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Boussinesq-type equations from nonlinear realizations of $W_3$ | | E. Ivanov
; S. Krivonos
; R.P. Malik
; | | Date: |
10 Oct 1992 | | Journal: | Int. J. Mod. Phys. A8 (1993) 3199-3222 | | Subject: | hep-th | | Abstract: | We construct new coset realizations of infinite-dimensional linear $W_3^{infty}$ symmetry associated with Zamolodchikov’s $W_3$ algebra which are different from the previously explored $sl_3$ Toda realization of $W_3^{infty}$. 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 the 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 $W_3$ and two spin 1 $U(1)$ Kac- Moody currents as well as two spin 0 fields related to the $W_3$currents via Miura maps, come out as the only essential parameters-fields of these cosets. The remaining coset fields are covariantly expressed through them;iii.The Miura maps get a new geometric interpretation as $W_3^{infty}$ 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 $W_3^{infty}$ 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. The approach proposed could provide a universal geometric description of the relationship between $W$-type algebras and integrable hierarchies. | | Source: | arXiv, hep-th/9210058 | | Services: | Forum | Review | PDF | Favorites | |
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