Non-trivial Darboux solutions of Classical Painlev\'e second equation

Abstract

In this article an other equivalent linear representation of classical Painlev\'e second equation is derived by introducing a gauge transformation to old Lax pair. The new linear system of that equation carries similar structure as other integrable systems possess in AKNS scheme. That system yields non-trivial Darboux solutions of classical Painlev\'e second equation which are further generalized to the $N$-th form in terms of Wranskian. Finally we present the exact solutions of that equation through its associated Riccati system.