For any affine Lie algebra g, we show that any finite dimensional representation of the universal dynamical R matrix R(λ) of the elliptic quantum group Bq,λ(g) coincides with a corresponding connection matrix for the solutions of the q-KZ equation associated with Uq(g). This provides a general connection between Bq,λ(g) and the elliptic face (IRF or SOS) models. In particular, we construct vect...

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanhfunction type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type m...

In previous work the author found solutions to the Toda equations that were expressed in terms of determinants of integral operators. Here it is observed that a simple variant yields solutions to the matrix Toda equations. As an application another derivation is given of a differential equation of Sato, Miwa and Jimbo for a particular Fredholm determinant. During the last twenty years, beginnin...

The dependence of the sixth equation of Painlevé on its four parameters (2α,−2β, 2γ, 1− 2δ) = (θ ∞ , θ 0 , θ 2 1 , θ 2 x) is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar “Lax” pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on θ∞ (and a holomorphic dep...

Exact solutions to conformable time fractional (3+1)dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE. The resultant polynomial equation is solved by using algebrai...