BOUSSINESQ-TYPE EQUATIONS FROM NONLINEAR REALIZATIONS OF W3

Abstract

We construct new coset realizations of infinite-dimensional linear [Formula: see text] symmetry associated with Zamolodchikov's W3 algebra which are different from the previously explored sl3 Toda realizations of [Formula: see text]. We deduce the Boussinesq and modified Boussinesq equations as constraints on the geometry of the corresponding coset manifolds. The main characteristic features of these realizations are: (i) among the coset parameters there are space and time coordinates x and t which enter the Boussinesq equations; all other coset parameters are regarded as fields depending on these coordinates; (ii) the spin 2 and 3 currents of W3 and two spin 1 U (1) Kac–Moody currents as well as two spin 0 fields related to the W3 currents via Miura maps, come out as the only essential parameter-fields of these cosets; the remaining coset fields are covariantly expressed through them; (iii) the Miura maps get a new geometric interpretation as [Formula: see text]-covariant constraints which relate the above fields while passing from one coset manifold to another; (iv) the Boussinesq equation and two kinds of the modified Boussinesq equations appear geometrically as the dynamical constraints accomplishing [Formula: see text]-covariant reductions of original coset manifolds to their two-dimensional geodesic submanifolds; (v) the zero-curvature representations for these equations arise automatically as a consequence of the covariant reduction; (vi) W3 symmetry of the Boussinesq equations amounts to the left action of [Formula: see text] symmetry on its cosets. The approach proposed could provide a universal geometric description of the relationship between W-type algebras and integrable hierarchies.