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

Under investigation in this paper is a ( 3 + 1 ) -dimensional variable-coefficient generalized shallow water wave equation. Bilinear forms, Bäcklund 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 Θ → 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 δ 1 ( t ) and δ 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.

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