The so called “Painlevé-Calogero correspondence” relates the sixth Painlevé equation with an integrable system of the Calogero type. This relation was recently generaized to the other Painlevé equations and a “multi-component” analogue. This paper reviews these results. 1 Historical background It was at the beginning of the twentieth century that Painlevé discovered what are nowadays called the...

Analytic-bilinear approach is used to study continuous and discrete non-isospectral symmetries of the generalized KP hierarchy. It is shown that Möbius symmetry transformation for the singular manifold equation leads to continuous or discrete non-isospectral symmetry of the basic (scalar or multicomponent KP) hierarchy connected with binary Bäcklund transformation. A more general class of multi...

In this paper, we establish exact solutions for nonlinear evolution equations. The homogeneous balance method is used to construct exact travelling wave solutions of the CalogeroDegasperis (CD) and Potential Kadomstev-Petviashvili (PKP) equations, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation. Many new f...

A meromorphic solution to the Burgers equation with complex viscosity is analysed. The equation is linearized via the Cole–Hopf transform which allows for a careful study of the behaviour of the singularities of the solution. The asymptotic behaviour of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerica...

Abstract We present new soliton equations related to the A -type spin Calogero–Moser (CM) systems introduced by Gibbons and Hermsen. These are generalizations of Benjamin–Ono (BO) equation recently non-chiral intermediate long-wave (ncILW) equation. obtain multi-soliton solutions these BO ncILW via a spin-pole ansatz where dynamics is governed CM system in rational hyperbolic cases, respectivel...

We quantize the spin Calogero-Moser model in the R-matrix formalism. The quantum R-matrix of the model is dynamical. This R-matrix has already appeared in Gervais-Neveu’s quantization of Toda field theory and in Felder’s quantization of the Knizhnik-Zamolodchikov-Bernard equation. PAR LPTHE 95-25 ∗L.P.T.H.E. Université Paris VI (CNRS UA 280), Box 126, Tour 16, 1er étage, 4 place Jussieu, 75252 ...

In this paper, we apply a relatively new technique which is called the exp-function method to construct generalized solitary and periodic solutions of Calogero-Degasperis-Fokas (CDF) equation which plays a very important role in mathematical physics, applied and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical r...

One-dimensional three-body Schrödinger equation is studied, with binding mediated by the power-law two-body long-range attraction V (x) = αx in superposition with a short-range repulsion V (x) = β/x (plus, possibly, further subdominant power-law components). Such an unsolvable and non-separable generalization of Calogero model (where one had L = K = 2) is shown to become separable and solvable ...

For the integrable N -particle Calogero-Moser system with elliptic potential it is shown that the Lax operator found by Krichever possesses a classical r-matrix structure. The r-matrix is a natural generalisation of the matrix found recently by Avan and Talon (hep-th/9210128) for the trigonometric potential. The r-matrix depends on the spectral parameter and only half of the dynamical variables...

We introduce an integrable time-discretized version of the classical CalogeroMoser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a discretized version of the Kadomtsev-Petviashvili equation, leading to a finite-dimensional symplectic mapping. Lax pair, symplectic structure and sufficient set of invariants of the discrete Cal...