Revisiting solutions of the Adler–Bobenko–Suris lattice equations and lattice Boussinesq-type equations

Solutions of all Adler–Bobenko–Suris equations except $Q4$, and of several lattice Boussinesq-type equations are reconsidered by using the Cauchy matrix approach. By introducing a “fake” nonautonomous plane-wave factor, we derive soliton solutions, oscillatory solutions, and semi-oscillatory solutions of the target lattice equations. Unlike the conventional soliton solutions, the oscillatory solutions take constant values on all elementary quadrilaterals on $\mathbb{Z}^2$, which demonstrates a periodic structure.