Two-dimensional Toda Lattice Hierarchy Research Articles

INTEGRABLE STRUCTURES OF UNITARY MATRIX MODELS

The unitary matrix model is considered from the viewpoint of integrability. We demonstrate that this is an integrable system embedded into a two-dimensional Toda lattice hierarchy which corresponds to an integrable chain (modified Volterra) under a special reduction. The interrelations between this chain and other chains (like the Toda one) are demonstrated to be given by Bäcklund transformations. The case of the symmetric unitary model is discussed in detail and demonstrated to be connected with the Hermitian matrix model. This connection as a discrete analog of the correspondence between KdV and MKdV systems is investigated more thoroughly. We also demonstrate that unitary matrix models can be considered as two-component systems as well.

Non-local integrable equations as reductions of the Toda hierarchy

We treat the hierarchies of non-local integrable equations like ILW n (generalized intermediate long wave equation) as reductions of the Kadomtsev-Petviashvili (KP) and two-dimensional Toda lattice hierarchies. The explicit form of these reductions is found. We obtain the complete set of Hirota's bilinear equations for the non-local integrable equation and also discuss their N-soliton solutions.

The conserved quantities and symmetries of the two-dimensional Toda lattice hierarchy

The conserved quantities and symmetries of the soliton equations included in the 2DTL (two-dimensional Toda lattice) hierarchy are considered by means of the theory of the τ function. A typical example is presented for the 2DTL equation that is the simplest nontrivial equation in the hierarchy. Then, by applying suitable reductions, the cases of the Toda lattice equation and the sine-Gordon equation are discussed.