The Boussinesq equation: Lax pair, Bäcklund transformation, symmetry group transformation and consistent Riccati expansion solvability

Abstract

The Boussinesq equation is a very important equation in fluid mechanics and some other disciplines. A Lax pair of the Boussinesq equation is proposed. With the help of the truncated Painlevé expansion, auto-Bäcklund transformation of the Boussinesq equation and Bäcklund transformation between the Boussinesq equation and the Schwarzian Boussinesq equation are demonstrated. Nonlocal symmetries of the Boussinesq equation are discussed. One-parameter subgroup invariant solutions and one-parameter group transformations are obtained. The consistent Riccati expansion solvability of the Boussinesq equation is proved and some interaction structures between soliton-cnoidal waves are obtained by consistent Riccati expansion.

Superposition formulas of multi-solution to a reduced (3+1)-dimensional nonlinear evolution equation

“…Finally, substituting f = w + v with above relationship expressions (15) into Eq. ( 2), we get 0, so f = w + v is an alternative form of the expression (12) with relations (14). It is worth noting that we get the new localized solutions u = −3(ln w) xx .…”

“…So finding exact solutions for NLEEs is significant. In nonlinear science, scholars have utilized various methods to obtain various solutions of NLEEs, such as the Darboux transformation, [5][6][7] the Hirota bilinear method, [8] Lie symmetry method, [9] the homogeneous balance method, [10] the tanh-coth method, [11] Bäcklund transformation, [12] consistent Riccati expansion method, [13,14] etc. It is quite efficient to use these methods to derive some solutions for NLEEs, such as variable separation solutions, [15] positon solutions, [16] algebraic-geometrical solutions, [17] N-soliton, [18] localized excitations, [19] dark solitary wave solutions, [20,21] etc.…”

Consistent Riccati expansion solvability, symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg–de Vries equation in fluid dynamics of internal solitary waves*

“…We will analyze the forced variable-coefficient extended KdV equation by means of the CRE method [20][21][22] in this section. The Riccati equation is written in the following form:…”

“…which can be applied in the internal solitray wave dynamics, but also can be used to describe the propagation of the weakly nonlinear waves in a composite medium. [13] Many effective methods have been proposed to analyze partial differential equations, such as Painlevé analysis, [14][15][16] symmetry group, [17][18][19] consistent Riccati expansion (CRE), [20][21][22] and so on. [23,24] To our knowledge, the CRE solvability, symmetries and cnoidal-solitary wave interaction solutions of the forced variable-coefficient extended KdV equation have not been studied.…”