N-soliton solutions and the Hirota conditions in (2+1)-dimensions

We compute N-soliton solutions and analyze the Hirota N-soliton conditions, in (2+1)-dimensions, based on the Hirota bilinear formulation. An algorithm to check the Hirota conditions is proposed by comparing degrees of the polynomials generated from the Hirota function in N wave vectors. A weight number is introduced while transforming the Hirota function to achieve homogeneity of the resulting polynomial. Applications to three integrable equations: the (2+1)-dimensional KdV equation, the Kadomtsev-Petviashvili equation, the (2+1)-dimensional Hirota-Satsuma-Ito equation, are made, thereby providing proofs of the existence of N-soliton solutions in the three model equations.