Quantum Painlevé Second Lax Pair and Quantum (Matrix) Analogues of Classical Painlevé II Equation

In this article, we present a quantum Painlevé second Lax pair that explicitly involves the Planck constant ħ and an arbitrary field variable v which distinguish it from the existing Flaschka–Newell Painlevé second Lax pair. We show that Flaschka–Newell pair appears as a special case of our quantum Painlevé second Lax pair. It is shown that the compatibility of quantum Painlevé second Lax pair simultaneously yields a quantum Painlevé equation and a quantum commutation relation between field variable v and independent variable z. We also show that the field variable v with different choices yields various analogs of classical Painlevé second equation as matrix Painlevé second equation, derivative matrix Painlevé second equation, and noncommutative Painlevé second equation. Further, we construct the gauge equivalence of quantum Painlevé second Lax pair whose compatibility condition generates a quantum P34 equation involving ħ with power +1 that brings the classical P34 equation close to its quantum analogs as compared to the existing one which carries ħ with power +2.