On the Virasoro Structure of Symmetry Algebras of Nonlinear Partial Differential Equations

Abstract

Abstract. We discuss Lie algebras of the Lie symmetry groups of two generically nonintegrable equations in one temporal and two space dimensions arising in different contexts. The first is a generalization of the KP equation and contains 9 arbitrary functions of one and two arguments. The second one is a system of PDEs that depend on some physical parameters. We require that these PDEs are invariant under a Kac-Moody-Virasoro algebra. This leads to several limitations on the coefficients (either functions or parameters) under which equations are prime candidates for being integrable.

Symmetries and Lie Algebra of the Differential–difference Kadomstev–petviashvili Hierarchy

“…First, the new time-dependence (2.11) of the spectral parameter η leads to the new algebra structure (3.21), which is different from the centreless Kac-Moody-Virasoro algebra (cf. [20]) of the D∆KP equation given in [18], and also different from the centreless Kac-Moody-Virasoro algebra of the KP hierarchy obtained in [15]. Besides, {K 1 , K 2 , τ s 1 } compose a subalgebra.…”

Analysis of the symmetry group and exact solutions of the dispersionless KP equation in n + 1 dimensions

“…This typical symmetry structure arises in completely integrable evolutionary equations in 2 + 1-dimensions like KP equation [7,8], three-wave resonance equations [9], Davey-Stewertson system [10] and a few others (see [11,12] for a further discussion). Indeed, this is the case here.…”

Infinite dimensional symmetry group, Kac-Moody-Virasoro algebras and integrability of Kac-Wakimoto equation

“…In classical sense, c = 0, but it plays crucial role in quantum mechanics (for more details, see the ref. [15][16][17]). The presence of Virasoro algebra is a good predictor of integrability that can be seen in the typically integrable equations in (2+1)-dimension [8,9,18,19].…”