A general method for constructing vector integrable lattice systems

We introduce an integrable differential-difference KdV equation with a source. A bilinear Bäcklund transformation and the associated nonlinear superposition formula are thereby obtained. And the multisoliton solution of the equation is also presented.

A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.

Basic invariants, such as conserved quantities, symmetries, mastersymmetries, and recursion operators are explicitly constructed for the following nonlinear lattice systems: The modified Korteweg–de Vries lattice, the Ablowitz–Ladik lattice, the Brusci–Ragnisco lattice, the Ragnisco–Tu lattice and some cases of the class of integrable systems introduced by Bogoyavlensky. The algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the Lax pair analysis is discussed. By explicit presentation of the higher-order members of the corresponding hierarchies new completely integrable lattice flows are found. For all systems, multi-Hamiltonian formulations are given.

We provide a new natural interpretation of the Lax representation for an integrable system; that is, the spectral problem is the linearized form of a Miura transformation between the original system and a modified version of it. On the basis of this interpretation, we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by Miura transformations. Thus, this method can be used to generate new integrable systems from known systems through inverse Miura maps; it can be applied to both continuous and discrete systems in 1 + 1 dimensions as well as in 2 + 1 dimensions. The effectiveness of the method is illustrated using examples such as the nonlinear Schrödinger (NLS) system, the Zakharov–Ito system (two-component KdV), the three-wave interaction system, the Yajima–Oikawa system, the Ablowitz–Ladik lattice (integrable space-discrete NLS), and two (2 + 1)-dimensional NLS systems.

After a quench, observables in an integrable system may not relax to the standard thermal values, but can relax to the ones predicted by the generalized Gibbs ensemble (GGE) [M. Rigol et al., Phys. Rev. Lett. 98, 050405 (2007)]. The GGE has been shown to accurately describe observables in various one-dimensional integrable systems, but the origin of its success is not fully understood. Here we introduce a microcanonical version of the GGE and provide a justification of the GGE based on a generalized interpretation of the eigenstate thermalization hypothesis, which was previously introduced to explain thermalization of nonintegrable systems. We study relaxation after a quench of one-dimensional hard-core bosons in an optical lattice. Exact numerical calculations for up to 10 particles on 50 lattice sites (≈10(10) eigenstates) validate our approach.

This paper studies the structure of Lax pairs associated with integrable lattice systems (where space is a one-dimensional lattice, and time is continuous). It describes a procedure for generating examples of such systems, and emphasizes the features that are needed to obtain equations which are local on the spatial lattice.

Thresholds for soliton creation in the Ablowitz–Ladik lattice

An integrable coupling of the known integrable differential-difference equation and its Lax pair are presented. Based on the gauge transformation between the corresponding four-by- four matrix spectral problems, a Darboux transformation of Lax pair for the integrable coupling is established. As an application of the obtained Darboux transformation, an explicit solution is given.

In this paper, the homotopy analysis method is successfully applied to solve the systems of differential-difference equations. The Ablowitz-Ladik lattice system are chosen to illustrate the method. Comparisons between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective and simple in solving systems of differential-difference equations.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

In this paper, we applied the rational formal expansion method to construct a series of soliton-like and period-form solutions for nonlinear differential-difference equations. Compared with most existing methods, the proposed method not only recovers some known solutions, but also finds some new and more general solutions. The efficiency of the method can be demonstrated on Toda Lattice and Ablowitz—Ladik Lattice.

A lattice Green's function theory is formulated to give a band-theoretical interpretation of stationary nonlinear localized modes (SNLM) in pure nonlinear lattices. In one-dimensional lattices and in its simplest form in higher-dimensional cases, the theory yields the eigenfrequency ω of the SNLM in terms of the frequency ω 0 ( q ) ( q is the wave vector) of the corresponding linear lattice in the form ω=ω 0 ( q 0 + i K ), where q 0 is an edge point of the Brillouin zone of ω 0 ( q ), and K ≡ K ( A ) is a parameter depending on the localized mode amplitude A. Such a result and more general ones involving lattice Green's functions lead to the ubiquity of the existence of the SNLM irrespective of the lattice dimensionality as approximate nonlinear modes in nonintegable lattices and its similarity to force-constant defect modes. Stationary solitons in the Ablowitz-Ladik lattice can be considered as an example of the SNLM in integrable lattices.

Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang-Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories.

This is a very interesting collection of introductory and review articles on the theory and applications of classical and quantum integrable systems. The book reviews several integrable systems such as the KdV equation, vertex models, RSOS and IRF models, spin chains, integrable differential equations, discrete systems, Ising, Potts and other lattice models and reaction--diffusion processes, as well as outlining major methods of solving integrable systems. These include Lax pairs, Bäcklund and Miura transformations, the inverse scattering method, various types of the Bethe Ansatz, Painlevé methods, the dbar method and fusion methods to mention just a few.

With the help of three shift operators and r-matrix theory, a few discrete lattice systems are obtained which can be reduced to the well-known Toda lattice equation with a constraint whose Hamiltonian structures are generated by Poisson tensors of some induced Lie–Poisson bracket. The recursion operators of these lattice systems are constructed starting from Lax representations. Finally, reducing the given shift operators to get a simpler one and its expanding shift operators, we produce a lattice system with three vector fields whose recursion operator is given. Furthermore, we reduce the lattice system with three vector fields to get a lattice system whose Lax pair and conservation laws are obtained, respectively.