Dark Equations

Observing the Universe, astronomers have concluded that the motion of stars can not be accounted for unless one assumes that most of the mass in the Universe is carried on by a "dark matter", so far impervious to all attempts at being detected.There is now a similar concept of "dark energy".I shall discuss a different subject, "dark equations".These have never indicated that they influence anything or even exist, but if one supposes that they do exist, one can systematically discover them and study their properties, some of which turn out to be strange and others mysterious.These equations are similar in spirit to what one gets when linearizing a given system, or studies how an external linear wave interacts with a particular solution of a given system.We define and study linear extensions of dynamical systems in general, and integrable and Hamiltonian systems in particular.Systems discussed include the KdV and mKdV equations and the associated Miura maps, the Burgers hierarchy and the associated Hopf-Cole transformations, long wave equations, the Benney hierarchy, and the KP hierarchy.B A Kupershmidt 10.The Burgers hierarchy 11.The equation u t + uu x = 0 and its hierarchy 12.Long wave equations 13.The Benney hierarchy 14.The KP hierarchy 15.One-dimensional linear extensions of linear Hamiltonian matrices 16.Nonlinear derivations of Lie algebras 17.Linear extensions as Symbols of nonlinear ones 18. Scalar extensions associated with scalar Lax operators of order ≥ 4 19.Composition of Hamiltonian extensions