Bell-polynomial construction of Bäcklund transformations with auxiliary independent variable for some soliton equations with one Tau-function

The Korteweg–de Vries (KdV)-type models are of significance in describing many physical situations in fluid flows (particularly for surface and internal waves), plasma physics, and solid state physics. In fluid dynamics, for example, the shallow water wave equation is utilized as a mathematical description of regular and generalized solitary waves in shallow water. Further, higher-order dispersive (e.g., the Lax fifth-order KdV equation) and higher-dimensional [e.g., the (2+1)- and (3+1)-dimensional breaking soliton equations] generalized nonlinear models are useful in analyzing and obtaining modulation theory, existence and stability of solitary waves, bores, and shocks, as well as other integrable properties. With symbolic computation, Bell-polynomial-typed Bäcklund transformations (BTs) are constructed for some single-field bilinearizable nonlinear evolution equations including the shallow water wave equation, Lax fifth-order KdV equation, and (2+1)- and (3+1)-dimensional breaking soliton equations. Bell-polynomial expressions are derived, which can be cast into the bilinear equations with one Tau-function. Key point lies in the introduction of certain auxiliary independent variable in the Bell-polynomial expression. With one auxiliary independent variable, the Bell-polynomial-typed BTs are then constructed according to the coupled two-field conditions between the primary and replica fields with both the fields satisfying the Bell-polynomial-expression equations. Auxiliary-independent-variable-involved Bell-polynomial-typed BTs are changed into their bilinear forms. Aforementioned equations turn out to be integrable in the sense of possessing the Bell-polynomial-typed BTs.