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

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

2004-12-27

arxiv.org/abs/nlin/0412067v2

Lett.Math.Phys. 72 (2005) 165-181

Nonlinear Sciences

Exactly Solvable and Integrable Systems

latex2e, a4 paper 15 pages, no figure; (v2) a few references are added

Scientific paper

10.1007/s11005-005-6782-5

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 $q$-analogue 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.

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