The multicomponent 2D Toda hierarchy: discrete flows and string equations

©IOP Publishing. The authors wish to thank the Spanish Ministerio de Ciencia e Innovación, research projects FIS2005-00319 and FIS2008-00200, and acknowledge the support received from the European Science Foundation (ESF) and the activity entitled Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM). This paper was finished during the research visits of one of the authors (MM) to the Université Catholique de Louvain and to the Scuola Internazionale Superiore di Studi Avanzati/International School for Advanced Studies (SISSA) in Trieste, MM wish to thanks Prof. van Moerbeke and Prof. Dubrovin for their warm hospitality, acknowledge economical support from MISGAM and SISSA and reckons different conversations with P. van Moerbeke, T. Grava, G. Carlet and M. Caffasso.
The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated with an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov-Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov-Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site-independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations.
Ministerio de Ciencia e Innovación (MICINN)
European Science Foundation (ESF)
MISGAM
SISSA
Depto. de Física Teórica
Fac. de Ciencias Físicas
TRUE
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