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

Abstract

In this article, we present a new quantum Painlevé II Lax pair which explicitly involves the Planck constant $\hslash$ and an arbitrary field variable $v$ so these two objects make this new pair different from Flaschka-Newell Painlevé II Lax pair and that pair appears as particular case of our's pair which consolidates the Painlevé II equation from quantum mechanical point of view. It is shown that the compatibility of quantum Painlevé II Lax pair simultaneously yields a quantum Painlevé II equation and a quantum commutation relation between field variable $v$ and independent variable $z$. We manifest that with the different choices of arbitrary field variable system reduces to its classical version, matrix Painlevé II equation and derivative matrix Painlevé II equation. Further, we construct the gauge equivalence of quantum Painlevé II Lax pair whose compatibility condition gives rise to quantum p34 equation that involves $\hslash$ with power $+1$ which makes our system more quantized as compared to the existed one that carries $\hslash$ with power $+2$.