Lax systems and nonlinear equations on a time-space scale

From the Lax system on a time-space scale, we derive Sine-Gordon, Burgers, and some other nonlinear equations. From the simplified two- and three-dimensional Lax systems on a time-space scale with variable nonvanishing graininess, we derive Korteweg–de Vries, Boussinesq, Krichever–Novikov, Hirota–Miwa, and other nonlinear equations on a time-space scale with soliton solutions. We also construct multi-soliton solutions for some equations by the direct method using the exponential functions on a time-space scale.