Multi-soliton and rational solutions for a modified Boussinesq equation

N-soliton solutions are investigated by the Hirota method for a modified Boussinesq equation, which can be related to the bad Boussinesq equation. The bilinear form of the modified Boussinesq equation is given through the bi-logarithmic transformation and two identical relations. N-soliton solutions and those in the Wronskian determinants are presented for the modified Boussinesq equation. Especially, the explicit one- and two-soliton solutions are discussed in detail. The complex solutions of the bad Boussinesq equation are correspondingly obtained through the Miura transformation simultaneously. From the structure of solutions and the difference of parameters, the solitons show the soliton or antisoliton type, which brings the interactions between the same and different types of waves. There exist three cases of the interactions between two solitons: soliton-soliton, soliton-antisoliton and antisoliton-antisoliton. Four types of interactions among three solitons/antisolitons are presented through graphics. The rational solutions, which admit the algebraic soliton, are derived by taking the limits of wave numbers and proper phase parameters.