Solitons and integrability for a (2 + 1)-dimensional generalized variable-coefficient shallow water wave equation

Under investigation in this paper is a (2+1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg-de Vries (KdV) equation and the Calogero-Bogoyavlenskii-Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.