Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one Tau-function

Abstract

In the framework of Bell-polynomial manipulations, under investigation hereby are three single-field bilinearizable equations: the (1+1)-dimensional shallow water wave model, Boiti-Leon-Manna-Pempinelli model, and (2+1)-dimensional Sawada-Kotera model. Based on the concept of scale invariance, a direct and unifying Bell-polynomial scheme is employed to achieve the Bäcklund transformations and Lax pairs associated with those three soliton equations. Note that the Bellpolynomial expressions and Bell-polynomial-typed Bäcklund transformations for those three soliton equations can be, respectively, cast into the bilinear equations and bilinear Bäcklund transformations with symbolic computation. Consequently, it is also shown that the Bell-polynomial-typed Bäcklund transformations can be linearized into the corresponding Lax pairs. C

On integrability of the extended (3+1)‐dimensional Jimbo‐Miwa equation

“…[15][16][17][18][19] According to this scheme, the bilinear form equations, the N-soliton solutions, the Bäcklund transforms, the Lax pairs, and the infinite conservation laws were obtained. [20][21][22][23][24] These results can help us better knowledge of the real world.…”

“…In this appendix, we briefly give certain basic knowledge of bilinear Bell polynomials approach, [14][15][16][17][18][19][20][21][22][23][24] which are required for the remaining part of the article.…”

A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg–de Vries-like model

“…Generally speaking, there exist some difficulties in applying the Hirota bilinear method to construct bilinear BTs for nonlinear equations. In some cases, employing the Bell-polynomial technique and introducing the auxiliary independent variable(s) are helpful [11][12][13].…”

“…In the study of nonlinear partial differential equations of mathematical physics, the Bäcklund transformation (BT) provides a means to construct new solutions from known ones [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. BTs can connect solutions of different equations or the same equation [2,3].…”

Generalized (2+1)-dimensional Gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms

“…By virtue of computerized symbolic computation [23,[27][28][29][30][31][32][33][34][36][37][38][39][40][41][42][43][44][45][46], the Painlevé integrability conditions of (3) among the coefficient functions have been derived [23], which reduce all the coefficient functions to be proportional only to γ (t), giving the special Painlevé-integrable case of (3) as [23] …”