We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding “transfer” matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix an...

Based on the recent formulation of a general scheme to construct boundary Lax pairs, we develop this systematic construction for the A (1) n affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sineGordon (A (1) 1 ) and the A (1) 2 models. The A (1) 2 Toda theory is the first non-trivial example of the hierarchy that exhibits two distinct typ...

We study the C2 Ruijsenaars-Schneider(RS) model with interaction potential of trigonometric type. The Lax pairs for the model with and without spectral parameter are constructed. Also given are the involutive Hamiltonians for the system. Taking nonrelativistic limit, we obtain the Lax pair of C2 Calogero-Moser model.

A new formulation of Calogero-Moser models based on root systems and their Weyl group is presented. The general construction of the Lax-pairs and the proof of the integrability applicable to all models based on the simply-laced algebras (ADE) are given for two types which we will call ‘root’ and ‘minimal’. The root type Lax pair is new; the matrices used in its construction bear a resemblance t...

The singular manifold method from the Painlevé analysis can be used to investigate many important integrable properties for the nonlinear partial differential equations. In this paper, the two-singular-manifold method is applied to the (2+1)-dimensional Gardner equation with two Painlevé expansion branches to determine the Hirota bilinear form, Bäcklund transformation, Lax pairs and Darboux tra...

In contrast to mono-constrained flows with N degrees of freedom, binary constrained flows of soliton equations, admitting 2 × 2 Lax matrices, have 2N degrees of freedom. By means of the existing method, Lax matrices only yield the first N pairs of canonical separated variables. An approach for constructing the second N pairs of canonical separated variables with additional N separated equations...

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P Es) is reviewed. The method assumes that the P Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for ...

We present a new, nonautonomous Lax pair for a lattice nonautomous modified Korteweg–deVries equation and show that it can be consistently extended multidimensionally, a property commonly referred to as being consistent around a cube. This nonautonomous equation is reduced to a series of q-discrete Painlevé equations, and Lax pairs for the reduced equations are found. A 2× 2 Lax pair is given f...

In this talk we introduce generalised Calogero–Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones based on crystallographic root systems, namely those associated wi...

The purpose of this study was to make use of physiological data to test whether there is a greater overlap between tense and lax vowels in Standard Austrian German (SAG) compared with those of Standard German (SG). In order to do so, movement data was analyzed of various tense/lax vowels pairs produced in symmetrical consonantal contexts using electromagnetic articulometry from seven SG and eig...