Abstract
In this paper, a more extended ( $$2+1$$ )-dimensional shallow water wave equation is considered, which is integrable in terms of the Hirota’s bilinear method. With the aid of the bilinear method, the N-soliton solutions are obtained. By applying the long-wave limit method to the N-soliton solutions, the multiple lump solutions are gained. The dynamical characteristics of the single lumps among the M-lump wave are investigated. From the point of view of mathematical mechanism, the propagation orbits, velocities and the interactions among the lumps are analyzed systematically. Furthermore, various types of elastic and inelastic interaction solutions among the lumps, breathers and solitons are constructed, which develop the types of the solutions of the ( $$2+1$$ )-dimensional extended shallow water wave model. The method presented in this paper can be applied to more nonlinear integrable equations to investigate the localized wave solutions and interaction solutions. The obtained results may provide some useful information to analyze the theory of the shallow water waves and solitons.
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