Bell-polynomial approach and soliton solutions for the Zhiber–Shabat equation and (2+1)-dimensional Gardner equation with symbolic computation

Abstract

Under investigation in this paper are the Zhiber–Shabat and (2+1)-dimensional Gardner equations in quantum fields, fluids and plasmas. Via the Hirota method and symbolic computation, the Bell-polynomial approach is performed to directly bilinearize those equations. For the Zhiber–Shabat equation, based on the bilinear form with an auxiliary variable, the bell-shaped soliton, upside-down bell-shaped soliton and breather-like solutions are obtained. Figures are plotted to illustrate the elastic interactions between two upside-down bell-shaped solitons and the interaction between the breather-like. As to the (2+1)-dimensional Gardner equation, bilinear form, Backlund transformation, one- and two-shock wave solutions are derived. Amplitude-compression and amplification interactions are investigated analytically and graphically.