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.
In this paper we consider a simplest two-dimensional reduction of the remarkable three-dimensional Hirota-Ohta system. The Lax pair of the Hirota-Ohta system was extended to a Lax triad by adding extra third linear equation, whose compatibility conditions with the Lax pair of the Hirota-Ohta imply another remarkable systems: the Kulish-Sklyanin system (KSS) together with its first higher commuting flow, which we can call as vector complex MKdV. This means that any common particular solution of these both two-dimensional integrable systems yields a corresponding particular solution of the three…
We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin--Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadom…
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.
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 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…
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 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.
Dispersionless version of multi-component Pfaff-Toda hierarchy
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；National Research University Higher School of Economics（国立研究高等经济学院）；NRC “Kurchatov institute”（库尔恰托夫国家研究中心）
The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy
在[Alexandrov A.，Kazakov V.，Leurent S.，坪井Z Zabrodin A.，J. High Energy Phys.2013（2013），no.9，064，65 pages，arXiv：1112.3310]，我们展示了如何构造具有扭曲边界条件的量子非均匀GL（N）XXX自旋链的主T算子。它满足经典mKP族的双线性恒等式和Hirota方程。我们还描述了一类的解决方案，对应于主T-运营商的特征值的mKP层次和研究动态的频谱参数的函数，他们的零点。这意味着量子自旋链和经典的鲁伊斯塞纳斯-施奈德质点系之间有着显著的联系。
With the extended logarithmic flow equations and some extended Vertex operators in generalized Hirota bilinear equations, extended bigraded Toda hierarchy(EBTH) was proved to govern the Gromov-Witten theory of orbiford $c_{NM}$ in literature. The generating function of these Gromov-Witten invariants is one special solution of the EBTH. In this paper, the multi-fold Darboux transformations and their determinant representations of the EBTH are given with two different gauge transformation operators. The two Darboux transformations in different directions are used to generate new solutions from k…
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.
We establish a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors. $τ$ functions in Hirota's form are derived and vertex operators that generate such $τ$ functions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational func…
General breather solution to the Sasa-Satsuma (SS) equation is systematically investigated in this paper. We firstly transform the SS equation into a set of three Hirota bilinear equations under proper plane wave background. Starting from a specially arranged tau-function of the Kadomtsev-Petviashvili hierarchy and a set of eleven bilinear equations satisfied, we implement a series steps of reduction procedure, i.e., C-type reduction, dimension reduction and complex conjugate reduction, and reduce these eleven equations to three bilinear equations for the SS equation. Meanwhile, general breath…
We study two members of the multi-component AKNS hierarchy. These are multi-NLS and multi-MKdV systems. We derive the Hirota bilinear forms of these equations and obtain soliton solutions. We find all possible local and nonlocal reductions of these systems of equations and give a prescription to obtain their soliton solutions. We derive also $(2+1)$-dimensional extensions of the multi-component AKNS systems.
