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In this work, the (2+1$2+1$)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation is investigated. Hirota’s bilinear method is used to determine the N-soliton solutions for this equation, from which the M-lump solutions are obtained by using long wave limit when N is even (i.e., N=2M$N=2M$). Then, taking N=5$N=5$ as an example, we discuss some novel mixed lump-soliton and lump-soliton-breather solutions by using long wave limit and choosing special conjugate complex parameters from the five-soliton solution. Figures are plotted to reveal the dynamical features of such obtained lump and mixed interaction solutions. These results may be useful for understanding the propagation phenomena of nonlinear localized waves.

Soliton, breather, lump and their interaction solutions of the ($2+1$)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation