Single-Soliton Solution of KdV Equation via Hirota’s Direct Method under the Time Scale Framework

Hirota’s direct method is one significant way to obtain solutions of soliton equations, but it is rarely studied under the time scale framework. In this paper, the generalized KdV equation on time-space scale is deduced from one newly constructed Lax equation and zero curvature equation by using the AKNS method, which can be reduced to the classical and discrete KdV equation by considering different time scales. What is more, it is the first time that the single-soliton solution of the KdV equation under the time scale framework is obtained by using the idea of Hirota’s direct method.