Equilibrium of charges and differential equations solved by polynomials

Abstract

We examine connections between rationality of certain indefinite integrals and equilibrium of Coulomb charges in the complex plane .Polynomiality of all solutions is assured by the rationality of (1).

Vortices and Polynomials

“…Adler and Moser [14] constructed the polynomials n (z) = N k=1 (z − z k ), which are defined in Definition 2 below, whose roots satisfy (9), and thus, all solutions of (9) can be obtained. These are constructed by Adler and Moser [14], using of the Darboux transformations of the operator d 2 dz 2 , following the work of Crum [36], though many of the results appear in the paper by Burchnall and Chaundy [35]; see also [37].…”

“…Substituting P(z) = z j + a j−1 z j−1 + · · · + a 0 and Q(z) = z k + b k−1 z k−1 + · · · + b 0 into Equation (31) yields the leading-order term 2µ( j − k)z j+k−1 and, thus, j = k, i.e., the polynomials P(z) and Q(z) have the same degree. Theorem 3 shows that the degrees of the polynomials P(z) and Q(z) are triangular numbers, though it appears not to be understood why this restriction holds, independent of knowing the solution, in contrast to the situation for stationary vortex patterns in Section 3.2.1 Adler and Moser [14] proved the following result concerning polynomial solutions of Equation (31); see also [6,7,37]. THEOREM 3.…”

“…Substituting P(z) = z m + a m−1 z m−1 + · · · + a 0 and Q(z) = z n + b n−1 z n−1 + · · · + b 0 into Equation (37) shows that the leading-order term is identically satisfied, and thus, the degrees of the polynomials P(z) and Q(z) are not restricted. Kadtke and Campbell [8] obtained some polynomial solutions of Equation (37) in the case when m = n = 2 and m = n = 4, though they also claimed that there were no solutions when m = n = 6. However, we have found six pairs of solutions of Equation (37) in the case when m = n = 6 and 12 pairs of solutions when m = n = 8 (for this case, two pairs of solutions are given in Ref.…”

Rational Solutions of the Boussinesq Equation

“…(3) The Yablonskii-Vorob'ev polynomials arise as partition functions in string theory [50]. (4) The Adler-Moser polynomials [7,10], which are generalizations of the Yablonskii-Vorob'ev polynomials and describe rational solutions of the Korteweg-de Vries equation (1.2), arise in the description of stationary vortex patterns [12][13][14]58]. …”

The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials

“…(1) The Yablonskii-Vorob'ev polynomials arise as partition functions in string theory [55]. (2) The Adler-Moser polynomials [56,57], which are generalizations of the Yablonskii-Vorob'ev polynomials and describe rational solutions of the Korteweg-de Vries equations, arise in the description of stationary vortex patterns [58][59][60][61]. (3) The generalized Hermite polynomials arise as multiple integrals in random matrix theory [62,63], and as coefficients of recurrence relations for orthogonal polynomials with weight (x − z) m exp (−x 2 ) [64].…”