Soliton Equations and Simple Combinatorics

Abstract

A systematic, elementary and pedagogical approach to a class of soliton equations, and to their spectral formulation, is presented. This approach, based on the use of exponential polynomials, follows naturally from a comparison of some simple results for two representatives of the class: the KdV-and the Boussinesq-equation.

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

“…Let g = g(x, y, z, t) be a C ∞ function, and the multi-dimensional Bell polynomials are written as [40][41][42][43][44] , , ,…”

Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

“…This equation can be viewed as a homogeneous P-condition [8] of weight 6 (the weight of each term being defined as minus its dimension, a weight 3 to ). That means (19) can be written as a linear combination of P-polynomials of weight 6:…”

“…In the early 1930s, the classical Bell polynomials were introduced by Bell which are specified by a generating function and exhibiting some important properties [7]. Recently, Lambert and coworkers have proposed a relatively convenient procedure based on Bell polynomials which enables us to obtain bilinear forms, bilinear BTs, Lax pairs, and Darboux covariant Lax pairs for NEEs [8][9][10][11]. It is shown that Bell polynomials play an important role in the characterization of bilinearizable equations and a deep relation between the integrability of an NEE and the Bell polynomials.…”

Soliton solutions and Bäcklund transformations of a (2 + 1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy

“…To understand the phenomena modeled by the NLEEs, some methods have been developed for people to search for the explicit solutions of the NLEEs, such as the inverse scattering transformation [1], Bell polynomials [10][11][12][13][14], Hirota method [15,16], Bäck-lund transformation (BT) [17,18], Darboux transformation [19], and Lie group method [20][21][22][23].…”