Painlevé analysis, Gram determinant solution and the dynamic behavior of the (2+1)-dimensional variable-coefficient asymmetric Nizhnik–Novikov–Veselov equation

In this paper, we focus on many types of solutions a (2+1)-dimensional variable-coefficient asymmetric Nizhnik–Novikov–Veselov (vcANNV) equation by the Kadomtsev-Petviashvili (KP) hierarchy reduction method. The derived constraint condition makes the KP hierarchy equation simplified into the target equation and the classical findings of the KP hierarchy equation is directly applied to solving. Under the specified constraint condition, the Gram determinant solutions for the asymmetric Nizhnik–Novikov–Veselov equation are systematically obtained via selecting suitable \(\tau \) $\tau$ -functions. A diverse array of solutions including solitons, breathers and lumps are generated, as well as the corresponding figures are presented for detailed analyses. Moreover, the Painlevé integrability of the vcANNV equation is examined. These results will be useful to further understand corresponding nonlinear wave phenomena and the related physical experiments.