On the combinatorics of the Hirota \(D\)-operators. (English) Zbl 0868.35101

Gilson, C.; Lambert, F.; Nimmo, J.; Willox, R.

Proc. R. Soc. Lond., Ser. A 452, No. 1945, 223-234 (1996).

Summary: A generic formula is presented which relates the Hirota \(D\)-operators to simple combinatorics. Particular classes of partition polynomials (Bell-polynomials and generalizations) are found to play an important role in the characterization of bilinearizable equations. As a consequence it is shown that bilinear Bäcklund transformations for single-field bilinearizable equations linearize systematically into corresponding Lax-pairs.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
05E35	Orthogonal polynomials (combinatorics) (MSC2000)
33C80	Connections of hypergeometric functions with groups and algebras, and related topics

Keywords:

Bell-polynomials; Hirota \(D\)-operators; partition polynomials; bilinearizable equations; Bäcklund transformations; Lax-pairs