New interaction solutions from residual symmetry reduction and consistent Riccati expansion of the (2 $$\varvec{+}$$ + 1)-dimensional Boussinesq equation

Abstract

In this paper, the (2 $$+$$ 1)-dimensional Boussinesq equation is studied by applying residual symmetry reduction method and consistent Riccati expansion (CRE) method, respectively. By introducing multiple new dependent variables to enlarge the (2 $$+$$ 1)-dimensional Boussinesq system, the residual symmetry is localized and the corresponding finite transformation is obtained by using Lie’s first theorem. The symmetry reduction solutions related to the residual symmetry of the (2 $$+$$ 1)-dimensional Boussinesq equation is obtained by using the standard Lie symmetry method, which includes complicated interaction models. Furthermore, the (2 $$+$$ 1)-dimensional Boussinesq equation is found to have CRE integrability, and new Backlund transformations (BTs) are consequently obtained. New interaction solutions are obtained from these BTs; particularly, the interaction solution between soliton and background cnoidal wave is given and analyzed explicitly.