Remarks on differential concomitants of densities

Remarks on the Conserved Densities of the Camassa-holm Equation

It is pointed out that the higher-order symmetries of the Camassa-Holm (CH) equation are nonlocal and nonlocality poses problems to obtain higher-order conserved densities for this integrable equation (J. Phys. A: Math. Gen. 2005, 38 869-880). This difficulty is circumvented by defining a nolinear hierarchy for the CH equation and an explicit expression is constructed for the nth-order conserve...

Two remarks on differential fields

We make two observations concerning differential fields. The first is a model-theoretic proof of the existence and uniqueness of a PicardVessiot extension for an iterative linear differential equation (in positive characteristic) answering a question of Hrushovski. The second observation is that definable sets of finite Morley rank in differentially closed fields of characteristic zero are (pos...

Remarks on Quantum Differential Operators

In the course of writing the book [9] and various papers [10, 11, 12, 13, 14, 15, 16] we encountered many q-differential equations but were frustrated by a lack of understanding about natural forms for such equations. One has operators of the type qKP or qKdV for example but even there, expressing the resulting equations (even via Hirota type equations or in bilinear form) seemed curiously diff...

Conformally invariant differential operators on tensor densities

Let Fλ be the space of tensor densities on R n of degree λ (or, equivalently, of conformal densities of degree −λn) considered as a module over the Lie algebra o(p+1, q+1). We classify o(p+1, q+1)-invariant bilinear differential operators from Fλ ⊗ Fμ to Fν . The classification of linear o(p + 1, q + 1)invariant differential operators from Fλ to Fμ already known in the literature (see [6, 9]) i...

Some Remarks on Linear Differential Systems*