In [1], Airault, McKean and Moser observed that the motion of poles of a rational solution to the K-dV or Boussinesq equation obeys the Calogero-Moser dynamical system [2, 3, 4] with an extra condition on the configuration of poles. In [8], Krichever observed that the motion of poles of a solution to the KP equation which is rational in t1 obeys the Calogero-Moser dynamical system. In this note...

In this paper, we present explicit Painlevé test for the potential Boussinesq equation, The murrary equation, The (2 + 1) Calogero equation, The Rosenau – Hyman equation (RH), Cole – Hopf (CH) equation, The Fornberg – Whitham equation (FW), Some of these equations have shown to possess Painlevé property, therefore, are Painleve integrable while the rest did not pass the test and reasons for tha...

The choice of a suitable random matrix model of a complex system is very sensitive to the nature of its complexity. The statistical spectral analysis of various complex systems requires, therefore, a thorough probing of a wide range of random matrix ensembles which is not an easy task. It is highly desirable, if possible, to identify a common mathematcal structure among all the ensembles and an...

All known today integrable scalar (1+1)-dimensional evolution equations with time-independent coefficients possess infinite-dimensional Abelian algebras of time-independent higher order symmetries (see e.g. [1, 2]). However, the equations of this kind usually do not have local timedependent higher order symmetries. The only known exceptions from this rule seem to occur [3] for linearizable equa...

Olshanetsky and Perelomov proposed the family of integrable quantum systems, which is called the Calogero-Moser-Sutherland system or the Olshanetsky-Perelomov system ([5]). In early 90’s, Ochiai, Oshima and Sekiguchi classified the integrable models of quantum mechanics which are invariant under the action of a Weyl group with some assumption ([4]). For the BN (N ≥ 3) case, the generic model co...

We construct non-Abelian analogs for some KdV type equations, including the (rational form of) exponential Calogero--Degasperis equation and generalizations of Schwarzian equation. Equations differential substitutions under study contain arbitrary parameters.

The Inozemtsev model is considered to be a multivaluable generalization of Heun’s equation. We review results on Heun’s equation, the elliptic Calogero–Moser–Sutherland model and the Inozemtsev model, and discuss some approaches to the finite-gap integration for multivariable models.

We relate two integrable models in (1+1) dimensions, namely, multicomponent Calogero-Sutherland model with particles and antiparticles interacting via the hyperbolic potential and the nonrelativistic factorizable S-matrix theory with SU(N)invariance. We find complete solutions of the Yang-Baxter equations without implementing the crossing symmetry, and one of them is identified with the scatter...