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Under investigation in this article is a (2+1)-dimensional generalised variable-coefficient shallow water wave equation, which describes the interaction of the Riemann wave propagating along the y axis with a long-wave propagating along the x axis in a fluid, where x and y are the scaled space coordinates. Bilinear forms, Bäcklund transformation, Lax pair, and infinitely many conservation law are derived based on the binary Bell polynomials. Multi-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 multi-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.

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Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili equation for the shallow water wave in a fluid or electrostatic wave potential in a plasma. Bilinear form, Bäcklund transformation and Lax pair are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota’s method. Propagation and interaction of the solitons are illustrated graphically: (i) Through the asymptotic analysis, elastic and inelastic interactions between the two solitons are discussed analytically and graphically, respectively. The elastic interaction, amplitudes, velocities and shapes of the two solitons remain unchanged except for a phase shift. However, in the area of the inelastic interaction, amplitudes of the two solitons have a linear superposition. (ii) Elastic interactions among the three solitons indicate that the properties of the elastic interactions among the three solitons are similar to those between the two solitons. Moreover, oblique and overtaking interactions between the two solitons are displayed. Oblique interactions among the three solitons and interactions among the two parallel solitons and a single one are presented as well. (iii) Inelastic–elastic interactions imply that the interaction between the inelastic region and another one is elastic.

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In this article, under the investigation on the interactions among multiple solitons for an eighth-order nonlinear Schrödinger equation in an optical fibre, oscillations in the interaction zones are observed theoretically. With different coefficients of the operators in this equation, we find that (1) the oscillations in the solitonic interaction zones have different forms with different spectral parameters of this equation; (2) the oscillations in the interactions among the multiple solitons are affected by the choice of spectral parameters, the dispersive effects and nonlinearity of the eighth-order operator; (3) the second-, fifth-, sixth-, and seventh-order operators restrain oscillations in the solitonic interaction zones and the higher-order operators have stronger attenuated effects than the lower ones, while the third- and fourth-order operators stimulate and extend the scope of oscillations.