We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, φxt + φxxxz/4 + φxφxz + φxxφz/2 + ∂ −1 x φzzz/4 = 0. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modifie...

A generalisation of the classical Calogero-Moser model obtained by coupling it to the Gaudin model is considered. The recently found classical dynamical r-matrix [E. Billey, J. Avan and O. Babelon, PAR LPTHE 93-55] for the Euler-Calogero-Moser model is used to separate variables for this generalised Calogero-Moser model in the case in which there are two Calogero-Moser particles. The model is t...

We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace’s equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials and the constraints on the coefficients of expansion are derived. These constraints involve generalized binomial coefficients defined through Jack polynomials...

There exist certain nonlinear evolution PDEs which can be mapped to linear evolution PDEs. The prototypical such linearizable PDE is the so-called Burger's equation u t − u xx = 2uu x , (1.1) which can be written in the conservation form ∂ t u = ∂ x (u x + u 2). Employing the so-called Cole-Hopf transformation u = q x q (1.2) and using the identity ∂ t q x q = ∂ t ∂ x ln q = ∂ x ∂ t ln q = ∂ x ...

In this paper, we investigate some innovative and exact travelling wave solutions to the time-fractional phi-four equation (2+1) dimensional Calogero-Bogoyavlanskil schilf (CBS) using ( )-expansion method. Periodic are displayed in hyperbolic, trigonometric, rational function solutions. Subsequently, construct new connecting free parameters of CBS equation, which characterized into three comple...

A bstract We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, depends on parameter ? > 0, describes time evolution of two coupled spin densities propagating real line, in limit ? ? it reduces to decoupled half-wave maps (HWM) equations opposite chirality. show that ncIHF is related -type hyperbolic Calogero...

We give the hypergeometric solutions of some algebraic equations including the general fifth degree equation. 1. It is well known that the general algebraic equation of degree n ≥ 5 cannot be solved by radicales [Ab 1826]. However, it may be solved if we use wider classes of functions. For example, the fifth degree equation may be solved in modular functions [He 1858], [Kr 1858], [Kl 1884]; the...