Hirota Bilinear Equations, Block Symmetries And Solutions Of Bigraded Toda Hierarchy

In this paper, firstly we generalize the Sato theory to the extended bi-graded Toda hierarchy (EBTH). We revise the definition of the Lax equations, give the Sato equations, wave operators, Hirota bilinear identities (HBIs) and show the existence of tau functionΤ(t). Meanwhile we prove the validity of its Fay-like identities and Hirota bilinear equations (HBEs) in terms of vertex operators whose coefficients take values in the algebra of differential opera-tors. In contrast with HBEs of the usual integrable system, the current HBEs are equations of product of operators involving e(?)x andΤr(t).Secondly, we define Orlov-Schulman's operators ML, MR, and then use them to construct the additional symmetries of the bigraded Toda hierar-chy (BTH). We further show that these additional symmetries form an in-teresting infinite dimensional Lie algebra known as a Block type Lie alge-bra, whose structure theory and representation theory have recently received much attention in literature. By acting on two different spaces under the W-constraints we find in particular two representations of this Block Lie algebra B.Then we give the definition of dipersionless bigraded Toda hierarchy (dBTH) and introduce some Sato theory on dBTH. We define Orlov-Schulman's ML, MR operator and give the additional Block symmetry of dBTH. Mean-while we give tau function of dBTH and some some related dipersionless bilinear equations.After that, we show that there exists a natural symmetry between the (N, M)-and (M, N)-bigraded Toda hierarchies. We then derive the Hirota bilinear form for those commuting flows, which consists of two-dimensional Toda hierarchy, the discrete KP hierarchy and its Backlund transformations. We also discuss the solution structure of the (N, M)-bigraded Toda equation in terms of the moment matrix defined via the wave operators associated with the Lax operator, and construct some of the explicit solutions. In particular, we give the rational solutions which are expressed by the products of the Schur polynomials corresponding to non-rectangular Young diagrams. At last, we give finite dimensional exponential solutions of the bigraded Toda Hierarchy(BTH). As an specific example of exponential solutions of the BTH, we consider a regular solution for the (1,2)-BTH with 3×3 Lax matrix, and discuss some geometric structure of the solution.