On the soliton solutions of the two-dimensional Toda lattice

The two-dimensional Toda lattice (2DTL) is a well-known (2+1)-dimensional integrable system that admits a large class of line-soliton solutions. We classify these soliton solutions by exploiting the structure of the Casoratian expression for its tau function. In general, these solutions consist of unequal numbers of 'incoming' and 'outgoing' line solitons. We classify the incoming and outgoing line solitons based on asymptotic analysis of the tau function of the 2DTL as the discrete variable tends to infinity. We also identify various subclasses of solutions and characterize some of them in terms of the amplitudes and directions of the interacting solitons. As a special case, we obtain the reduction in the soliton solutions of the one-dimensional Toda lattice. Throughout, we point out the similarities with-and the differences from-the corresponding results for the Kadomtsev-Petviashvili equation.