Bernoulli Numbers and Solitons — Revisited

Abstract

In the present paper we propose a new proof of the Grosset-Veselov formula connecting onesoliton solution of the Korteweg-de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.

On some connections between the Gompertz function and special numbers

“…which has been demonstrated in [18]. For another proofs of ( 17) see [13] and [23]. Formula (17) shows the connection between 1-soliton solution of the KdV equation and the Bernoulli numbers.…”

A Generalized Logistic Function and Its Applications

“…and B 2n is the 2nth Bernoulli number. Other proofs of the Grosset-Veselow formula can be found in [2], [20]. Bernoulli numbers have the following generating function (see Graham, Knuth, Patashnik [8])…”

“…The saturation levels of waves #4 and #5 in (20) were calculated according to formula (17). However, for wave #3 there is no clear maximum of the Index.…”