On the algebraic dependence of solutions of the second Painlevé equation

Abstract

Consider the second Painlev´e equation w′′ = 2w3 + zw + α. (P2) It is analyzed with the use of the B¨acklund transformations T : wα−1 → wα = −wα−1 − (α − 1/2)/(w′ α−1 + w2 α−1 + z/2) , (1) T −1 : wα → wα−1 = −wα + (α − 1/2)/(w′ α − w2 α − z/2) , (2) S : w(z, α) → −w(z, −α), (3) on which the B¨acklund autotransformations are based [1].