q-analogue of modified KP hierarchy and its quasi-classical limit

A q-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear q-difference equations. These bilinear equations are translated to the language of wave functions, which turn out to satisfy a system of linear q-difference equations. These linear q-difference equations are used to formulate the Lax formalism and the description of quasi-classical limit. These results can be generalized to a qanalogue of the Toda hierarchy. The results on the q-analogue of the Toda hierarchy might have an application to the random partition calculus in gauge theories and topological strings. Mathematics Subject Classification: 35Q58, 37K10, 58F07