We generalize the construction of Gelfand-Dikii brackets to the case of nonstandard Lax equations. We also discuss the possible origin of Kac-Moody algebras present in such systems.

We propose interpreting the zero curvature condition associated with an integrable model as an anomaly equation. This can lead to the WZWN action and the associated current algebra quite readily and clarifies further the connections found between the integrable models and 2D gravity theories. We analyze, in detail, the cases SL (2, R) (KdV hierarchy), OSp (2/1) (sKdV hierarchy) and SL (3, R) (Boussinesq hierarchy) and obtain the operator product expansions of the appropriate fields. We also make some observations on the generalization of our method to SL (n, R).

Articles you may be interested inDegenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equationsThe Hamiltonian structures of the superKP hierarchy associated with an even parity superLax operator It is shown that with every Lax operator, which is a pseudodifferential operator of nonzero leading order, is associated a KP hierarchy. For each such operator, we construct the second Gelfand-Dikii bracket associated with the Lax equation and show that it defines a Hamiltonian structure. When the leading order is positive the corresponding compatible first Hamiltonian structure, which turns out, in general, to be different from the naive first Gelfand-Dikii bracket is derived. The corresponding Hamiltonian structures for the constrained Lax operator, where the next to leading-order term vanishes or has a constant coefficient, is discussed.

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We investigate theoretically the transition temperature of a weakly interacting Fϭ1 spinor Bose-Einstein condensate ͑BEC͒ formed in an optical trap. The Hartree-Fock-Bogoliubov formalism for the spinor BEC is developed based on the mean-field theory for inhomogeneous systems at finite temperatures. The shift of the transition temperature due to finite-size effects and the two-body interaction is obtained simultaneously by analytically solving the coupled eigenvalue equations for quasiparticle excitations and amplitudes under Hartee-Fock-Bogoliubov and other relevant approximations. When applying this to the recent experiment of the all-optical formation of a spinor BEC with a condensate of 3.5ϫ10 4 87 Rb atoms ͓Phys. Rev. Lett. 87, 010404 ͑2001͔͒, our calculations show that the condensation occurs at a temperature about 177 nK. An analytic result for the shift of the transition temperature in the presence of a weak magnetic field is also derived.