Related papers: Quantum Lax Pair From Yang-Baxter Equations
We consider a general class of boundary terms of the open XYZ spin-1/2 chain compatible with integrability. We have obtained the general elliptic solution of $K$-matrix obeying the boundary Yang-Baxter equation using the $R$-matrix of the…
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…
Determining whether a dynamical system is integrable is generally a difficult task which is currently done on a case by case basis requiring large human input. Here we propose and test an automated method to search for the existence of…
We present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a Lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect…
In the framework of the theory of differential coverings \cite{KV}, we discuss a general geometric construction that serves the base for the so-called Lax pairs containing differentiation with respect to the spectral parameter \cite{OS}.…
In this paper, we present multi parametric quadgraph equations which are consistent around the cube. These equations are obtained by applying a `double twist' to known integrable equations. Furthermore, we perform a limit to one of these…
It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations…
We formulate $Q$-systems for the closed XXZ, open XXX and open quantum-group-invariant XXZ quantum spin chains. Polynomial solutions of these $Q$-systems can be found efficiently, which in turn lead directly to the admissible solutions of…
We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter…
The ground state of Shor-Movassagh chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states, the model is not solvable. We prove the integrability of the model without interacting…
We present a simple but explicit example of a recent development which connects quantum integrable models with Schubert calculus: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated…
We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and B\"acklund transformations. We show that these concepts, which are associated with integrability, are not strictly…
Using an algebraic framework we solve a problem posed in [5] and [7] about the axiomatizability of a quantum computational type logic related to fuzzy logic. A Hilbert-style calculus is developed obtaining an algebraic strong completeness…
The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures. We restrict…
In the present work, we study Hamiltonian systems on (co)adjoint orbits and propose a Lax pair formalism for Gelfand-Tsetlin integrable systems defined on (co)adjoint orbits of the compact Lie groups ${\rm{U}}(n)$ and ${\rm{SO}}(n)$. In the…
These lecture notes are devoted to the integrability of discrete systems and their relation to the theory of Yang-Baxter (YB) maps. Lax pairs play a significant role in the integrability of discrete systems. We introduce the notion of Lax…
A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice…
The Lax pair representation in Fourier space is used to obtain a list of integrable scalar evolutionary equations with quadratic nonlinearity. The famous systems of this type such as KdV, intermediate long-wave equation (ILW), Camassa-Holm…
In this paper, we construct a Lax pair for the classical hyperbolic van Diejen system with two independent coupling parameters. Built upon this construction, we show that the dynamics can be solved by a projection method, which in turn…
In this series of lectures we present the universal method based on the use of the functional integrals to derive the bound state equations for different two-body systems in elementary particle physics as well as in condensed matter theory:…