BKP 与CKP 可积系列的递归算子

借助于新引进的算子B, 本文给出了BKP 与CKP 可积系列约束条件在其Lax 算子L中的动力学变量上的具体体现, 即奇数阶动力学变量u_2k+1 能被偶数阶动力学变量u_2k 显式表达. 同时本文给出了BKP 与CKP 可积系列的流方程以及(2n + 1)- 约化下递归算子的统一公式, 揭示了BKP 可积系列和CKP 可积系列的重要区别. 作为例子, 本文给出了BKP 与CKP 可积系列在3- 约化下的递归算子的显式表示, 并验证了u₂ 的t₁ 流通过递归算子的确可以产生u₂ 的t₇ 流, 该流方程与3- 约化下产生的对应流方程是一致的.     

Recursion relations for the constrained multi-component KP hierarchy

In this paper, we define a new constrained multi-component KP(cMKP) hierarchy which contains the constrained KP(cKP) hierarchy as a special case. We derive the recursion operator of the constrained multi-component KP hierarchy. As a special example, we also give the recursion operator of the constrained two-component KP hierarchy.     

Locality of Symmetries Generated by Nonhereditary, Inhomogeneous, and Time-Dependent Recursion Operators: A New Application for Formal Symmetries

Using the methods of the theory of formal symmetries, we obtain new easily verifiable sufficient conditions for a recursion operator to produce a hierarchy of local generalized symmetries. An important advantage of our approach is that under certain mild assumptions it allows to bypass the cumbersome check of hereditariness of the recursion operator in question, what is particularly useful for the study of symmetries of newly discovered integrable systems. What is more, unlike the earlier work, the homogeneity of recursion operators and symmetries under a scaling is not assumed as well. An example of nonhereditary recursion operator generating a hierarchy of local symmetries is presented.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part II. Integrable Discrete Nonlinear Schrödinger Equations and Discrete AKNS Hierarchy

In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.     

Symmetries and Their Lie Algebra of a Variable Coefficient Korteweg-de Vries Hierarchy

Isospectral and non-isospectral hierarchies related to a variable coefficient Painlev′e integrable Korteweg-de Vries(Kd V for short) equation are derived. The hierarchies share a formal recursion operator which is not a rigorous recursion operator and contains t explicitly. By the hereditary strong symmetry property of the formal recursion operator, the authors construct two sets of symmetries and their Lie algebra for the isospectral variable coefficient Korteweg-de Vries(vc Kd V for short) hierarchy.     

Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.     

A HIERARCHY OF INTEGRABLE HAMILTONIAN SYSTEMS WITH NEUMANN TYPE CONSTRAINT

A hierarchy of integrable Hamiltonian systems with Neumann type constraint isobtained by restricting a hierarchy of evolution equations associated with λφ_(xx)+u_iλ~iφ=λ~mφ to aninvariant subspace of their recursion operator.The independentintegrals of motion and Hamiltonian functions for these Hamiltonian systems areconstructed by using relevant reeursion formula and are shown to be in involution.Thusthese Hamiltonian systems are completely integrable and commute with each other.     

Construction of a hierarchy of negative‐order integrable Burgers equations of higher orders

In this work, we employ the recursion operator, the Burgers equation and its inverse operator, for constructing a hierarchy of negative‐order integrable Burgers equations of higher orders. The complete integrability of each established equation emerges by virtue of the correlation between integrability and recursion operators. We use the simplified Hirota's method to obtain multiple kink solutions for some of the derived equations, and in particular, for the generalized negative‐order Burgers equation.     

On the Lie algebra structures connected with Hamiltonian dynamical systems

We construct the hierarchies of master symmetries constituting Virasoro-type algebras for the Hamiltonian vector fields preserving a recursion operator. Similarly, repeatedly contracting a Hamiltonian vector field with the corresponding recursion operator, we define an Abelian Lie algebra of the thus obtained hierarchy of vector fields. The approach is shown to be applicable for the Volterra and Toda lattices. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 699–705, May, 1997.     

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.     

A recursion operator for the universal hierarchy equation via Cartan’s method of equivalence

We apply Cartan’s method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.     

Multi‐component integrable couplings for the Ablowitz–Kaup–Newell–Segur and Volterra hierarchies

A kind of N × N non‐semisimple Lie algebra consisting of triangular block matrices is used to generate multi‐component integrable couplings of soliton hierarchies from zero curvature equations. Two illustrative examples are made for the continuous Ablowitz–Kaup–Newell–Segur hierarchy and the semi‐discrete Volterra hierarchy, together with recursion operators. Copyright © 2014 John Wiley & Sons, Ltd.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .     

Two-peak soliton in the CKP hierarchy

We present a systematic approach to the construction of soliton solutions for the 5-reduction of the C-type sub-hierarchy for the Kadomtsev–Petviashvili (CKP) hierarchies starting from the general τ-function τ^(n+k) of the Kadomtsev–Petviashvili (KP) hierarchy. We obtain the one-soliton and two-soliton solutions for the bi-directional Kaup–Kupershmidt (bKK) equation, i.e. the 5-reduction of CKP hierarchy.     

A study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

We derive a new ( 2 + 1)‐dimensional Korteweg–de Vries 4 (KdV4) equation by using the recursion operator of the KdV equation. This study shows that the new KdV4 equation possess multiple soliton solutions the same as the multiple soliton solutions of the KdV hierarchy, but differ only in the dispersion relations. We also derive other traveling wave solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Symmetric q-Deformed KP Hierarchy

Based on the analytic property of the symmetric q-exponent e_q(x),a new symmetric q-deformed Kadomtsev-Petviashvili(q-KP for short) hierarchy associated with the symmetric q-derivative operator α_q is constructed.Furthermore,the symmetric q-CKP hierarchy and symmetric q-BKP hierarchy are defined.The authors also investigate the additional symmetries of the symmetric q-KP hierarchy.     

Canonical and Noncanonical Recursion Operators in Multidimensions

The hierarchies of evolution equations associated with the spectral operators ?_x?_y ? R?_y ? Q and ?_x?_y ? Q in the plane are considered. In both cases a recursion operator Ф₁₂, which is nonlocal and generates the hierarchy, is obtained. It is shown that only in the first case does the recursion operator satisfy the canonical geometrical scheme in 2 + 1 dimensions proposed by Fokas and Santini. The general procedure proposed allows one to derive, at the same time, the evolution equations associated with a given linear spectral problem and their Backlund transformations (if they exist), with no need to verify by long and tedious computations the algebraic properties of Ф₁₂. Two equations in the first hierarchy can be considered as two different integrable generalizations in the plane of the dispersive long wave equation. All equations in this hierarchy are shown to be both a dimensional reduction of bi-Hamiltonian n × n matrix evolution equations in multidimensions and a generalization in the plane of bi-Hamiltonian n × n matrix evolution equations on the line.     

一类孤子方程族及其多个Hamilton结构

本文建立了一个含11个位势的新的等谱问题,得到了一组新的Lax对,由此得到一类新的孤子方程族.该族是Liouville可积的,具有4-Hamilton结构,且循环算子的共轭算子是一个遗传对称算子.另外,为确切说明所得方程族是一个4-Hamilton结构,在附录中证明了所得的4个Hamilton算子的线性组合恒为Hamilton算子.     

Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy

Based on the stationary zero-curvature equation and the Lenard recursion equations, we derive the coupled modified Korteweg–de Vries (cmKdV) hierarchy associated with a 3×3$3\times 3$ matrix spectral problem. Resorting to the Baker–Akhiezer function and the characteristic polynomial of Lax matrix for the cmKdV hierarchy, we introduce a trigonal curve with three infinite points and two algebraic functions carrying the data of the divisor. The asymptotic properties of the Baker–Akhiezer function and the two algebraic functions are studied near three infinite points on the trigonal curve. Algebro-geometric solutions of the cmKdV hierarchy are obtained in terms of the Riemann theta function.     

String Equations of the q-KP Hierarchy

Based on the Lax operator L and Orlov-Shulman’s M operator, the string equations of the q-KP hierarchy are established from the special additional symmetry flows, and the negative Virasoro constraint generators {L _−n , n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.