Soliton solutions of q-Toda lattice by Hirota direct method

This paper presents the q -analogue of Toda lattice system of difference equations by discussing the q -discretization in three aspects: differential- q -difference, q -difference- q -difference and q -differential- q -difference Toda equation. The paper develops three- q -soliton solutions, which are expressed in the form of a polynomial in power functions, for the differential- q -difference and q -difference- q -difference Toda equations by Hirota direct method. Furthermore, it introduces q -Hirota D -operator and presents the q -differential- q -difference version of Toda equation. Finally, the paper presents its solitary wave like a solution in terms of q -exponential function and explains the nonexistence of further solutions in terms of q -exponentials by the virtue of Hirota perturbation.