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Binary Bell polynomials are applied to construct two kinds of bilinear derivative equations of a (3+1)$(3+1)$-dimensional nonlinear equation. Based on one of the bilinear forms, we derive a Bäcklund transformation, the corresponding Lax pair, infinite conservation laws, and explicit solutions with an arbitrary function in y. In the meantime, from the other bilinear form, we get another bilinear Bäcklund transformation and exact solutions by utilizing the exchange formulas for Hirota’s bilinear operators.

Bilinear Bäcklund transformations and explicit solutions of a ( 3 + 1 ) $(3+1)$ -dimensional nonlinear equation