Lattice Hierarchy Research Articles

A lattice hierarchy associated with RTL[formula omitted] system: Hamiltonian structures, discrete [formula omitted]-fold Darboux transformation, soliton elastic interactions and dynamics

Starting from a 2 × 2 discrete matrix spectral problem, we construct an integrable lattice hierarchy associated with RTL+(α) system which is a one-parameter perturbation of the usual Toda lattice system describing the particle motion in a lattice. Based on the obtained RTL+(α) hierarchy, some integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Furthermore, we establish the discrete N-fold Darboux transformation (DT) of RTL+(α) system. As applications of the obtained DT, multi-soliton solutions are derived on constant seed backgrounds, and discrete soliton propagation and elastic interaction structures are shown graphically and analyzed via asymptotic analysis. Numerical simulations are utilized to demonstrate the dynamical behaviors of such soliton solutions. Numerical results show that soliton evolutions are stable against a small noise. Results given in this paper might be helpful for understanding the propagation of nonlinear waves in soliton theory.

Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model

We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians which govern the dynamics of poles with respect to the m-th hierarchical times t_m and \bar t_m of the 2D Toda lattice hierarchy are obtained from expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points.

Discrete nonlocal N-fold Darboux transformation and soliton solutions in a reverse space-time nonlocal nonlinear self-dual network equation

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of [Formula: see text]-fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal [Formula: see text]-symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals.

Formal context reduction in deriving concept hierarchies from corpora using adaptive evolutionary clustering algorithm star

It is beneficial to automate the process of deriving concept hierarchies from corpora since a manual construction of concept hierarchies is typically a time-consuming and resource-intensive process. As such, the overall process of learning concept hierarchies from corpora encompasses a set of steps: parsing the text into sentences, splitting the sentences and then tokenising it. After the lemmatisation step, the pairs are extracted using formal context analysis (FCA). However, there might be some uninteresting and erroneous pairs in the formal context. Generating formal context may lead to a time-consuming process, so formal context size reduction is require to remove uninterested and erroneous pairs, taking less time to extract the concept lattice and concept hierarchies accordingly. In this premise, this study aims to propose two frameworks: (1) A framework to review the current process of deriving concept hierarchies from corpus utilising formal concept analysis (FCA); (2) A framework to decrease the formal context’s ambiguity of the first framework using an adaptive version of evolutionary clustering algorithm (ECA*). Experiments are conducted by applying 385 sample corpora from Wikipedia on the two frameworks to examine the reducing size of formal context, which leads to yield concept lattice and concept hierarchy. The resulting lattice of formal context is evaluated to the standard one using concept lattice-invariants. Accordingly, the homomorphic between the two lattices preserves the quality of resulting concept hierarchies by 89% in contrast to the basic ones, and the reduced concept lattice inherits the structural relation of the standard one. The adaptive ECA* is examined against its four counterpart baseline algorithms (Fuzzy K-means, JBOS approach, AddIntent algorithm, and FastAddExtent) to measure the execution time on random datasets with different densities (fill ratios). The results show that adaptive ECA* performs concept lattice faster than other mentioned competitive techniques in different fill ratios.

Integrability, multi-soliton and rational solutions, and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter

Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter α abbreviated as RTL_(α) system by Suris, which may describe the motions of particles in lattices interacting through an exponential interaction force. First of all, an integrable lattice hierarchy associated with an RTL_(α) system is constructed, from which some relevant integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Secondly, the discrete generalized (m, 2N − m)-fold Darboux transformation is constructed to derive multi-soliton solutions, higher-order rational and semi-rational solutions, and their mixed solutions of an RTL_(α) system. The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis. Finally, soliton dynamical evolutions are investigated via numerical simulations, showing that a small noise has very little effect on the soliton propagation. These results may provide new insight into nonlinear lattice dynamics described by RTL_(α) system.

ON THE INTEGRATION OF A HIRERARCHY FOR THE PEREODIC TODA LATTICE WITH A FREE TERM

In this article, we have explored the Toda lattice hierarchy in the class of periodic functions with a free term. We have given an effective method of constructing of the periodic Toda lattice hierarchy with a free term. We have discussed the complete integrability of the constructed systems that is based on the inverse spectral problem of an associated discrete Hill`s equation with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy.

Explicit Solutions of the Coupled Bogoyavlensky Lattice 1(2) Hierarchy

By means of the discrete zero curvature representation, the coupled Bogoyavlensky lattice 1(2) hierarchy related to a $$3\times 3$$ matrix problem is derived. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a trigonal curve $$\mathcal {K}_{m-1}$$ of arithmetic genus $$m-1$$ and construct the related Baker–Akhiezer function and meromorphic function. By analyzing the asymptotic expansion of the meromorphic function, the algebro-geometric solutions to the stationary coupled Bogoyavlensky lattice 1(2) are obtained. Moreover, the explicit theta function representations for the coupled Bogoyavlensky lattice hierarchy are given with the help of the Abelian differential.

A generalized integrable lattice hierarchy associated with the Toda and modified Toda lattice equations: Hamiltonian representation, soliton solutions

In this paper, we give an integrable four-field lattice hierarchy associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. The Hamiltonian representation of the four-field lattice hierarchy are constructed by using the trace identity. We consider reductions of our hierarchy, and find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, and some new integrable three-field lattice hierarchies. In addition, the modified Toda lattice equation is studied, the figures of one-, two-soliton and periodic solutions with properly parameters are shown graphically to illustrate the propagation of solitary waves, and we find that the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The elastic and inelastic interactions among the two-soliton for the modified Toda lattice equation are also discussed. Furthermore, we generate the relationships between the structures of exact solutions and parameters, and present soliton solutions with free parameters. The results in this paper might be helpful for interpreting certain physical phenomena.

Modulational instability and dynamics of discrete rational soliton and mixed interaction solutions for a higher-order nonlinear self-dual network equation

In this paper, the higher-order nonlinear self-dual network equation is investigated. Firstly, an integrable lattice hierarchy associated with this equation is constructed from a discrete matrix spectral problem without the denominator. Secondly, the condition of modulational instability for this equation is given. Thirdly, the infinitely many conservation laws are constructed on the basis of its new Lax representation. Finally, the discrete generalised $$(m, N-m)$$ -fold Darboux transformation (DT) with non-zero constant as seed solution is used to derive new rational soliton (RS) and mixed interaction solutions. As an application, RS solutions can be obtained by the discrete generalised $$(1, N-1)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=1$$ ), and mixed interaction solutions of the usual sech-type soliton (US) and RS can be derived by the discrete generalised $$(2, N-2)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=2$$ ). We also perform the numerical simulations for such soliton solutions to explore their dynamical behaviours. Results given in this paper may have some prospective applications for understanding the propagation of electrical signals.

Toda lattice hierarchy and soliton equations on square lattice

Lattice soliton equations of KdV-family and Boussinesq-family are investigated from the viewpoint of the Toda lattice hierarchy. As a consequence, special solutions such as soliton solutions and algebro-geometric solutions are constructed in a unified manner.

Matrix Spectral Problems and Integrability Aspects of the Błaszak-Marciniak Lattice Equations

A method to derive the matrix spectral problems of the Blaszak–Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Blaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Blaszak–Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed based on the matrix spectral problem of this lattice equation, the N-fold Darboux transformation and exact solutions are derived, the solitary wave structures and interaction behaviours of these exact solutions are displayed graphically, and finally the infinitely many conservation laws are listed in a standard way

Positive and negative integrable lattice hierarchies: Conservation laws and [formula omitted]fold Darboux transformations

Positive and negative nonlinear integrable lattice hierarchies are established starting from a discrete matrix spectral problem with three potential functions, particularly the corresponding Hamiltonian structures are presented respectively with the help of the trace identity, all these facts show that these two hierarchies are integrable in Liouville sense. By using Lax pair we derive infinite number of conservation laws and N−fold Darboux transformation (DT) for the first nontrivial system in the two hierarchies. Comparing with the usual 1−fold DT, this kind of N−fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications, we derive N−fold explicit exact solutions from seed solutions and plot their figures with properly parameters to analyze and illustrate the propagation of solitary waves.

Matrix Resolvent and the Discrete KdV Hierarchy

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered.

An Ablowitz-Ladik Integrable Lattice Hierarchy with Multiple Potentials

Within the zero curvature formulation, a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type. The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity. When the involved two potential vectors are scalar, all the resulting integrable lattice equations are reduced to the standard Ablowitz-Ladik hierarchy.

Notes on “A integrable lattice hierarchy based on Suris system: N-fold Darboux transformation and conservation laws” [Nonlinear. Dyn. (2018) 91:625–639

We point out that the Hamiltonian structure of the integrable lattice hierarchy in “A integrable lattice hierarchy based on Suris system: N-fold Darboux transformation and conservation laws” [Nonlinear. Dyn. (2018) 91:625–639] is not correct. Then, we establish a correct Hamiltonian structure by the trace identity. Further, we also prove that the integrable lattice hierarchy has bi-Hamiltonian structure. Thus, the Liouville integrability of the integrable lattice hierarchy is obtained.

On tau-functions for the Toda lattice hierarchy

We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the generating series of $k$-point correlation functions of the solution. Applications to computing GUE correlators and Gromov--Witten invariants of the Riemann sphere are under consideration.

Algebro-Geometric Integration of the Modified Belov—Chaltikian Lattice Hierarchy

Using the Lenard recurrence relations and the zero-curvature equation, we derive the modified Belov—Chaltikian lattice hierarchy associated with a discrete 3×3 matrix spectral problem. Using the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a tri gonal curve Km−2 of arithmetic genus m−2. We study the asymptotic properties of the Baker—Akhiezer function and the algebraic function carrying the data of the divisor near \(P_{\infty_{1}}\), \(P_{\infty_{2}}\), \(P_{\infty_{3}}\), and P0 on Km−2. Based on the theory of trigonal curves, we obtain the explicit theta-function representations of the algebraic function, the Baker—Akhiezer function, and, in particular, solutions of the entire modified Belov—Chaltikian lattice hierarchy.

ABJM matrix model and 2D Toda lattice hierarchy

It was known that one-point functions in the ABJM matrix model (obtained by applying the localization technique to one-point functions of the half-BPS Wilson loop operator in the ABJM theory) satisfy the Jacobi-Trudi formula, which strongly indicates the integrable structure of the system. In this paper, we identify the integrable structure of two-point functions in the ABJM matrix model as the two-dimensional Toda lattice hierarchy. The identification implies infinitely many non-linear differential equations for the generating function of the two-point functions.

Decomposing the Toda hierarchy by an implicit symmetry constraint

An implicit symmetry constraint of the famous Toda lattice hierarchy is presented. Using this symmetry constraint, every lattice equation in the Toda hierarchy is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.

Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi‐Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.