Painlevé test and the first Painlevé hierarchy

Abstract

Starting from the first Painlevé equation, Painlevé type equations of higher order are obtained by using the singular point analysis. 4 j=0 c j (z)y j = 0, (1.3)

Integration via Modification: A Method of Reduction of Order for Systems of Ordinary Differential Equations

“…Solving (15) for u and substituting into (14) yields (13). On the other hand, solving (14) for U and substituting in (15) yields…”

“…[10], and provides an alternative to classical Painlevé classification, e.g. as undertaken in [11,12,13,14,15]. However, having obtained an ODE together with its underlying linear problem, there then remains the question of whether that ODE is of the minimum order possible.…”

Third order differential equations with fixed critical points

“…In this class, no new Painlevé transcendents were discovered, and all of them were solvable either in terms of the known functions or one of the six Painlevé transcendents. The case in which F is a polynomial in y and its derivatives was also investigated in [15,16]. Eq.…”

“…(1.6) with F analytic in z and rational in its other arguments, was considered in [17][18][19][20][21]. Fourth and higher order equations with the Painlevé property were investigated in many articles [14][15][16][22][23][24][25][26][27][28][29][30][31][32]. Kudryashov [23], Clarkson et al [33], and Gordoa et al [34,35] obtained first, second and fourth Painlevé hierarchy, by using the non-isospectral scattering problems.…”

Non-polynomial Third Order Equations which Pass the Painlevé Test

“…The third order Painlevé type equations y = F (z, y, y , y ), (1.1) where F is polynomial in y and its derivatives, were considered in [4,5,6,7]. Some fourth and higher order polynomial-type equations with the Painlevé property were investigated in [5,6,7,8,9,10].…”