On General Representation of the Meromorphic Solutions of Higher Analogues of the Second Painleve Equation

Abstract

„On general representation of the meromorphic solutions of higher analogues of the second painleve equation" Mathematical Modelling Analysis, 2(1), p.61-65

Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: II

“…Remark 4.4. According to [17], the solutions enumerated in Corollary 4.1 are the only algebraic solutions of equation (1.1) with a = 0 (resp., equation (1.6)). Here, we show how this fact can be deduced from our asymptotic results.…”

“…We now mention some works that are related to the topic of our study. Gromak [17] proved that the general third Painlevé equation has algebraic solutions iff it reduces (with, perhaps, the help of the transformation u(τ ) → 1/u(τ )) to the degenerate case (1.1) with ia = n ∈ Z: for each n, equation (1.1) has exactly three solutions of the form R(x), where R is a rational function and (2εx) 3 = b 2 τ . From the functional point of view, we have one multi-valued function, and the three solutions are obtained via a cyclic permutation of the sheets of the Riemann surface (2εx) 3 = b 2 τ .…”

Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: I

“…Recently, equation (1) has appeared in a number of physical [1][2][3][4] and geometrical applications [5] (in contexts independent of equation ( 2)) where knowledge of asymptotic properties of its solutions is of special importance. The Bäcklund transformation for equation (1) was obtained by Gromak [6]: he also proved [7] that the only algebraic solutions of the complete third Painlevé equation are rational functions of τ 1/3 . Actually, these functions are solutions of equation (1) for ai = n ∈ Z and arbitrary, non-vanishing values of ε and b.…”

On General Representation of the Meromorphic Solutions of Higher Analogues of the Second Painleve Equation

“…One of the important questions of the nonlinear ordinary di erential equations theory is representation of the meromorphic solutions as the ratio of the entire functions similarly to the Weierstrass function (z), which is a solution of an equation 0 2 = 4 3 + g 2 + g 3 and it has representation through the entire function (z) (z) = 0 2 ; 00 2 = ; 0 (z) = ( l n ( (z)) 0 We shall consider a reduction of the higher -Korteweg-de Vries equations (2m ; 1)u t = X m u (1) where X 1 u = Du= D @H 1 @u D = @ @x X m u = ( 2 u+ 2 DuD ;1 ;D 2 )X m;1 u =…”

“…become the Korteweg -de Vries equation and the second Painleve equation (P 2 ) correspondingly. Let us call equation ( m P 2 ) the higher analogue of the second Painleve equation similarly to equation (1). For m = 3 w e h a ve w (4) = 1 0 w 2 w 00 + 1 0 ww 0 2 ; 6w 5 ; zw; :…”