nLS类方程。
Hirota bilinear formalism and Supersymmetry
将标准双线性形式的$τ$函数的规范不变性原理推广到超对称情形，定义了${\cal N}=1$超对称Hirota双线性算子.利用它们，我们双线性化超对称非线性发展方程。讨论了超孤子解。作为一个奇怪的悖论，本文证明了Manin-Radul-Mathieu方程的Lax可积超对称KdV对任意参数的$N\geq 3$不具有N个超孤子解.只有对它们中的一个特定的选择，N超孤子解存在。
Hirota bilinear approach to GUE, NLS, and Painlevé IV
Tracy和Widom证明了高斯幺正系综的能级间距函数与第四潘勒韦方程的一个特解有关。本文从孤子理论中广田双线性方法的观点重新考虑了这个问题，并给出了另一个证明。我们还考虑了能级间距函数为$s\to\infty$的渐近行为，以及它与Painlevé IV方程的“Clarkson-McLeod解”的关系。
The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or half-integers, which produce bilinear ordinary difference equations of Somos/Gale-Robinson type. Here it is explained how to obtain Lax pairs and presymple…
Akns Hierarchy, Self-Similarity, String Equations and the Grassmannian
本文利用无限维格拉斯曼理论，对AKNS族的伽利略、标度和平移自相似条件进行了几何分析。本文对单矩阵模型进行了无标度极限分析，发现弦方程符合伽利略自相似条件。我们描述，在零曲率1-形式的AKNS系列的初始数据，这些自相似的解决方案在佐藤格拉斯曼的模空间。作为一个副产品，我们在Segal-威尔逊格拉斯曼对应的萨克斯合理的解决方案的AKNS方程和中村-Hirota合理的解决方案的NLS方程的特点。确定了AKNS方程的一个显式单参数族伽利略自相似解和NLS方程的相应解。
Solitons of the Resonant Nonlinear Schrodinger Equation with Nontrivial Boundary Conditions and Hirota Bilinear Method
在Hirota双线性方法中考虑了最近提出的用于描述冷无碰撞等离子体中单轴波的具有非平凡边界条件的共振非线性薛定谔（RNLS）方程的物理相关孤子解。通过Madelung表示，将模型转化为NLS方程的反应扩散模拟，研究了其双线性表示、孤子解及其相互作用.
We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.
Generalized Hirota bilinear identity and integrable q-difference and lattice hierarchies.
引入了Cauchy-Baker-Akhieser（CBA）核的Hirota双线性恒等式，作为构造包含格和q-差时的可积族的基本工具.导出了亚纯函数在CBA核上作用的行列式公式。这个公式为构造可积格方程的通解提供了机会。
Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3x3 stencil
Hirota的双线性方法（“直接方法”）在构造许多可积方程的孤子解方面非常有效。甚至对于不可积的双线性方程，单孤子解和双孤子解的构造也是可能的，但是一般的三孤子解的存在性施加了严格的限制，并且实际上等价于可积性。这个属性已被用于搜索可积偏微分方程，在本文中，我们将其应用到二维偏差分方程上定义的3x3模板。我们还讨论了所得到的方程如何与Hirota和Miwa的三维主方程的投影和极限有关，并发现有时需要一个奇异极限。
The Vortex Filament Equation (VFE) is part of an integrable hierarchy of filament equations. Several equations part of this hierarchy have been derived to describe vortex filaments in various situations. Inspired by these results, we develop a general framework for studying the existence and the linear stability of closed solutions of the VFE hierarchy. The framework is based on the correspondence between the VFE and the nonlinear Schrödinger (NLS) hierarchies. Our results show that it is possible to establish a connection between the AKNS Floquet spectrum and the stability properties of the s…
A novel Hirota bilinear approach to $N=2$ supersymmetric equations
本文提出一种新的 Hirota 双线性方法，直接处理 $N=2$ 超对称 KdV 和 Burgers 方程，避免分裂为 $N=1$ 方程，利用超 Bell 多项式获得双线性表示和多孤子解。
A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction
本文在反自对偶Yang-Mills约简框架下，统一处理AKNS、DNLS、KP和mKP层次结构，通过限制规范群为GL(2)，利用平方特征函数对称约束推导KP和mKP层次结构，并通过准行列式构造相关方程的Gram型精确解。
A new relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions, gauge equivalent to Papanicolau spin model, defined on the one sheet hyperboloid is introduced. By using the double numbers, the model is represented as hyperbolic complex valued relativistic massive Thirring type model. By Hirota's bilinear method, an exact one and two dissipative soliton solutions of this model are constructed. Calculation of first three integrals of motion for one dissipaton solution shows that the last one represents a particle-like nonlinear excitation, with relativistic…
Extension of the bilinear formalism to supersymmetric KdV-type equations
将标准双线性形式中τ函数的规范不变性原理推广到超对称情形，定义了N=1超对称Hirota算子.使用它们，我们双线性化超对称KdV型方程（KdV，泽田小寺，Hirota-Satsuma）。讨论了超孤子多重碰撞的解和超对称sine-Gordon的推广。
Bilinear Bäcklund transformations and Lax pair for the Boussinesq equation
Hirota的双线性方法是构造孤子系统解的一种非常有效的方法。该方法将非线性方程组转化为线性方程组，用摄动法求解。本文研究了双线性Boussinesq方程，得到了它的双线性贝克隆变换.从这个双线性贝克隆变换出发，我们还导出了它的拉克斯对，并检验了它的可积性。
Higher order Hirota bilinear forms
本文研究了高阶 Hirota 双线性形式的三孤子解条件及不同形式的非线性偏微分方程的解结构。
We introduce a class of reductions of the two-component KP hierarchy, which includes the Hirota-Ohta system hierarchy. The description of the reduced hierarchies is based on the Hirota bilinear identity and an extra bilinear relation characterising the reduction. We derive the reduction conditions in terms of the Lax operator and higher linear operators of the hierarchy, as well as in terms of the basic two-component KP system of equations.
Bilinear Approach to N=2 Supersymmetric KdV equations
在Hirota双线性方法的框架下研究了N=2超对称KdV方程.对于两个这样的方程，即$N=2, a=4$和$N=2, a=1$超对称KdV方程，我们得到了相应的双线性形式.使用它们，我们为这两种情况下构建特定的解决方案。特别地，给出了$N=2, a=1$超对称KdV方程的双线性贝克隆变换.
We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, eac…
Multisoliton solution and super-bilinear form of lattice supersymmetric KdV equation
本文给出了Xue，Levi和Liu [1]所发现的超对称KdV方程的半离散和全离散（差分-差分）形式的Hirota双线性形式和多孤子解.孤子相互作用项显示在连续超对称的情况下，费米子敷料因子。利用双线性方程，它也表明，可以构建一个新的可积半离散（和全离散）版本的超对称KdV，它具有更简单的双线性形式，但更复杂的相互作用敷料。计算了它的连续极限。
Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies
本文提出利用统一双线性形式离散递归算子的方法，以 AKNS 族和 KdV 族为例推导离散递归算子，并证明其在标准极限下收敛到连续形式。
Writing the Hirota-Satsuma (HS) system of equations in a symmetrical form we find its local and new nonlocal reductions. It turns out that all reductions of the HS system are Korteweg-de Vries (KdV), complex KdV, and new nonlocal KdV equations. We obtain one-soliton solutions of these KdV equations by using the method of Hirota bilinearization.
Hirota bilinear form and soliton solutions for super-KdV of Kuperschmidt (Kuper-KdV) are given. It is shown that even though the collision of supersolitons is more complicated than in the case of supersymmetric KdV of Manin-Radul, the asymptotic effect of the interaction is simpler. As a physical application it is shown that the well known FPU problem having a phonon-mediated interaction of some internal degrees of freedom expressed through grassmann fields, goes to the Kuper-KdV equation in a multiple-scale approach.
A Matrix Model Solution of Hirota Equation
给出了三元Hirota双线性差分方程通解的厄米矩阵链表示。在大N极限下，该矩阵模型给出了三元连续微分Hirota方程的显式特解。讨论了这种表示与二维量子可积模型转移矩阵本征值的关系。
We first study coupled Hirota-Iwao modified KdV (HI-mKdV) systems and give all possible local and nonlocal reductions of these systems. We then present Hirota bilinear forms of these systems and give one-soliton solutions of them with the help of pfaffians. By using the soliton solutions of the coupled HI-mKdV systems for $N=2,3,$ and $N=4$ we find one-soliton solutions of the local and nonlocal reduced equations.
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 present some nonlocal integrable systems by using the Ablowitz-Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schrödinger (NLS) and modified Korteweg-de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to nonlocal Fordy-Kulish equations by utilizing the nonlocal reduction to the Fordy-Kulish system on symmetric spaces. We also consider the super AKNS system and then show that Ablowitz-Musslimani nonlocal reduction can be extended to super integrable equa…
Soliton and breather interactions in the integrable discrete focusing Manakov system via Hirota's method
本文应用Hirota双线性方法于聚焦色散区的可积离散Manakov系统，构造并分析了孤子和呼吸子解，包括基本孤子、基本及复合呼吸子，以及两体相互作用，重点给出了显式公式、可视化和长时间渐近行为。
Alternative integrable discretisation of Korteweg de Vries equation
基于Hirota双线性形式，提出了微分差分KdV方程的另一种可积离散化方法。它示出，使用两个tau函数的双线性方程的直接离散化立即给出众所周知的离散KdV方程。我们还评论的可积性和关系的经典双线性形式只涉及一个tau函数。
The Lamé functions and elliptic soliton solutions: Bilinear approach
本文用 Hirota 双线性方法研究 Lamé 函数构造的椭圆孤子，回顾了 KdV 和 KP 方程的相关进展，并推导出离散势 KdV 与 KP 方程的 τ 函数，证明其与连续方程族共享相同顶点算子。
We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to waves moving with rational speed N/M on the lattice, where N,M are coprime integers, we prove the Liouville integrability of the maps when N + M is odd, and prove various properties of the general case. There are two main ingredients to our construction: the cluster algebra associated with each of the Hirota bilinear equations, which provides invariant (pre)sym…
A new transformation $u=4 ({\rm ln}f)_x$ that can formulate a quintic linear equation and a pair of Hirota's bilinear equations for the (2+1)-dimensional Sawada-Kotera (2DSK) equation is reported firstly, which enables one to obtain a new hierarchy of multiple soliton solutions of the 2DSK equation. It tells a crucial fact that a nonlinear partial differential equation could possess two hierarchies of multiple soliton solutions and the 2DSK equation is the first and only one found in this paper. The quintic linear equation is solved by a pair of Hirota's bilinear equations, of which one is the…
Explicit quasi-periodic wave solutions and asymptotic analysis to the supersymmetric Ito's equation
基于Riemann theta函数和超Hirota双线性形式，给出了超空间$\mathbb{C}_Λ^{2,1}$中超对称Ito方程的拟周期波解的显式构造公式.一旦一个非线性方程被写成双线性形式，那么准周期波解就可以直接由我们的公式得到。严格地建立了周期波解与孤立子解之间的关系。结果表明，在小振幅极限下，准周期波解趋于孤子解。
Reduced Trilinear Reformulation of the Nakamura Conjecture
本文引入约化三线性框架，将Nakamura猜想中的Hirota双线性导数和普通导数统一为三线性算子形式，并继承Hirota直接方法，为Tomimatsu-Sato族的可积结构提供新视角。
Analytic-bilinear approach to integrable hierarchies. I.Generalized KP hierarchy
讨论了构造和研究可积族，特别是KP族的解析双线性方法。它是基于广义广田恒等式。这种方法允许表示广义层次的可积方程的压缩形式的有限函数方程。解这些函数方程得到$τ$ -函数及其加法公式，并给出了$τ$ -函数的一般离散变换的行列式形式。封闭的一种形式和其他公式也自然出现在所提出的方法。用不同的Combescure对称变换不变量表示的广义KP族与通常的KP族和mKP族一致。
A Riemann theta function formula with its application to double periodic wave solutions of nonlinear equations
基于Riemann theta函数和Hirota的双线性形式，给出了一种清晰而直接的方法来显式构造非线性微分方程和差分方程的双周期波解.一旦将这样的方程写成双线性形式，就可以利用统一的θ函数公式直接得到它的周期波解。严格地建立了周期波解与孤子解之间的关系。我们所提出的方法的有效性可以证明对一类非线性方程，如本文所考虑的，应水波方程，（2+1）维Bogoyavlenskiii-Schiff方程和微分差分KdV方程。
Lagrangian multiform structure for the lattice KP system
给出了双线性离散KP（或Hirota-Miwa）方程的拉格朗日量。此外，我们表明，这个拉格朗日可以扩展到拉格朗日3-形式时，嵌入在一个更高的维晶格，服从封闭关系。因此，我们在高维情况下建立了arXiv：0903.4086v1 [nlin.SI]中提出的多形式结构。
We propose a new bilinear Hirota equation for $τ$-functions associated with the $E_8$ root lattice, that provides a "lens" generalisation of the $τ$-functions for the elliptic discrete Painlevé equation. Our equations are characterized by a positive integer $r$ in addition to the usual elliptic parameters, and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the $E_8$ root lattice. We construct explicit $W(E_7)$-invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/in…
Multicomponent DKP hierarchy and its dispersionless limit
本文通过自由费米子与玻色化引入多分量DKP层次，推导Hirota-Miwa型双线性方程，并揭示其无色散极限中内置的椭圆曲线结构，实现方程简化。
Determinant Structure of the Rational Solutions for the Painlevé IV Equation
利用Hirota双线性形式研究了Painlevé IV方程的有理解.结果表明，其中一个解族用3-约化Schur函数表示，另两个解族用Hermite多项式的Casorati行列式或Schur多项式的特殊情况表示.
Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation
我们引入了一个扩展的\ell减少KP层次，我们称之为\ell -Bogoyavlensky层次。在\ell =2的情况下，KdV方程的Bogoyavlensky的2+1维扩展是层次的最低方程。我们在2-环面李代数sl_ \ell^{tor}的基础上给出了这一族的群论刻画.这基本上再现了与最近由Billig和Iohara等人引入的Hirota双线性方程相同的Hirota双线性方程。我们可以进一步从该族的Lax形式导出这些Hirota双线性方程。这种Lax形式还使我们能够构造一族特殊解，推广了Bogoyavlensky的破缺孤子解。这些解包含了N-孤子解，而N-孤子解通常是用顶点算子构造的。
Sato theory on the $q$-Toda hierarchy and its extension
本文构造了一个新的$q$ -变形Toda族（QTH）的包含Hirota双线性方程和tau函数的Sato理论.同时给出了该族的Block型附加对称性和双Hamilton结构。从Hamilton τ对称性出发，给出了该族的τ函数的另一种定义.然后，我们将$q$ -Toda族推广到一个扩展的$q$ -Toda族（EQTH），它满足一个广义Hirota二次方程.广田二次方程可能在Gromov-Witten理论中有进一步的应用。并构造了相应的包含多重达布变换的Sato理论。最后，我们构造了$q$ -Toda族的多分量扩张，并证明了它的可积性，包括它的双Hamilton结构、τ对称性和守恒密度.
On the dynamics of rational solutions for 1-D generalized Volterra system
给出了广义Volterra系统的Hirota双线性形式和孤子解。从孤子解出发，利用Ablowitz和Satsuma的长波极限方法和适当的规范变换，得到了一类非奇异的有理解。它们的性质也进行了讨论，它表明，这些解决方案弹性相互作用没有相移。
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