Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebañski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present recursion relations for symmetries for thes...

most of the theologians and muslim philosophers, at least in accordance with tradition and narrative, believe in bodily resurrection as a necessary faith; but they have different conceptualizations of the “body of resurrection” such as: elemental body, heavenly body, intermediary body and imaginary body. the presentation of these conceptualizations will help us to find guidelines for more accur...

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them for both equations. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to syst...

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them for both equations. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to syst...

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equat...

Yes, we are dreamers, because, like children, we don’t like nightmares. Yes, we are utopian. This utopia is not a heavenly paradise come to earth. Neither is it a return to a mythical Golden Age. This ‘other’ place is a symbolic territory, based on our revolutionary refusal to put up with a world founded on the violence of class and ethnic or sexual domination, of the exploitation of labour and...

Contestants have to choose whether to initiate a contest or war, or whether to remain peaceful for another period. We find that agents wait and initiate the contest once their rival is sufficiently weak to be an easy target.

The ∂̄-dressing scheme based on local nonlinear vector ∂̄-problem is developed. It is applicable to multidimensional nonlinear equations for vector fields, and, after Hamiltonian reduction, to heavenly equation. Hamiltonian reduction is described explicitely in terms of the ∂̄-data. An analogue of Hirota bilinear identity for heavenly equation hierarchy is introduced, τ -function for the hierarchy...