The Painlev\'e--Kovalevskaya test is applied to find three matrix versions of the Painlev\'e II equation. All these equations are interpreted as group-invariant reductions integrable evolution equations, which makes it possible construct isomonodromic Lax pairs for them.

We construct a Lax pair for the E (1) 6 q-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices – the q-linear lattice – through a natural generalisation of the big q-Jacobi weight. As ...

The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation with arbitrary functions in time t for the KPESCS which, in contrast with the binary Darboux transformation of the ...

This project investigates the articulatory properties of German vowels on the basis of tagged Cine-MRI data. German has 15 monophthongs, which are classified into seven tense-lax pairs. Understanding the phonetic correlate of vowel tenseness has proven elusive, partly due to the difficulty of obtaining global shape information for the tongue, especially the pharyngeal portion. Tagged Cine-MRI a...

A systematic approach to Liouville integrable defects is proposed, based on an underlying Poisson algebraic structure. The non-linear Schrödinger model in the presence of a single particle-like defect is investigated through this algebraic approach. Local integrals of motions are constructed as well as the time components of the corresponding Lax pairs. Continuity conditions imposed upon the ti...

The modified Kadomtsev-Petviashvili (mKP) equation is shown in this paper to be decomposable into the first two soliton equations of the 2N -coupled Chen-Lee-Liu and Kaup-Newell hierarchies by respectively nonlinearizing two sets of symmetry Lax pairs. In these two cases, the decomposed (1+1)-dimensional nonlinear systems both have a couple of different Lax representations, which means that the...