Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.
Nonlinear self-adjointness and conservation laws of the variable coefficient combined KdV equation with a forced term