On aq-deformation of the discrete Painlevé I equation andq-orthogonal polynomials

We present aq-analog of the discrete Painlevé I equation, and a special realization of it in terms ofq-orthogonal polynomials.     

Variational symmetries and Lagrangian multiforms

Letters in Mathematical Physics - By considering the closure property of a Lagrangian multiform as a conservation law, we use Noether’s theorem to show that every variational symmetry of a...     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

Integrable time-discretisation of the Ruijsenaars-Schneider model

An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 XYZ Heisenberg magnet. We present a Lax pair, the sympletic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.Supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)     

Separable interactions and liquid 3He

A rigorous expression is given for the free energy of a system of fermions in the presence of a magnetic field interacting via a pairing hamiltonian in the weak-coupling limit. Starting from this expression Landau expansions are derived in two special cases, i.e. the case of general l-wave pairing in zero field and the case of l = 1 pairing, which applies to liquid ³He, in non-zero field.     

A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Bäcklund transformation, and the method of solving a Riccati map by exploiting two known particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler’s lattice equation (or Q4).     

Bäcklund transformations and three-dimensional lattice equations

A (nonlocal) linear integral equation is studied, which allows for Bäcklund transformations in the measure. The compatibility of three of these transformations leads to an integrable nonlinear three-dimensional lattice equation. In appropriate continuum limits the two-dimensional Toda-lattice equation and the Kadomtsev-Petviashvili equation are derived as special examples.     

Separable interactions and liquid He : III. Landau expansion in the presence of a Hubbard interaction

In this paper we derive the Landau expansion for the ordered phases of liquid ³He in the presence of a magnetic field starting from a model hamiltonian which contains an l = 1 pairing interaction of the BCS type and a contact term of the Hubbard type. In the derivation use is made of an exact theorem in the theory of separable interactions and the contribution from the Hubbard interaction is evaluated in second order perturbation calculation. Some typical differences with the results based on a spin fluctuation theory are mentioned.