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Bell-polynomial approach and soliton solutions for the Zhiber-Shabat equation and (2+1)-dimensional Gardner equation with symbolic computation. (English) Zbl 1263.35177
Summary: 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, Bäcklund transformation, one- and two-shock wave solutions are derived. Amplitude-compression and amplification interactions are investigated analytically and graphically.

MSC:
35Q30 Navier-Stokes equations
35Q40 PDEs in connection with quantum mechanics
35C08 Soliton solutions
33B10 Exponential and trigonometric functions
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