Bilinear Bäcklund transformation, soliton and periodic wave solutions for a $$\varvec{(3 + 1)}$$ ( 3 + 1 ) -dimensional variable-coefficient generalized shallow water wave equation

Abstract

Under investigation in this paper is a \((3 + 1)\)-dimensional variable-coefficient generalized shallow water wave equation. Bilinear forms, Backlund transformation and Lax pair are obtained based on the Bell polynomials and symbolic computation. One-, two- and three-soliton solutions are derived via the Hirota method. One-periodic wave solutions are obtained via the Hirota–Riemann method. Discussions indicate that the one-periodic wave solutions approach to the one-soliton solutions when \(\varTheta \rightarrow 0\). Propagation and interaction of the soliton solutions have been discussed graphically. We find that not the soliton amplitudes, but the velocities are related to the variable coefficients \(\delta _{1}(t)\) and \(\delta _{2}(t)\). Phase shifts of the two-soliton solutions are the only differences to the superposition of two one-soliton solutions, so the amplitudes of the two-soliton solutions are equal to the sum of the corresponding two one-soliton solutions.