Infinite-Dimensional Frobenius Manifolds for 2+1 Integrable Systems

Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

“…We anticipate that the results of this paper will find applications in the theory of infinitedimensional Frobenius manifolds, see [8] for the first steps in this direction.…”

“…respectively. Our main result is a description of Hamiltonian densities h(v, w) for which the corresponding systems ( 6) - (8) are integrable by the method of hydrodynamic reductions as proposed in [13]. Let us point out that the integrability conditions for the general class of twocomponent systems (1) were derived in [14] in the coordinates where the first matrix A is diagonal.…”

“…However, it is not clear how to isolate Hamiltonian cases within these general descriptions. In this paper we adopt a straightforward approach and derive the integrability conditions by directly applying the method of hydrodynamic reductions to the systems ( 6) - (8). For Hamiltonian systems of type (6) this was done in [13].…”

“…4). However, it possesses a remarkable SL(3, R)-invariance which reflects the invariance of the Hamiltonian formalism (8) under linear transformations of the independent variables x, y, t. Based on this invariance, we were able to classify integrable potentials.…”

Block type symmetry of bigraded Toda hierarchy

“…In [11], the authors generalize the Sato theory to the EBTH and give the Hirota bilinear equations in terms of vertex operators whose coefficients take values in the algebra of differential operators. In [12], a geometric structure associated with Frobenius manifold of 2D Toda hierarchy was introduced. Furthermore, motivated by the potential applications of the BTH, which is also defined by omitting the extended logarithmic flows of the EBTH, in the theory of the matrix models, it is necessary and interesting to explore its algebraic structure from the point of view of the additional symmetry.…”