LAUSR

Under investigation in this work is a generalized $$(2 + 1)$$(2+1)-dimensional Boussinesq equation. By employing the Bell’s polynomials, bilinear formalism of this generalized $$(2+1)$$(2+1)-dimensional Boussinesq equation is succinctly derived. With the aid of the obtained bilinear formalism, general high-order breather solutions are constructed by using the Hirota’s bilinear method combined with the perturbation expansion. The breathers only periodically propagate along the x-direction. Taking a long-wave limit of the obtained breather solutions and then making further parameter constraints, general smooth rational solutions to the generalized $$(2 + 1)$$(2+1)-dimensional Boussinesq equation would be succinctly constructed. These smooth rational solutions are high-order lumps and mixed solutions comprising a line rogue wave and lumps. These results exhibit the dynamical behavior of the generalized $$(2+1)$$(2+1)-dimensional nonlinear wave fields.