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

TL;DR

This paper derives a new linear representation of the classical Painlevé II equation using gauge transformations, enabling the construction of non-trivial Darboux solutions and their generalizations.

Contribution

It introduces an equivalent linear system for Painlevé II via gauge transformation, leading to novel Darboux solutions and their N-th order generalizations using Wronskian techniques.

Findings

Derived a new linear system for Painlevé II

Constructed non-trivial Darboux solutions

Presented exact solutions via Riccati system

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.