Trace Identity Research Articles

Higher-Order Matrix Spectral Problems and Their Integrable Hamiltonian Hierarchies

Starting from a kind of higher-order matrix spectral problems, we generate integrable Hamiltonian hierarchies through the zero-curvature formulation. To guarantee the Liouville integrability of the obtained hierarchies, the trace identity is used to establish their Hamiltonian structures. Illuminating examples of coupled nonlinear Schrödinger equations and coupled modified Korteweg–de Vries equations are worked out.

Wadati-Konno-Ichikawa-Type Integrable Systems and their Constructions

A standard form of Wadati-Konno-Ichikawa(WKI)-type integrable systems is derived from an sl(2,mathbb {R})-valued spectral problem. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the trace identity. Then, the higher grading affine algebraic construction of some special cases is proposed. We also show that the generalized short pulse equation arises naturally from the negative WKI flow.

Lotka–Volterra lattice system: N-fold Darboux transformation, the corresponding integrable lattice family and bi-Hamiltonian structure

An one-fold Darboux transformation for the Lotka–Volterra lattice system is first established using a proper gauge transformation matrix. Then, as a result of the N times one-fold Darboux transformation, the corresponding N-fold Darboux transformation of the Lotka–Volterra lattice system is presented, and two exact solution are obtained by the resulting Darboux transformation. Hereafter, its the corresponding iso-spectral integrable lattice family is derived. Using the trace identity, bi-Hamiltonian structure of the Lotka–Volterra integrable family is established, and its Liouville integrability is proven.

A multi-component integrable hierarchy and its integrable reductions

We present a multi-component integrable hierarchy via the zero curvature formulation. The trace identity is applied to construction of its Hamiltonian structure, and two integrable reductions are generated under similarity transformations. The adopted matrix spectral problem is associated with a special Lie sub-algebra of the general linear algebra.

Algebro-geometric Constructions of a Hierarchy of Integrable Semi-discrete Equations

A hierarchy of integrable semi-discrete equations is deduced in terms of the discrete zero curvature equation as well as its bi-Hamiltonian structure is gotten through the trace identity. The above hierarchy is separated into soluble ordinary differential equations according to the relationship between the elliptic variables and the potentials, from which the continuous flow is straightened out via the Abel–Jacobi coordinates resorting to the algebraic curves theory. Eventually, the meromorphic function and the Baker–Akhiezer function are introduced successively on the hyperelliptic curve and the algebro-geometric solutions which are expressed as Riemann theta function can be obtained through the two functions mentioned above.

Multi-component generalized Gerdjikov–Ivanov integrable hierarchy and its Riemann–Hilbert problem

In this paper, the multi-component generalized Gerdjikov–Ivanov integrable hierarchy is obtained by means of the corresponding spectral problem and the zero curvature equation. With the help of trace identity, the multi-Hamiltonian structures of this integrable hierarchy are investigated. Based on the spectral problem, a specific Riemann–Hilbert problem is formulated for the generalized Gerdjikov–Ivanov integrable hierarchy. When the jump matrix is a unit matrix, through solving this Riemann–Hilbert problem, the N-soliton solutions can be derived.

The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution

It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures.

A modified Suris hierarchy and N-fold Darboux -Bäcklund transformation

A modified Suris hierarchy is derived by discrete zero curvature equation. Bi-Hamiltonian structure of the whole hierarchy is established through the discrete trace identity. And we prove that the obtained hierarchy is Liouville integrable. Then a one-fold Darboux- Bäcklund transformation for the modified Suris system is established by means of a proper gauge transformation matrix. As application, an explicit solution is given. Finally, as a result of the N times one-fold Darboux–Bäcklund transformation, we derive N -fold Darboux–Bäcklund transformation.

Storied Answers to Questions about Interethnic Marriage in Multiethnic Qinghai Province, PR of China

The article discusses the use of stories as answers to direct questions. The author encountered this pattern while conducting fieldwork tracing identity contexts among young Chinese Muslims of the Hui ethnicity in the city of Xining, a cultural crossroads in Qinghai Province in northwest China, between 2016 and 2018. Regardless of a respondent’s ethnicity and gender, respondents shared the same pattern of sharing stories from the lives of unnamed friends and relatives. This pattern appeared predominantly when asking about interethnic relations and personal views on interethnic marriage and can thus be understood as a high-context cultural strategy for politely and indirectly expressing personal attitudes. The article analyzes this pattern in terms of Chinese culture and the national image of harmonious coexistence in a multiethnic society.

"Tang Ping" of Chinese Youth: Origin Tracing and Social Identity Survey

Following the popularity of “involution ", the word "Tang Ping" has rapidly become a popular word in China's Internet in recent years. What are the etymology, meaning, philosophical origin of “Tang Ping ", and the future trend and social identity of "Tang Ping” culture? This is an interesting question. By tracing literatures, the etymology and philosophical source of "Tang Ping" are explored; According to the method of "species difference + adjacent genus" , the meaning of " Tang Ping " is defined; then, the social identity of " Tang Ping " is analyzed by questionnaire and statistical methods; the future development trend and conditions of "Tang Ping" are inferred through logical deduction; Finally, based on the critical thinking of " Tang Ping " from the perspective of individual and society, this paper puts forward the social strategy to deal with the " Tang Ping ", which is of certain significance to understand and correctly guide the changes of Chinese youth values.

Chebyshev polynomials and inequalities for Kleinian groups

The principal character of a representation of the free group of rank two into [Formula: see text] is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen’s inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

Matrix algebras with degenerate traces and trace identities

In this paper we study matrix algebras with a degenerate trace in the framework of the theory of polynomial identities. The first part is devoted to the study of the algebra Dn of n×n diagonal matrices. We prove that, in case of a degenerate trace, all its trace identities follow by the commutativity law and by pure trace identities. Moreover we relate the trace identities of Dn+1 endowed with a degenerate trace, to those of Dn with the corresponding trace. This allows us to determine the generators of the trace T-ideal of D3.In the second part we study commutative subalgebras of Mk(F), denoted by Ck of the type F+J that can be endowed with the so-called strange traces: tr(a+j)=αa+βj, for any a+j∈Ck, α, β∈F. Here J is the radical of Ck. In case β=0 such a trace is degenerate, and we study the trace identities satisfied by the algebra Ck, for every k≥2. Moreover we prove that these algebras generate the so-called minimal varieties of polynomial growth.In the last part of the paper, devoted to the study of varieties of polynomial growth, we completely classify the subvarieties of the varieties of algebras of almost polynomial growth introduced in ([7]).

The Gross–Zagier–Zhang formula over function fields

We prove the Gross–Zagier–Zhang formula over global function fields of arbitrary characteristics. It is an explicit formula which relates the Néron-Tate heights of CM points on abelian varieties and central derivatives of associated quadratic base change L-functions. Our proof is based on an arithmetic variant of a relative trace identity of Jacquet. This approach is proposed by Zhang. We apply our results to the Birch and Swinnerton–Dyer conjecture for abelian varieties of \({\mathrm {GL}}_2\)-type. In particular, we prove the conjecture for elliptic curves of analytic rank 1.

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

The conformal anomaly action to fourth order (4T) in d=4 in momentum space

We elaborate on the structure of the conformal anomaly effective action up to 4-th order, in an expansion in the gravitational fluctuations (h) of the background metric, in the flat spacetime limit. For this purpose we discuss the renormalization of 4-point functions containing insertions of stress-energy tensors (4T), in conformal field theories in four spacetime dimensions with the goal of identifying the structure of the anomaly action. We focus on a separation of the correlator into its transverse/traceless and longitudinal components, applied to the trace and conservation Ward identities (WI) in momentum space. These are sufficient to identify, from their hierarchical structure, the anomaly contribution, without the need to proceed with a complete determination of all of its independent form factors. Renormalization induces sequential bilinear graviton-scalar mixings on single, double and multiple trace terms, corresponding to Rsquare ^{-1} interactions of the scalar curvature, with intermediate virtual massless exchanges. These dilaton-like terms couple to the conformal anomaly, as for the chiral anomalous WIs. We show that at 4T level a new traceless component appears after renormalization. We comment on future extensions of this result to more general backgrounds, with possible applications to non local cosmologies.

Solitons in lattice field theories via tight-binding supersymmetry

Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), and closely related solitonic solutions to the Bogoliubov-de Gennes equations in the theory of superconductivity. These solutions rely on the inverse scattering method, which reduces these seemingly unrelated problems to identifying reflectionless potentials of an auxiliary one-dimensional quantum scattering problem. There are several ways of constructing these potentials, one of which is quantum mechanical supersymmetry (SUSY). In this paper, motivated by recent experimental platforms, we generalize this framework to develop a theory of lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to lattice versions of interacting field theories. Our analysis hinges on the use of trace identities, which are relations connecting the potential of an equation of motion to the scattering data. For a discrete Schrödinger operator, such trace identities had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities in a lattice Gross-Neveu and chiral Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond to reflectionless potentials associated with the discrete scattering problem. These models are of significance as they are equivalent to a mean-field theory of a lattice superconductor. To explicitly construct these solitons, we generalize supersymmetric quantum mechanics to tight-binding models. We show that a matrix transformation exists that maps a tight-binding model to an isospectral one which shares the same structure and scattering properties. The corresponding soliton solutions have both modulated hopping and onsite potential, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.

A Degasperis–Procesi equation II with multi-peakon solutions

Based on a 3 × 3 matrix spectral problem, a new hierarchy of nonlinear evolution equations is obtained by means of the zero-curvature equation and the Lenard recursion equation. With the help of the trace identity, the Hamiltonian structures of this hierarchy are established. Then a new nonlinear wave equation, called the Degasperis–Procesi equation II, is derived from a negative flow, which admits exact solutions with N-peakons. Finite-dimensional dynamical systems related to N-peakon solutions and an infinite sequence of conserved quantities of the Degasperis–Procesi equation II are obtained.

A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

Trace identities and direct sums of triangular matrices

We construct a counterexample to the conjecture that all trace identities of An are consequences of the ordinary polynomial identities of A and the trace identities of We also describe the trace identities of any direct sum of upper triangular matrix algebras.

Some generalized isospectral-nonisospectral integrable hierarchies

In this article, with the aid of the Lie algebra A1 composed of second order matrices and Lie algebra A2 composed of third order matrices, some new soliton hierarchies of evolution equations are deduced and the corresponding Hamiltonian structures are also worked out by utilizing the trace identity. Specially, one of the integrable soliton hierarchicy is reduced to the generalized Fokker–Plank equation (gFP) and special bond pricing equation. Next, the nonlinear self-adjointness of the generalized Fokker–Plank equation is verified and conservation laws are constructed with the aid of Ibragimov’ method. Moreover, we investigate the coverings and the nonlocal symmetries of the generalized Fokker–Plank equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product. Besides, we apply the Li’s method to obtain two basic Darboux transformations of (gFP) with two different forms of T.