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

Abstract

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.

arXiv commentlatex2e, a4 paper 15 pages, no figure; (v2) a few references are added

References in corpus (5)

• SDiff(2) Toda equation -- hierarchy, $\tau$ function, and symmetries
• Quasi-classical limit of Toda hierarchy and W-infinity symmetries
• The algebra of q-pseudodifferential symbols and the q-W_{KP}^n algebra
• The solution to the q-KdV equation
• On a c-number quantum $\tau$-function

Cited by in corpus (6)

• Classical R-matrix theory for bi-Hamiltonian field systems
• Functional representations of integrable hierarchies
• The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons
• A q-analogue of gl_3 hierarchy and q-Painleve VI
• Integrable discrete systems on R and related dispersionless systems
• Integrable structure of melting crystal model with two q-parameters