On an integrable discretisation of the Ablowitz-Ladik hierarchy

Several integrable semi-discretizations are known in the literature for the massive Thirring system in characteristic coordinates. We present for the first time an integrable semi-discretization of the massive Thirring system in laboratory coordinates. Our approach relies on the relation between the continuous massive Thirring system and the Ablowitz-Ladik lattice. In particular, we derive the Lax pair for the integrable semi-discretization of the massive Thirring system by combining together the time evolution of the massive Thirring system in laboratory coordinates and the Bäcklund transformation for solutions to the Ablowitz-Ladik lattice.

“…It is known [37,42] (24) and (25) for the same spectral parameter λ. The relation between ϕ m and ϕ m can be written in the form used in [35]:…”

Section: Bäcklund-darboux Transformation For the Al Latticementioning

confidence: 99%

Section: Choice Of Parametersmentioning

confidence: 99%

Integrable semi-discretization of the massive Thirring system in laboratory coordinates

Joshi

Pelinovsky

2018

“…For real orbits, upon noting that |x| = max(x, −x), the latter is seen to be "tropical" (it is defined in the (max, +) algebra), and it turns out that for this tropical map all orbits are periodic with period 9 [2]. If we would wish to apply Definition 2.1 to (22), then a condition of the form…”

Section: The Cohen Mapmentioning

confidence: 99%

Algebraic entropy for algebraic maps

Hone¹,

Ragnisco²,

Zullo³

2015

Self Cite

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Bäcklund transformations.

“…Classical Bäcklund transformations for the Ablowitz-Ladik model have been considered in different works (see e.g. [23], [30]). It is possible to compose elementary parametric transformations to get more complex multi-parametric maps.…”

Section: Classical Bäcklund Transformationsmentioning

confidence: 99%

“…The parameter µ can be seen as an evolution parameter for the Bäcklund transformations. Then the maps are the integral curves of a non autonomous Hamiltonian system of equations, the flow being generated by the variable conjugated to µ expressed in the variables (r, q) [29]- [30], that is by Φ = ∂F ∂µ r=r(r,q)…”

Section: Classical Bäcklund Transformationsmentioning

confidence: 99%

A q-difference Baxter operator for the Ablowitz–Ladik chain

Zullo¹

2015

Self Cite

We construct the Baxter's operator and the corresponding Baxter's equation for a quantum version of the Ablowitz Ladik model. The result is achieved by looking at the quantum analogue of the classical Bäcklund transformations. For comparison we find the same result by using the well-known Bethe ansatz technique. General results about integrable models governed by the same r-matrix algebra will be given. The Baxter's equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Bäcklund transformations gives a trace formula representing the classical analogue of the Baxter's equation. An explicit q-integral representation of the Baxter's operator is discussed.