## Multicomponent *q-P*_{IV} and its ultradiscrete limit

### C. M. Field and C. M. Ormerod

#### Abstract

We generalize the symmetric fourth q-Painleve equation q-P_{VI}
to the noncommutative setting. Considering the symmetric
q-P_{IV} to be matrix valued, well-defined multicomponent
systems are obtained. The ultradiscrete limit of these systems
yields coupled multicomponent ultradiscrete systems that
generalize ultradiscrete P_{IV}. The dynamics, and specifically
the integrability, of the newly introduced multicomponent
ultradiscrete systems is studied.

Keywords:
Integrable, cellular automata, Painlevé, ultradiscrete.

AMS Subject Classification:
Primary 39A13; secondary 33C70, 37J35, 16Y60.

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