Matrix Lax Representation and Darboux Solutions of Classical Painlev´e Second Equation

In this chapter, we discuss various integrable aspects of classical Painlev´e second as the linear representations of its symmetric form and with a brief review on its connections to well know physical solitoinc equation the Korteweg-de Vries equation. This chapter encloses the derivation of Darboux solutions of classical Painlev´e second equation by transforming its matrix Lax pair in new setting under the gauge transformations to yield its Darboux expression in additive form may be applied to calculate its non-trivial solutions. The new linear system of that equation carries similar structure as other integrable systems possess in AKNS scheme. Finally, we generalize the Darboux solutions of classical Painlev´e second equation to the N-th form in terms of Wranskian.