A note on Bernoulli polynomials and solitons

Abstract

The dependence on time of the moments of the one-soliton KdV solutions is given by Bernoulli polynomials. Namely, we prove the formula ∫ℝ xn sech2(x - t) dx = 2πn (-i)n Bn(1/2 + π/i), expressing the moments of the one-soliton function sech2(x-t) in terms of the Bernoulli polynomials Bn (x). We also provide an alternative short proof to the Grosset-Veselov formula connecting the one-soliton to the Bernoulli numbers ∫ℝ (Dm-1sech2x)2 dx = (-1)m-1 22m-1 B2m, (D = d/dx) published recently in this journal.